Being Alongside

Being Alongside

For the Teaching and Learning of Mathematics

Alf ColesUniversity of Bristol, UK

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DEDICATION

To three generations of my family:

Iona, Arthur, Iris~

Niki~

Walt and Flo

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TABLE OF CONTENTS

Preface ix

Timeline of Teaching and Research Activity xi

Acknowledgements xiii

Part One

1. Introduction 3

2. Enactivism 11

3. A Cyclical Enquiry 25

Part Two

4. On Using Video 37

5. On Teacher Learning 53

6. Re-looking at Teacher Discussions 65

Part Three

7. On Metacognition 79

8. The Story of Teacher A 89

9. Heightened Listening 101

Part Four

10. Conclusion 115

Appendix 1 121

Appendix 2 129

Appendix 3 147

Appendix 4 149

References 163

Index 169

vii

PREFACE

This is a book of a research project in one mathematics department in a secondary school in the UK. It is written in four parts. In the first, I set up the context for the study, my research stance and the methodology used. In the second part, I report on and analyse the use made, by the mathematics department, of video recordings of mathematics lessons in the school. In the third part, I analyse a set of video recordings of one teacher (Teacher A) who taught mathematics in the school. Through looking at the practice of Teacher A, I arrive at a re-framing of the concept of metacognition. I then look at the similarities and differences between the way meetings were run when working on video with teachers and the classroom practice of Teacher A. This comparison leads me to the idea of a ‘heightened listening’ as a description of a way of paying attention to discussions. In the final part of the book, I look back over a variety of different distinctions made in the book and draw out some similarities, differences and issues.

Appendix 1 contains lesson write-ups of four of the mathematical activities referred to in this book. I suggest you familiarise yourself with these activities and work on them yourself for a while, before reading.

In the book, I refer to people, positions and projects I have been involved with as a teacher and researcher since 1994. I therefore also suggest reading the timeline on the following page, to help make sense of the chronology of events.

I have adopted the following notation and conventions. Italics used in quotations are always in the original. Italics as part of normal text indicate emphasis. In order to preserve gender anonymity for students, I use ‘they’ and ‘them’ for individuals as well as in the plural.

The teacher who is central to this study I label ‘Teacher A’ or TA. Other teachers are labelled TC, TD, etc, and I use initials (AC) for myself. I label students S1, S2, etc. If any student re-appears in the same transcript I give them the same label, but not across different transcripts (unlike the teacher labels which consistently refer to the same person throughout the book).

I adopted the following transcript notation, avoiding all other punctuation:

[ ] indecipherable speech[text] my best guess at speech[text] note to the reader(.) pause less than 1 second(2) pause of 2 seconds? rising intonation/ / interrupted speech… some words left out from transcription

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x

PREFACE

text speech said as if in the voice of another-: more than one speaker talking at the same timeLine endings are chosen, where possible, to indicate phrasing.I used line numbers when this was useful reference for analysis. Where it is not already made explicit in the text, transcripts are dated and labeled either “Lesson” or “TD/TB” (which would indicate a Teacher Discussion, focused on a lesson given by Teacher B).

TIMELINE OF TEACHING AND RESEARCH ACTIVITY

– 1992 Appointed to a position in a school in Eritrea to teach English – 1993 Accepted onto mathematics teaching training course in the UK – 1994 Qualified as a teacher in the UK – Appointed to first teaching post in the UK – 1995 Began research collaboration with Laurinda Brown – 1996 Appointed as a full-time mathematics teacher in a new school – the one on

which this study in based – Began part-time master’s studies at the University of Bristol – 1998 Awarded a Teacher Training Agency, Teacher Research Grant, with Laurinda

Brown as University Steerer – 1999–2001 Involved as a researcher and teacher on an Economic and Social

Research Council (ESRC) project, ‘Developing algebraic activity in a ‘community of inquirers’’ with Laurinda Brown as Principal Investigator

– 2000 Completed master’s degree – 2001 Appointed Head of Mathematics at the same school – 2006 Awarded an ESRC Studentship (+3) for doctoral research at the University

of Bristol – 2007–2009 Involved as a researcher and teacher on a Teacher Development

Agency Grant (& Continuation Grant), ‘Enquiring Schools Project’ with Laurinda Brown as director

– 2008 Appointed Assistant Headteacher (i.e., no longer Head of Mathematics) at the same school

– 2010 Appointed as part time Senior Lecturer at University of Bristol (i.e., no longer employed in schools)

– 2011 Appointed full time Senior Lecturer at University of Bristol

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ACKNOWLEDGEMENTS

I would like to acknowledge the support of the Economic and Social Research Council in funding this study. I had to pinch myself every time I woke up with a whole day ahead of me to read, think and write.

Thank you to my colleague Laurinda Brown and her (late) mentor Dick Tahta for the many conversations two-way and three-way going back seventeen years that have shaped the writing that follows. The title of this book ‘Being Alongside’ is a reference to one such three-way conversation that we wrote up for the journal Mathematics Teaching (Brown and Coles, 2007; Tahta and Williams, 2007) and which I draw on in Chapter 9. Thank you also to Laurinda for the detailed read and invaluable comments on a late draft of this text.

Thank you to the teacher who figures centrally in this book, for allowing me such privileged access to your teaching practice. I hope my writing makes it apparent how much I enjoyed dwelling in the detail of recordings of your lessons and learning from what you do.

Thank you to the students and other teachers I recorded and to the wider community of the school where this study is based, all of you were consistently supportive of this project. I want to mention in particular David, without whose wisdom I doubt this study could have taken place.

And thank you to my family, and especially Niki, for your faith in what I have been doing.

The cover photo is a section of tiling from the Alhambra palace in Granada. These tilings were loved by Dick Tahta: “[i]n contemplating the complex interlacing patterns of some of the Alhambra tilings, the eye has no reason to pause anywhere; moreover, the flow comes back on itself, so that there is no start or end. Such interlacing was felt to be a direct expression of the idea of a divine unity behind the enormous multiplicity of the world.” (Tahta, 2006b, p. 196). Or, I might say, the invitation from the fourteenth century Islamic craftsmen, through contemplation of these tilings, is to experience ‘being alongside’ what we typically see as separate to us.

Some sections of Chapters 7 and 8 are taken from an article “On metacognition”, originally published in the journal ‘For the Learning of Mathematics’ (FLM), in 2013, volume 33(1), pp. 21–26, they are re-produced here with kind permission of FLM Publishing Association.

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PART ONE

3

CHAPTER 1

INTRODUCTION

I speak of it, in order to begin somewhere, and also to remove from you the delusion that somewhere within or without there is something absolutely firm and definite. (Jung, 1982, p.44)

STORY 1: AN INCIDENT IN A SECONDARY SCHOOL

Teacher F: (she is pregnant, and visibly upset) Alf, will you come and remove a student from my classroom, he has just said that he hopes my baby dies.

I go to her room and ask the student to leave, he appears agitated and loudly expresses his innocence to me as he leaves the room and when outside. I walk him away from the classroom. I notice rising anger in myself as the student interrupts me with more protestations of being hard done by. I feel myself close to shouting at the student, the anger upon me. I tell the student to move to ‘the bridge’, a space nearby which connects the mathematics and science departments. It is a glass corridor at first-floor level with double doors at either end. As I stand in this space facing the student, I have a sudden awareness that it will be simply the words we exchange which will determine how this interaction goes, whether it escalates, or is resolved. I also register an awareness that things are currently escalating.

Student: I was going to say as a joke that maybe the baby will die to save the world, she didn’t let me finish.

Alf: (shouting) This is not something you joke about, my son almost died when he was born (now with tears welling to my eyes) you do not joke about this.

Student: (also with tears welling in his eyes) My brother died as a baby (pause) I was saying maybe the baby would die to save the world (now said more quietly) she didn’t let me finish.

A Transition

Story 1 was my memory, written two years after the event and so somewhat stylized and condensed, of what is still a vivid incident. Tears welled again for me in the writing. Something happened in the conversation on the bridge – there was a shift. I am aware

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now of the bizarreness of the place in which we were talking, a glass corridor between two departments, a space with the function of allowing transition from one place to another. And yet how appropriate it was for the transition that occurred between us. As I described it soon after the event, it seemed as though we suddenly saw each other’s humanity. Something, somehow, opened between us that allowed us to step out of the roles we had been playing. The story captures the nub of my interest in writing this book. I want to explore the question, what can be said about conversation that allows the new, the unexpected, the creative to arise? What can be done in conversation that offers individuals possibilities of new or different responses to situations, or that allows individuals to view a situation differently, or in Davis’ (2004) phrase, to expand ‘the space of the possible’ (p.184). And, in a group context, how do patterns of interaction get established which either do, or do not, serve such expanding of possibilities? The roots of these concerns, of course, lie in my past and I offer below some of my personal history, which serves to introduce the key themes and figures of this book.

BECOMING A RESEARCHER

At university, I received a philosophy training in the English analytic tradition and I accepted, unquestioning, assumptions about the separateness of, for example, subject/object, mind/body, consciousness/brain states. My thinking has a tendency to slip into such dualities or dichotomies. I associate this tendency with a need for certainty and with the taking up of a judgmental stance (things being right/wrong). I also recognise that these modes of thought are not productive for me in the context of teaching. When I am in an overly analytic mode, in a lesson, it feels as though I fail to make any connection with the students in front of me.

I have noticed in myself, at times, a tendency to believe that certain individuals have all the answers to questions of importance to me. I think I find some comfort in the idea of an all-knowing or all-wise ‘other’ to whom I can aspire. I see myself slipping into duality; person X has the ‘truth’, against a background of others who do not. Without ever meeting him, Caleb Gattegno (1911–1988) was such a figure for me when I took up my first teaching post in the UK in 1994. I had an image of what I thought he must have been like in a classroom, to which I aspired without, initially, any sense of what I could do in action to help move towards this ideal. I came across Gattegno’s ideas on my Postgraduate Certificate in Education (PGCE) course and his use of the terms ‘an awareness’ and ‘awarenesses’ (e.g., 1988, p.10). Gattegno turned ‘awareness’ into a countable noun and commented we must use these awarenesses, ‘to illuminate our fields of action’ (1987, p.25). Mason similarly sees awarenesses as ‘the bases for action’ (2008, p.62). It was a link to action I knew I was missing in my own classroom at the time.

I began a journey of engaging in research about my teaching, through a collaboration with Laurinda Brown. Laurinda pointed me towards the work of Gregory Bateson (1904–1980), whose intellectual standpoint or perspective I found both thrilling and challenging. I aspired to his insights but knew I did not share them.

INTRODUCTION

5

Bateson explicitly wrote about the need (for our own survival as a species) to move away from a standard scientific viewpoint to a more recursive, or reflexive, one (1972, p.468). There was something I found compelling about this possibility of an alternative vision.

In 1992, while teaching in Eritrea, I wrote the plan for a book in which I was going to reconcile mathematics, philosophy and religion. I planned to show how each religion in its spiritual guise was speaking about the same state of mind, or way of relating to the world. I was then going to show how the analytic methods of western philosophy were self-defeating since mathematics and reason ultimately lead to contradiction (or at least un-decidability, as proved by Gödel, see Nagel and Newman, 2001). The problems with the purely analytic outlook would then force a need to adopt the spiritual outlook pointed to by the mystical strands of all religions.

Despite cringing now at the hubris of this book plan, in some ways I can see my intellectual journey since 1992 as a continuation of the search to reconcile the analytic and intuitive within me (without, now, projecting this on to the whole of religion, mathematics and philosophy!). I think the reason I felt such connection with the writing of Bateson was that he had so patently made this reconciliation himself. Photographs of his face spoke to me of wisdom, which perhaps is another word for the joint operation of intuition and analysis. So, in engaging in research I am seeking to become wiser. I am searching for a position that is somehow beyond the splits of subject/object or mind/body or intuition/analysis, in which I can bring all of myself to all that I experience. Perhaps I am searching for the wisdom of the man of negative capability, in the sense of Keats, in a letter to his brother in 1817:

At once it struck me, what quality went to form a Man of Achievement … I mean Negative Capability, that is when man is capable of being in uncertainties, Mysteries, doubts, without any irritable reaching after fact and reason (Keats and Buxton Forman, 2004, p.57).

I notice there is a part of me that would like to be recognised as a ‘Man of Achievement’, part of me that would like to be known and admired. I also recognise in this thought the very ‘irritable reaching’ which precludes Negative Capability. I aim to engage in research in a principled manner that respects the desire to avoid being caught between abstract dichotomies.

RESEARCH CONTEXT

As I embarked on the research project that forms the basis for this book, I was head of the mathematics department in a mixed 11–18 school. That department is the focus of the study. The department had some claim to being innovative, for example mathematics was chosen as the leading subject for the school’s successful specialist status application and Ofsted, in 2006, cited outstanding features of the department’s approach to teaching students to think mathematically. We would get numerous visitors from teachers in the UK and overseas and the department was invited

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to showcase its way of working via guest editorship of a mathematics education website based in the UK.

As I began this study, I described the department’s way of working as a ‘hearing culture’, and I was initially interested in investigating the strategies by which such a culture was developed and sustained. I now hesitate to use the word ‘culture’ since it can imply a glossed uniformity, seeing similarity where there is perhaps wide difference. What I had meant at the time by a ‘hearing culture’ was that in many of the mathematics classes, students questioned both what the teacher said and what each other said – there was an expectation that voices would be heard. I would not claim this happened in every lesson, but I did believe that what was widely seen as innovative, or different, about what happened in mathematics classrooms in the department, was in part to do with the way students and teachers listened to each other. In many lessons, students posed (as well as answered) questions and there was a sense that mathematics was not about things being right/wrong but more about a way of approaching problems.

In this department, there were some quite specific articulations of a desired classroom practice. Appendix 1 contains four lesson write-ups that were part of the department’s scheme of work for teachers and which give some sense of the way of working. These activities could last for several weeks with a class, typically from six to ten one-hour lessons. A mural along one corridor of the mathematics department stated, in letters that were around half a metre tall, ‘THINK MATHEMATICALLY’. I painted this and it was a key phrase in terms of what I thought I was trying to develop as head of department. In Brown and Coles (2008), I described watching Laurinda Brown teach a class of my year 8 students (aged 12–13). This was powerful for me, as it gave me an experience of what it might mean for a class of my own students to be ‘thinking mathematically’, a very loosely defined concept for me at that time. The kinds of things I now mean by this phrase are students: making predictions; using algebra to explain their thinking; having their own ideas and lines of inquiry; testing out ideas and commenting on what they notice. It is such aspects of students’ work in mathematics that I was interested to foster as a teacher, head of department and now as a researcher.

In the first department meeting of the year, each year that I was head of department, the mathematics staff and I created a poster of responses to my asking what people meant by ‘thinking mathematically’. The common phrases on these posters, across the years, were:

– asking questions – spotting patterns – making conjectures or predictions – giving reasons or justifications.

When I mention ‘thinking mathematically’ or working as a ‘mathematician’, it is these activities I have in mind.

In this study, I wanted to explore what are effective strategies for a teacher to create a classroom in which students think mathematically, and what are effective

INTRODUCTION

7

strategies for a teacher leader to create an analogous department amongst staff. A quotation neatly sums up the classroom aspect of my interest:

What is the role of the teacher? How can he (sic) create the conditions in which creative and independent work can take place? (Banwell, Saunders and Tahta, 1986, p.18).

To adapt these words for the other half of my initial interest: what is role of the teacher leader? How can I create the conditions in which teachers are able to work creatively and independently?

I am interested in how ‘shifts’ such as in Story 1 can take place in relation to mathematics or mathematics teaching, so that students and teachers come to see an issue, the subject, or themselves, in a new light with new or different possibilities for action.

DICHOTOMIES AND DISTINCTIONS

I separated, above, the dichotomies of mind/body, subject/object with a slash, which stands for a distinction, where one side is distinguished from another. Every word, idea or concept contains within it a distinction, if only from its background, or from everything else that is not captured by this word or idea. Bateson (1972) argued that there is an infinitude of potential distinctions that can be made in relation to any object or entity. He considered something as simple as a piece of chalk and the potentially infinite number of differences between it and the rest of the universe (p.459). From this multitude of potential distinctions, we select the differences that matter to us. It is our role as observers to be active in selecting these differences, not the role of the chalk. Although the notion of drawing a distinction may appear simple, the concept of ‘difference’ is problematic. When we draw a distinction, where is the difference? Is it in the original object, in us, in some space in between, or does this question not make sense? From introspection, it seems the only reasonable answer is that the differences arise through my interaction with the object. The differences are an aspect of my relationship to the object. There are further complexities, however: is the difference between, say, a dog and a cat the same kind of thing as the difference between a dog, and the set of all dogs? I leave this unanswered, while noting that the question appears to be one about the logical level of what is being considered (i.e., is it an object, or category of objects). Logical levels will become a crucial part of the arguments I put forward later in this book.

Within mathematics education, Reid and Brown (1999) have addressed the issue of dealing with dichotomies. They discuss three mechanisms for escaping them and moving to a position where they are not seen to be in conflict. The first mechanism is ‘both’ (p.17). Conflict only arises when extremes are seen as either/or, whereas they can be seen as both/and. In Reid and Brown’s example they avoid conflict between their roles as teacher and researcher in a classroom through seeing them as complementary. A second mechanism is ‘fork in the road’ (p.17). Sometimes a

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choice does have to be made but there are always routes between the forks at a later time. The third mechanism is ‘star’ (p.18), which draws on an image created by Varela, cited in Hampden-Turner (1981, p.193).

Mind

Whole

Context

Territory

Being

Intuition

Right brain

Environment

Body

Part

Text

Map

Becoming

Logic

Left brain

System

Figure 1.1. Varela’s ‘star’.

If the left hand set of words are considered as ‘it’, the right hand set can be seen as ‘the process of becoming it’. Hampden-Turner states:

Varela proposes that dualisms or dialectical ‘contradictions’ such as mind/body, whole/part, context/text, territory/map, being/becoming, intuition (right brain)/logic (left brain) or environment/system should be conceived of as ‘stars’. All stars consist of ‘the it’/‘the process of becoming it’ where the slash or oblique stroke means ‘consider both sides of’. Hence, we must consider both it and the process leading to it … By looking at the star from one side or the other, ‘it’ or ‘the process of becoming’ are seen as emerging from the context of its opposite. (1981, p.192)

Unlike the one-dimensional view of terms and their negation, Varela’s star presents conflicting ideas as being at different logical levels, ‘it is a way to proceed from disjoint pairs to their unity at a metalevel’ (Varela, 1976, p.62). The position I seek as a researcher, in which I am not caught in dichotomies, is at a metalevel to wherever the dichotomies lie.

ABOUT STORY

My writing to this point has already given away some of my commitments in terms of how to approach the research task. I see my history as relevant, and I am alert to issues around logical levels and metalevels. I have also included the first person voice of the ‘story’ (p.1). Story is a technical term for me, which I ground in the work of Bruner and Bateson (see Brown and Coles, 1997; Coles, 2004).

Bruner (1990) takes the ‘folklore’ conception of story as his starting definition. Characters and context react and changes give rise to an unexpected ‘predicament’ which requires actions and ‘[t]he response to this predicament brings the story to its

INTRODUCTION

9

conclusion’ (Bruner, 1990, p.44). The notion of story as a predicament and response is perhaps applicable to this book as a whole, or the chapters within it. Bruner (1990) felt our very experience was framed in narrative terms and, furthermore, that, ‘what does not get structured narratively suffers loss in memory’ (p.56). He distinguished this broad conception of narrative, as structuring the whole of experience, from stories which are ‘well-formed narratives’ (p.51). He gives as examples of stories, ‘[t]he perpetual revisionism of historians, the emergence of “docudramas,” the literary invention of “faction,” the pillow talk of parents trying to make revised sense of their children’s doings’ (1990, p.55). Speedy (2000) draws on Bruner (among other authors), to make a similar distinction between ‘‘narrative’ as an overarching world-view’ from ‘‘story’ for the many tales we tell and re-tell about our endeavours and ourselves’ (p.363). However, this is not quite the notion of story I want, initially, to invoke. Bateson states, ‘[a] story is a little knot or complex of that species of connectedness which we call relevance’ (1979, p.12). Bateson goes on to use, as a synonym for ‘species of connectedness’, the phrase, ‘connecting pattern’ (p.15). I find this linking of story to a connecting pattern is one that fits better with my purposes. I will be looking at micro incidents in the classroom, where the fuller sense of Bruner’s story may be hard to interpret. For Bateson, a story is a connecting pattern and to make sense of any communication we need to be able to place it within such a pattern. The task of the researcher is to seek out ‘a pattern of patterns of connection’ (1979, p.16), which, along with seeking a position at a metalevel to dichotomies, is another way I view my task in this study.

Bateson’s notion of story applies to a wider sphere of phenomena (all communication) than Bruner’s. For Bateson, ‘thinking in terms of stories must be shared by all mind or minds, whether ours or those of redwood forests and sea anemones’ (1979, p.12), a description I take as being essentially an overarching world-view, and hence close to Bruner and Speedy’s ‘narrative’. I want to invoke this broader sense of story as central to connectedness and meaning.

A further entwining of Bateson and Bruner’s ideas about story and narrative comes from a link, that both of them see one aspect of story as communication about other communications, i.e., metacommunication, a word that Bateson (1979, p.107) acknowledged taking from Whorf (1956). Since coming across it, I have been intrigued by this quotation from Bruner, ‘the existence of story as a form is perpetual guarantee that humankind will “go meta” on received versions of reality’ (1990, p.55). I take it from this that Bruner sees story as a kind of metacommunication, or metacommentary on experience. Metacommunication and metacommentary will come to play an important role for me in this study and I find this Bruner quotation a useful reminder that a metacommentary is but another story, another pattern, hopefully connecting connecting patterns. This brings me back to Bateson, who comments, ‘[m]y central thesis can now be approached in words: The pattern which connects is a metapattern’ (1979, p.10). It is metapattern (patterns of patterns) I explore in this book.

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METACOMMENTARY

This chapter has served to introduce the themes of logical levels, avoiding dichotomy, story, pattern, connection, context and metacommunication, as well as the figures of Bateson and Gattegno.

The book is written in three voices. The first two I have introduced already, the personal, autobiographical voice of the story (Story 1), and the researcher’s voice of analysis (all other sections up to this one). The second voice, in part, looks at the first. This will be the unmarked voice as I write. A third voice will look at them both and comment, as I am now, about what is said. I call this third voice a ‘metacommentary’, following Bateson’s use of ‘metacommunication’ (1979, p.107) to denote communication that is about communication, for example, establishing the context of an exchange. As someone approaches me on the street and makes eye contact, we communicate with each other about the kind of exchange we anticipate. Holding eye contact is likely to be a mark of anticipating either a friendly or hostile exchange. Glancing away may be a clue that no further interaction is expected. All these gestures are forms of metacommunication. As will become clearer as this book unfolds, I am particularly interested in verbal metacommunication, i.e., when something is said about the type of communication that is taking place. I call a verbal metacommunication a metacomment.

There is a paradox in starting the write-up of any study. I feel the need to justify every decision I made, but cannot justify the first decision without falling into an infinite regress. Hence the quotation at the start; I begin somewhere and hope to remove from you the thought that I will arrive at something absolutely firm and definite.

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CHAPTER 2

ENACTIVISM

How will you look for it, Socrates, when you do not know at all what it is? How will you aim to search for something you do not know at all? If you should meet with it, how will you know that this is the thing that you did not know? (Plato, Meno, 80d)

FINDING A PLACE TO STAND

One context of this study, as I set out in Chapter 1, was the aim I had as head of the mathematics department to encourage students’ mathematical thinking. Also, it was part of my role to organise and run department meetings for all mathematics staff. In some meetings, we would look at small sections of video recordings of each others’ lessons. This was a practice that began through my involvement in an earlier research project (Brown et al., 2001) that used video. I personally found watching video recordings of lessons was powerful in terms of provoking a motivation to try out new actions in my classroom. As head of department, I continued the practice of taking video recordings of different teachers’ lessons and would give time in meetings to watching these recordings. I became interested in the discussions of video at these teacher meetings, and also in the video recordings of the lessons themselves (and in particular, discussions on these video recordings involving the whole class and teacher). I had a sense that in both classrooms and teacher meetings these whole group discussions were sometimes significant in terms of shaping particular ways of working that I valued. I wanted to investigate, to learn more about what was or was not happening in these discussions.

There is a paradox about learning, however, encapsulated in the quotation above from Plato, how do we ever find out about the possibilities for action or knowing that we do not know exist? This paradox applies equally to me, engaging in research, to the teachers I study who are learning about teaching, and to the students in the classrooms who are learning mathematics. If I am not careful I will only see, in my data, that which I already think and believe. Thinking through this paradox involves reflecting on what I mean by knowledge, being explicit about my epistemological stance. In this chapter, I set out the key elements of enactivism as both a theory of knowing (an epistemology) and a way of approaching the research task (a methodology). I begin with two stories.

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Story 2: Standing Stones

A Parish Magazine is delivered to my house every month. One month I noticed, as part of an article, that our Parish Council had a logo, an arrangement of three standing stones. I reflected at the time that for this to be the Parish Council logo these standing stones must be near where I live, yet I had never seen nor heard of them. The next day, I was driving down a road outside our village that I must have gone down and back on several hundred times. Whilst travelling close to 60mph, and not done deliberately, I turned my head to the right and glimpsed for a fraction of a second, at a break in the wall by the road, the configuration of standing stones.

Story 3: Unheard Noise

I was walking home and feeling in my pocket for my keys. I could not get at the keys, so I pulled out my handkerchief first. As I did so, I was aware of a flash of light, consistent in time and space with a set of keys catching the sun as they fell to the floor and I “heard” the sound of the keys dropping. A moment later I realised the flash of light had not come from keys, but was reflected from a nearby car and in that moment I recognised that I had not actually heard any sound. My keys were still in my pocket.

Reflections

I cannot believe that seeing the standing stones the day after seeing the logo for the first time occurred by chance. The only sense I can make of this experience is that I had an unconscious, or unknown, awareness as to the possible location of the standing stones, and that I directed my attention there when I next passed by. This story links with the experience that I believe is familiar to others that on getting a new car suddenly you are aware of many more of the same make on the roads than you had previously imagined. What I notice as I live my life is that to which I direct my attention (consciously or otherwise).

In retelling the second story it is hard to explain my experience of ‘hearing’ a noise and then recognising I had not heard it. I am not sure how to justify that I knew I had not heard it. A similar incident, with vision, is perhaps more familiar. When my youngest daughter was twenty months old, she and I were walking down a lane and had just gone past a corner with a car parked on the verge. I heard a car approaching from behind us and turned around. In that instant I ‘saw’ a car moving towards us, before the image resolved itself into the parked car we had just walked by. I now notice myself on a fairly regular basis ‘filling in’ my experience in these kinds of ways. It appears as though my expectations of what will occur next play a role in some kind of preparation to perceive. In the cases where expectations are met, this goes un-noticed. In cases such as the ones above, my expectations can mean I perceive what I later come to recognise as illusion.

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ENACTIVISM AS AN EPISTEMOLOGY

Kuhn (1970) brought to awareness the idea that there is a ‘paradigm’ under which research is conducted; he defined the word in a deliberately circular manner, ‘A paradigm is what members of a scientific community share and, conversely, a scientific community consists of men (sic) who share a paradigm’ (p.176). He also distinguished between ‘paradigms’ and ‘schools’ (p.15). In a pre-paradigm era in the development of a science, there will be a multiplicity of schools, which collectively will share some aspects of a paradigm, but with no consensus about methods, procedures and what constitutes its core problems. What distinguishes a mature paradigm, for Kuhn, is that the field, ‘identifies challenging puzzles, supplies clues to their solution, and guarantees that the truly clever practitioner will succeed’ (1970, p.179). I hesitate to place myself in a particular paradigm (e.g., within the classification of Lincoln and Guba, 2000, p.170: positivism; post-positivism; critical theory; constructivism; participatory). One reason I hesitate, is that I do not see the kind of consensus mentioned by Kuhn at, for example, mathematics education conferences run by the British Society for Research into Learning Mathematics (BSRLM) or the International Group for the Psychology of Mathematics Education (PME). It is perhaps only in retrospect that paradigms can be identified, once a sufficient consensus of activity is reached. I prefer to label my overall research stance an ‘epistemology’. I take this word in the broad sense used by Bateson, as ‘the great bridge between all branches of the world of experience – intellectual, emotional, observational, theoretical, verbal, and wordless’ (1991, p.231). What I take Bateson to mean by the bridge between different aspects of our experience is that we are able, in each category, to be recursive (to observe our observations, theorise about our theories) and in this shift from doing, to an awareness of our doing whilst doing, we face the issue of how we come to know about our world, hence the label ‘epistemology’ for the title of this section.

The experiences described in stories 2 and 3 point to a view of learning and knowledge that is intimately linked to action, intention and expectation. Such a viewpoint is central to the ‘enactive’ (Maturana and Varela, 1987, p.256) approach to cognition, which has informed this study.

I have engaged in research into mathematics education since 1995, largely through a long-term collaboration with Laurinda Brown. This research has involved funded projects and publications in books and journals all of which have adopted an enactivist epistemology and methodology (e.g., see Brown and Coles, 1997, 2000, 2008, 2010, 2011, 2012). When I embarked on my PhD studies, I re-looked at this commitment and questioned my enactivist stance (see Coles, 2007). It was, in part, recognising how much Bateson’s writing has influenced how I act as a teacher and researcher, that led to my continuing commitment to enactivism.

The word ‘enactive’ was appropriated by Bruner (1966, p.11) to describe a mode of representation and knowing based in action. We know how to ride a bicycle not through images or symbols but through our actions. This knowledge is, therefore,

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represented enactively (according to Bruner). Varela used the term to label a radical view of cognition, which was articulated in Varela, Thompson and Rosch (1991):

We propose as a name the term enactive to emphasize the growing conviction that cognition is not the representation of a pregiven world by a pregiven mind but is rather the enactment of a world and a mind on the basis of a history of the variety of actions that a being in the world performs (p.9).

Cognition is taken to be fundamentally historical, enacted and embodied, which has implications for knowing and knowledge. The embodied nature of knowing was taken up by Lakoff, Johnson and Núñez (Lakoff and Johnson, 1999; Lakoff and Núñez, 2000) who put forward the hypothesis that all our abstract thoughts are based on metaphors learnt, through action, in the first years of life. In Lakoff and Núñez (2000), the authors give a detailed analysis of high-level mathematical concepts, showing their bodily basis. While these authors may not call themselves enactivist, their views of cognition are consonant with enactivism (such that Varela, Thompson and Rosch (1991, p.7) claim them as enactivists). Within the enactivist viewpoint, practical knowledge linked to action is foregrounded, with more propositional knowledge arising from awareness of action. Maturana and Varela state that, ‘cognition is effective action, an action that will enable a living being to continue its existence in a definite environment’ (1987, p. 29). So an action is effective if it allows me to continue operating in a specific context. If this context is being asked a question in a classroom by a teacher, an effective action might mean giving a response that is deemed correct. In other classrooms, effective actions might be ones that are interpreted as being ‘mathematical’, for example, making a prediction. It is in linking ‘doing’ to ‘effective action’ that I understand the aphorism, ‘All doing is knowing and all knowing is doing’ (Maturana and Varela, 1987, p.27).

A key distinction, in Maturana’s thinking, is between perception and illusion. Story 3 illustrates that we are not able to tell the difference between perception and illusion in the moment of perceiving. Maturana (1987, 1988) uses this fact to problematise any objectivist view that we can gain access to a world independent from the observer. The point of his argument is that there is nothing in our experience that marks out perceptions as being ‘true’, or ‘illusory’, let alone certain. The only way we are able to make a distinction between perception and illusion is via a different (meta) experience (of the same observer, or somebody else) about the first experience. And being subject to the same restrictions as the original experience, there is no guarantee this metaexperiential authority will not later be seen as illusion, hence we can never arrive at certainty or ‘objective’ truth.

Maturana and Varela were biologists and I assume this training is one source of another radical insight, that our biological structure determines what counts for us as perception and knowledge. ‘Structure’ is a technical term for Maturana and Varela, linked to the ‘organisation’ of a system or ‘unity’ (1987, p.47). The organization of a system denotes the relations that must exist between components for that system to be a member of a specific class (e.g., a human, or a table). The structure of a system

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denotes the actual components and relations that comprise a specific unity. I can saw the end off a table and it retains its organization (it is still a table) but alters its structure; if I saw it in two its organisation and structure have altered. In the course of our lifetime as humans, although our organisation remains largely the same (we are still humans, even if we lose our hair), our structure evolves and is altered by every interaction. Our organisation determines the kinds of things we have the potential to perceive (e.g., what spectrum of light) but it is our structure, carrying the history of our interactions in the world, which determines what counts for us as a perception.

Bateson is an often-cited figure in the historical development of the enactive view of cognition (and the more general position of ‘embodied cognition’). Bateson met another central proponent of the embodiment of mind, Warren McCulloch, at the 1942 Macy Foundation conference on the new discipline of cybernetics (which is broadly speaking the study of systems and their interaction), after which ‘they became close friends, corresponding and discussing ideas until McCulloch’s death’ (Kobayshi, 1988, p.347). McCulloch, in 1959, collaborated with a group including the young Chilean biologist, Maturana (see Lettvin, Maturana, McCulloch and Pitts, 1968). The group performed experiments that overturned assumptions previously held about visual perception in frogs, and Maturana attributes this experiment as the start of his journey to an alternative epistemology, although denying any direct intellectual influence from McCulloch (see Maturana and Poerksen, 2004, p.148). The standard view of perception, at the time of the experiments, was that, ‘the eye mainly senses light, whose local distribution is transmitted to the brain in a kind of copy by a mosaic of impulses’ (Lettvin, et al, 1968, p.237). In contrast, Lettvin et al, started from the hypothesis that the eye senses pattern and relations in the world. They found compelling evidence to support their hypothesis.

The experiments indicated there is no correlation between differences in neural activity and differences in light or colour intensity in the perceptual field of the frog. What these scientists did find were internal correlations between neural activity and behaviour. In discussing the workings of the nerves leading from the frog’s eye, they concluded these operations, ‘have much more the flavour of perception than of sensation’ (p.253) and, ‘the language in which they are best described is the language of complex abstractions from the visual image’ (p.254). In other words, it is not the case that the frog’s eye provides a representation of the world, which is then processed in its brain, for example, by de-coding the representation to conclude (not in awareness presumably) ‘bug to the right’, which then triggers activity. It seems as though the processing has already been done by the eye. What the frog’s eye tells the frog’s brain appears to be already categorised. The patterns of activity coming directly from the frog’s eye can only be correlated with something as high level as (to an outside observer) ‘bug to the right’, which then triggers a response directly.

Maturana was led (by this and subsequent experiments that confirmed the findings in other species) to the view that our nervous systems function as informationally closed (they clearly receive energy from outside) and do not contain representations or transforms of the environment. All living systems therefore are ‘structure

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determined’ (Maturana, 1988, p.12), by which he means it is the structure of the nervous system and body of an animal that determines its response to something, not the stimulus itself. Maturana argues, the only thing the environment can do to a system is to trigger a response. How the system responds is a function of its structure rather than the trigger.

To return to the example of the frog, in order to understand how correlations between neural activity in the eye and abstractions from the visual field become established, one must take account of the history of interaction of system (frog) and medium (which sometimes includes bugs). This history is relevant both on the time-scale of the frog’s lifetime (determining its particular structure), and the species (determining frog organisation).

Maturana introduced the notion of ‘structural coupling’ to refer to what happens in the living of a system involved in repeated interactions with a medium (which, of course, will usually include other structure-determined systems). System and medium ‘change together congruently as a spontaneous consequence of these recursive interactions’ (Maturana and Verden-Zoller, 2008, p.26–7) leading to a structural congruence or structural ‘coupling’.

It is structural coupling that allows organism and medium to go on living in recursive interaction, i.e., it is structural coupling that allows effective action. The organization of the frog, and its history of structural coupling with its environment, ensure the effective behaviour that certain patterns of stimuli in its visual field lead to ‘bug-catching’ action.

We credit the frog with intelligence and knowledge about its surroundings if it is able to act effectively (catch the bug). Its knowing does not consist in an ability to de-code a model of the world but simply in acting effectively within it. Maturana holds this view of knowledge for even the most abstract of human thought. I credit someone as knowing something if I judge their behaviour to be effective, in a given situation.

In an intriguing parallel to the way the frog’s eye seems, through its biology, to be carrying a significant cognitive load, Clarke (2008) in a book sub-titled, ‘Embodiment, Action, and Cognitive Extension’ describes how humans are able to walk using less than one tenth the neural processing of Honda’s Asimov robot, through making use of ‘ecological control’ (Clarke, 2008, p.5). In such cases:

part of the “processing” is taken over by the dynamics of the agent-environment interaction, and only sparse neural control needs to be exerted when the self-regulating and stabilizing properties of the natural dynamics can be exploited (Pfeifer et al, 2006, p.7, cited in Clarke, 2008, p.6).

This is what I take to have happened through the history of structural coupling of frogs and bugs. The history of the dynamics of the bug/frog interaction, which are expressed in the organization and structure of the frog’s eye, mean that only sparse neural control needs to be exerted for the frog to catch the bug. The frog catches the bug without needing to create an internal representation of the field of perception.

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METACOMMENTARY

Ideas about cognition can seem a long way from day-to-day concerns as a teacher, yet I see direct implications. In a classroom, the most significant features of the ‘environment’ for any individual are the other individuals in the room. Any interaction between teacher and student or between student and student must, in the process, alter the structure of both. Enactivism is a profoundly social theory, we are quite literally changed through interaction with others, or more precisely, we change ourselves through interaction with others who likewise change themselves. We cannot not change, however minimally, in every encounter. Equally, we cannot specify any change we want to provoke in others. It is in this sense that I understand Stewart (2010), in a book on enactivism, when he writes, ‘instruction, in the strict sense of the word, is radically impossible’ (p.9). Any learning that may take place in an interaction is a result of the structure of the individual who learns, not the trigger that provided the provocation. A teacher may occasion the learning of students and a discussion facilitator may occasion the learning of other teachers by providing the context and possibility for such learning, but there is no causality.

The enactive view of perception also has profound implications for the doing of research. It implies that what I see is literally shaped by how I have lived. In order to act differently, to learn, I need to perceive differently. In the context of research in the complex space of a school, this implies bringing a careful attentiveness to bear on my habitual ways of speaking and writing and thinking about what I see, if I am to become aware of anything new. Equally it implies (since knowing is doing and doing is knowing) that if I am interested in the development of mathematical thinking in classrooms then I need to look at the ways students and teachers speak and write. I need, therefore, to look at the actions in a classroom.

ENACTIVISM AS A METHODOLOGY

Reid (1996) wrote a paper setting out the methodological implications of enactivism for mathematics education. There were two key features: multiple perspectives, and theories that are good enough for.

Multiple perspectives are often achieved through the involvement of more than one researcher.

Enactivism foregrounds the role of the observer in any event, and acknowledges the complexity of interactions. For research methodology, this implies that we can never ignore the perspective or filter that each observer brings … [a]n enactivist approach … attempts to preserve the diversity of interpretations on the grounds that each interpretation says something significant about the observer-event (Reid and Brown, 1999, p.19).

It is possible to have multiple perspectives with a single researcher taking on different frameworks or ‘filters’ for interacting with data and revisiting data

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multiple times. Varela’s star (Figure 1.1) in which there is a movement between viewing the ‘it’ and the ‘process of becoming it’, offers one mechanism for how to try to take two perspectives on a phenomenon. Bateson also offers a mechanism for taking up multiple perspectives in his notion of ‘double description’ (1979, p.134). In a double description, any phenomenon is viewed both in its own right and as part of a wider relationship, allowing two perspectives and hence, ‘the possibility of a new order of information’ (ibid). The sense of a new order of information is one I recognise from, for example, being one of a number of different researchers offering multiple views on the same data. Sections of data that are significant from different perspectives are of particular value. Another source of perspective comes from communicating about research to others and inviting their interpretations.

While enactivist research is interested in learning, knowing and theory building, these only make sense with a purpose in mind, ‘clarifying our understanding of the learning of mathematics for example, and it is their usefulness in terms of that purpose which determines their value’ (Reid, 1996, p.209). Theories are not judged in terms of ‘truth’ but whether they are ‘good enough for’ the purposes of their users. The type of knowing enactivist research is aiming for is practical, informing our practice as teachers and researchers.

These ideas can be built into research design, for example by different researchers viewing the same piece of data together from different perspectives or emerging findings being fed-back to participants as soon as possible, to generate further, multiple views.

A third feature of enactivist methodology, that has become increasingly important in my practice as a teacher and researcher, is the privileging of the notion of difference or distinction as our primary cognitive function. As a result of the importance of this notion for the whole project, I place my ideas about ‘difference’ in some kind of context in the next section.

ABOUT DIFFERENCE

In chapter 1, I set out my desire in this work not to get caught in dichotomies and to think differently about the distinctions from which they arise. The philosophical consideration of difference goes back to at least Plato (Phaedo, 74c), and was taken up more recently by thinkers such as Spencer-Brown (1972), Keeney (1983) and Flemons (1991), to the point where the French philosopher, Jacques (1991) wrote of ‘difference’ as being, ‘hailed as a cure-all, a fundamental value without which philosophical thought of any kind was impossible’ (p.xii).

I first came across a theorisation about difference in the writing of Bateson and his aphorism that an idea (or in other places ‘information’) is ‘a difference that makes a difference’ (e.g., 1972, p.272).

[I]t is not too much to say that ‘difference’ was that analogue of a Newtonian particle for which [Bateson] had searched for so long in the social sciences (Harries-Jones, 1995, p.168).

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It is differences that, for Bateson, re-enter any system-environment interaction and form the basis of recursion. In his example of a man with an axe, he exemplifies this idea:

Consider a man felling a tree with an axe. Each stroke of the axe is modified or corrected, according to the shape of the cut face of the tree left by the previous stroke. This self-corrective (i.e., mental) process is brought about by a total system, tree-eyes-brain-muscles-axe-stroke-tree …

More correctly, we should spell the matter out as: (differences in tree) – (differences in retina) – (differences in brain) – (differences in muscles) – (differences in movement of axe) – (differences in tree), etc. What is transmitted around the circuit is transforms of differences. And, as noted above, a difference which makes a difference is an idea or unit of information. (1972, p.317)

At the end of the first paragraph of this quotation, Bateson calls the total system ‘mind’. In another famous example, a blind man with a stick, Bateson asks where the ‘man’ begins, ‘At the tip of the stick? At the handle of the stick? Or at some point half way up the stick?’ (1972, p.318). He rightly judges such questions to be nonsense, since man, stick, immediate environment form a ‘systemic circuit’ (ibid) that can only be cut (e.g., to delimit the ‘man’ part) in an arbitrary way. Seemingly, in opposition to Maturana’s operational closure of the nervous system, Bateson sees humans as open systems. Differences ‘outside’ become transformed into differences ‘inside’ and back out again in such complex and repeating cycles that it makes no sense to consider the ‘human’ part of the system in isolation.

The reading, above, of what is involved in a man felling a tree has led Wolfe (2000, p.177) and Dell (2007, p.1) to criticise Bateson as harbouring a vestige of objectivism. In other words, they read Bateson as implying there is something ‘real’ in the outside world to which we can gain direct access and which gets transformed to the inside. It is certainly the case that Bateson believed our epistemology can be ‘objectively’ correct and he attributed pathologies of modern life, e.g., alcoholism, environmental disregard and schizophrenia, to ‘errors’ of epistemology. However, I believe a focus on difference need not entail maintaining an untenable objectivism. In discussing the difference between kicking a dog and kicking a stone (1972, p.409), Bateson makes the distinction between the dog responding as a result of its metabolism and the stone as a result of the laws of physics. In other words, Bateson is committed to the view that, with a system such as a dog, all the environment can do is trigger and any changes that occur will be the result of the internal structure of the dog. This is precisely Maturana’s view of the closure of the nervous system (see above). So, I can look at a man cutting a tree and observe the ‘units of information’, or differences, moving around the system, but these differences are triggers and do not, in themselves, determine the changes that will be produced.

To make the point more clearly, that ‘difference’ does not entail objectivity, involves introducing another aspect of Bateson’s thinking, his distinction between

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the digital and the analogue. A digital difference is all or nothing. With an analogue difference the degree of difference also counts. In Bateson’s thinking (1972, p.291), he sees analogue communication as more primitive and often analogue differences in the environment get transformed to digital differences within the internal structure of an animal. An example of this would be a trigger of a reflex mechanism. There are gradations of heat that will burn my hand to different degrees. At a distinct point on that analogue scale, a reflex response is invoked and I draw away my hand. My reflex response is either active or not, it is a digital response to an analogue trigger. This distinction, between analogue and digital types of difference, is picked up by writers who might be classified as postmodern.

Davies, drawing on the work of Deleuze (1968), distinguishes between ‘difference’ and ‘differentiation’ (Davies, 2009, p.17). Davies writes that for Deleuze, ‘difference comes about through a continuous process of becoming different, of differentiation’ (ibid). Massey (2005) also distinguishes discrete difference from continuous differentiation and sees Deleuze as wishing, ‘to instate the significance, indeed the philosophical primacy, of the second (continuous) form of difference over the first (discrete) form’ (p.21). The distinction between difference/differentiation mirrors Bateson’s digital/analogue divide and also the ‘star’ (Figure 1.1), where Varela distinguishes between ‘it’ and the ‘process of becoming it’.

The continuous form, differentiation, the process of becoming different, is also a central concept for Derrida:

differences, thus, are ‘produced’ – deferred – by différance … what is written as différance, then, will be the playing movement that “produces” – by means of something that is not simply an activity – these differences, these effects of différance (Derrida, 1981, p.11).

This is a complex piece of writing, but I take it to be making a distinction between the production of difference (différance) and the differences, the effects, that are thus produced. To take a macro example, the man with the axe begins to make cuts in the tree. The depth of cut increases in a continuous manner; at some point a threshold is reached and a digital difference (the tree falling) is produced. I believe that by foregrounding the production of difference over its effects, any objectivism, which may be interpreted as lurking in Bateson’s concept of a ‘difference that makes a difference’, can be eliminated. Taking a postmodern reading of Bateson, ‘[t]he result is a radicalization of the stable notion of “information” so that it becomes not a “unit of difference” but rather a “process of differentiation” ’ (White, 1998, p.6) and it is this notion of difference I wish to invoke. To stay with the example of the man felling the tree, in this radicalized view, there are not units of information being transmitted through the system, but rather each element of the system (tree-eyes-brain-muscles-axe-stroke-tree) can be seen as under-going a process of differentiation (of continuous change), determined by their structures. At the point when the tree eventually falls, that particular system ceases to exist. In fact, I believe this radicalisation was always present in Bateson’s work, within his distinction between analogue and digital

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difference and his view that the digital came out of the analogue. There may be times when a system does ‘unitise’ a difference in its environment (e.g., my example above of a reflex response) but Bateson is not guilty of projecting such units back into the environment, which would be the case with objectivism.

This reading of Bateson brings his views in line with those of Maturana and the stance that the nervous system is informationally closed to its environment. It is via a mutual production of differences, or an ongoing process of differentiation, that system and environment are able to engage in the structural coupling that leads to effective behaviour.

METACOMMENTARY

Again, this discussion of postmodern and cybernetic views of difference might appear far from the concerns of educators. Yet, taking seriously the primacy of distinction and differentiation, has direct pedagogic implications. My starting point for any teaching is to set up a context as quickly as possible in which I will be able to ask students the question, ‘what is the same, what is different?’ and get them making distinctions. I believe (both theoretically and from experience) everyone can do this. In my thinking about research design and data analysis, I am interested in procedures that will highlight difference over time and procedures that give access to patterns, a closely related notion (since a pattern implies a distinction between items that are the same to the exclusion of those that are different).

In looking for pattern and difference in my data, I do not assume that I am gaining access to anything stable or fixed. A difference that I notice says as much about me as about my data, or in Reid’s (1996) language, ‘there is no data only interpretations and interpretations of interpretation’ (p.208). I do, however, seek to collect data in a way that reduces intervention or interpretation, and to present it in a manner that offers the reader some access to both sides of the distinctions I draw, what is excluded as well as what is included. I have made many connections in the writing of this book and hope readers may find some connecting patterns as well.

EQUIFINALITY

The final methodological implication of enactivism I wish to highlight concerns the approach to data analysis. Enactivism, being a theory of cognition, has implications for what learning and knowing are, but does not provide constructs (such as ‘power’ or ‘community’) with which to analyse something like a classroom setting. Instead, it provides a suggestion for a manner of approach to analysis, for example in the idea of equifinality.

The concept of equifinality is associated with early cybernetic, or general systems, thinking (von Bertalanffy, 1969). Bateson was also involved in the early cybernetic movement, which aimed to look for generalities across systems, independent of their components. In the early years of this strand of thinking, it was assumed that systems

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move from one equilibrium-position to another. It was assumed that a system not in equilibrium was in danger of collapse, and would resolve itself as quickly as possible into an equilibrium state, before being triggered out of it by some other event, only to return again to equilibrium. The concept of ‘equifinality’ described how such stability seeking systems would often appear to reach the same equilibrium position from a wide variety of initial conditions.

Later systems-thinking, influenced by the mathematics of chaos and complexity, has come to recognise the incredible sensitivity on initial conditions that certain systems display. This sensitivity led Lorenz (1963), an early pioneer of the mathematics of chaos, to comment in relation to the atmosphere (or any other non-periodic system), ‘prediction of the sufficient distant future is impossible by any method, unless the present conditions are known exactly’ (p.141) and so, for the weather, ‘precise very long range forecasting would appear to be nonexistent’ (ibid). For a system such as the atmosphere, ‘present conditions’, being analogue, can never be known exactly. It has been recognised more recently how systems are able to exist and operate ‘far from equilibrium’ (Juarrero, 2002, p.119). One insight here is that complex systems are finite. Although existing ‘far from equilibrium’ is inherently unstable, systems (such as humans) are able to maintain such instability for a finite time before collapsing.

It might seem as though communication in classrooms, with the myriad complexities of individuals and relations, would be a complex system existing far from equilibrium. Yet a surprising finding from my on going collaboration with Laurinda Brown (e.g., see Brown and Coles, 2008) has been Laurinda’s awareness that year on year there was something recognisably the same about my year 7 (aged 11–12) classes. Individuals differ and particular classroom routines and language varies, yet despite these differences there is something identifiably the same, by the end of a year working together. One aspect of what is the same is the way that people talk to each other (but not the specifics of what they say). It is perhaps the case then that the patterns of language in a classroom do not form a far from equilibrium system, but rather consist of a simpler, equilibrium seeking system. There can be some sense of equilibrium that is achieved year on year by experienced teachers, independent of the varying initial conditions within each class of students. To re-emphasise, this is not implying there is a sameness in terms of the content of what is said (although there would be some key words that would recur year on year, in my own classroom, such as ‘conjecture’) but a sameness in the process of how talk unfolds in a lesson. Each year, the specific patterns of language would be different, but there would be some stability in those patterns and connections across years, i.e., the sameness Laurinda observed was a pattern in the patterns that developed each year.

For example, one year Laurinda observed students in my year 7 class referring in writing and conversation to the idea that for mathematicians it is ‘okay to make mistakes’. In a subsequent year, students referred more often to the notion that mathematicians ‘try to prove themselves wrong’. The idea itself changed, what

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stayed the same was a metapattern that there was one predominant guiding idea of what it meant in this classroom to be ‘becoming a mathamatician’. In this sense, Laurinda’s observations suggest that equifinality may operate across patterns of patterns of language in the classrooms of experienced teachers. Some plausibility is perhaps added to the suggestion that equifinality is relevant to classroom interaction, by noting that it is a concept still used today in some branches of family therapy to describe how family patterns can become constrained (Stroh Becvar and Becvar, 2000).

The insight into equifinality, i.e., the existence of some stable metapatterns of interaction that become established over the course of teaching the same group of students for an academic year, leads to a principle of enactivist data analysis. Analysis is cyclical. At the end of any phase of research (e.g., at the end of an academic year) analysis begins with the final piece of data. On the assumption of equifinality, the final piece of data should exhibit the most stable (meta)patterns, or perhaps the patterns that are most fundamental. Analysis proceeds therefore by identifying patterns in the final data item, and then tracing these patterns back through the rest of the dataset. The intention in the tracing back is not to tell any story of causality, but rather to trace the emergence of pattern. The cycle of interaction between theory and data continues in loops throughout the life of an enactivist inspired project, i.e., data is looked at and talked about, which informs future data collection. In this study, I took data over two years, hence the analysis of patterns from the first year informed my analysis of the second year of data.

METACOMMENTARY

I wrote, in chapter 1, about wanting to support individuals develop possibilities for new or different responses to situations, and supporting them in viewing situations differently. The principles behind enactivism I see as all leading to, or pointing to, activity that will do just that for the researcher as well. The principles of taking multiple views, of holding all theory as conjecture, of tracing patterns, help me as a researcher, ‘to see more and see differently’ (to quote an often used phrase of Laurinda’s) supporting the need, for me, to make the familiar setting of the classroom strange again. It is with these principles, and the enactive view of knowledge as linked to doing, that a way through Plato’s dilemma about learning (quoted at the start of this chapter) can be negotiated. We learn about the things we do not even know exist by staying alert to the detail of what we see, by sharing the detail of common experiences with others and by deliberately not seeking generalities beyond those that support effective action.

It is now possible to state the research questions I address in this book, keeping in mind this stance on learning, drawing on the language of pattern and difference:

– how can discussions of video occasion teacher learning? What patterns of interaction support teacher learning? What is the role of the discussion facilitator?

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– how can classroom discussion occasion mathematical thinking? What patterns of interaction get established and alter in a classroom, over an academic year, that support mathematical thinking? What is the role of the teacher?

– how are the roles of discussion facilitator and teacher the same or different?

I do not expect to get answers to these questions, but rather deepen my awareness of the issues and hope, in the process, this may provoke in readers some new ways of seeing and acting.

In the next chapter, I summarise the research methods used in this study, to conclude the first, introductory, section of the book.

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CHAPTER 3

A CYCLICAL ENQUIRY

It is, after all, an observer who observes the observing (Maturana and Poerksen, 2004, p.37).

TWO AREAS OF FOCUS

A key purpose of this study is to compare and contrast the approach to managing whole-class discussions with students, and facilitating discussions with teachers. I was interested in data about both the classroom practice in the department and about discussion between teachers of that practice. The aim of this study was therefore to investigate patterns of interaction in both teacher meetings and classrooms.

I collected data over two academic years as I was interested in patterns over time and also how classroom expectations are set up with students new to the school. I have experience from previous projects of using video to study classrooms and knew this was an effective tool.

Turning on the camera is an interpretive act, for example in the decision of where to place it. I knew I was interested in times of whole-class discussion in a lesson and wanted long sequences to be able to capture what can sometimes be thirty minutes of dialogue. I also knew from my previous work with video that a camera at the back of the room allows me to pick up enough of the sound to transcribe most of what is said. I also believe, from experience of video recording my own lessons, that, after the first one or two occasions, students’ behaviour appears not to be noticeably altered by the presence of an unattended camera at the back of the classroom. So, to make recordings of lessons, I would set up a camera before the lesson, set the tape recording before the class and leave. I took video recordings of lessons early in the academic year and then followed-up with recordings later on. These video recordings then became the focus for teacher discussion. Any recording, of course, brings with it ethical issues.

ETHICAL ISSUES

My views on ethics have been strongly influenced by Varela, who alongside his research into neuroscience and phenomenology, wrote a book (1999) detailing the ethical implications of the enactive stance. He argued for the importance of ‘ethical know-how’ (Varela, 1999, p.30), which I see as complementary to the enactivist foregrounding of knowing as doing and doing as knowing.

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Varela (1999, pp.21–24) wrote about ethical expertise being non-deliberate in general and also linked it to being a ‘full participant’ (p. 24) of a community. (What being a ‘full participant of community’ means could take considerable un-packing, I take this to indicate an engagement in stated or common practices evident within any group of individuals who meet repeatedly.) Ethics (in the West) is almost always conceived as being tied to some rational and analytical process, and this focus on rationality masks the on-going ethical dimension to our automatic and skilled behaviours. We spend most of our lives in un-reflexive skilled behaviour, including expert ethical behaviour. We frequently display ethical know-how in relatively mundane ways, for example listening empathetically to a friend’s distress, or reaching out a hand to steady someone who is about to trip. We perform these actions without recourse to rational reflection or calculation. Rational deliberation, despite what we (rationally) may think, is the exception rather than the rule in terms of our day-to-day, moment-by-moment behaviour. We call on our rational resources at breaks in our skilled behaviour, for example when we cannot decide what to write to a friend who has been bereaved, or when we are stuck on a mathematics problem.

So, in thinking about my own ethical behaviour as a teacher and researcher, it is important I fulfil university requirements (in becoming a full participant, as a researcher, of that institution) and also school requirements (in becoming a full participant, as a teacher, within the department).

From a university perspective, particular ethical issues arise in any study (such as mine) that involves taking recordings of children, and in the taking of a dual role of teacher and researcher.

The revised ethical guidelines for educational research (2004) adopted by the British Educational Research Association (BERA) state:

11. Researchers must take the steps necessary to ensure that all participants in the research understand the process in which they are to be engaged, including why their participation is necessary, how it will be used and how and to whom it will be reported. Researchers engaged in action research must consider the extent to which their own reflective research impinges on others, for example in the case of the dual role of teacher and researcher and the impact on students and colleagues. Dual roles may also introduce explicit tensions in areas such as confidentiality and must be addressed accordingly. (BERA, 2004, p.5)

In keeping with my enactivist standpoint, I do not believe I can ‘ensure’ anyone understands anything. However, I find it helpful splitting the issue of consent in to three: why participation is necessary; how it will be used; and to whom it will be reported. I sought and gained written permission from students’ parents via a letter that informed them about these issues, based on guidelines from the National Society for the Prevention of Cruelty to Children (NSPCC, 2006).

The issue of dual roles was more complex. I hoped my research role would have a beneficial impact on students and colleagues, through encouraging teachers in sharing and reflecting on practice. Given that my research interest was in the

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development of patterns of communication over the course of a year, and that I suspected the early lessons were important in setting up particular patterns, I was concerned that by undertaking this research the process of gaining informed consent might actually get in the way of the learning of the participants and potentially alter those very patterns of interaction I wanted to study. A story from my own classroom illustrates some of my concerns.

Story 5: An Unwelcome Visitor

In the early 1990’s in the UK, the government introduced a National Numeracy Strategy which included specific advice on how to structure lessons into three parts: a ‘starter’; ‘main activity’ and ‘plenary’. A high-level figure from the Local Education Authority (LEA) came to visit my school to observe how we were implementing this new policy (we weren’t). He spent an hour in my classroom when I was teaching a year 7 class (11–12 year olds). The class were working on a problem we call ‘Functions and Graphs’. The class were several lessons into this problem and at a stage where students were following different lines of enquiry, all based around the overall challenge of predicting what the graph of a function will look like just from the rule. For example, some students were looking at graphs of rules such as, n->2n, n->2n+1, n->2n+2, etc; other students were looking at different groups of rules. I overheard the LEA observer interact with one student, who was drawing (successfully) graphs of the rules n->n2 and n->n3, and ask the student “Did you choose those rules or did someone tell you to do them”.

I knew at the time of this incident that I was uncomfortable with the kind of questions this LEA observer was asking, without quite being sure why. What I now am aware of is concern with any question that seems to be coming from a place that is not a part of the classroom and lesson focus, i.e., where it is clear the questioner is not engaged with the task of the lesson, but more with issues about the lesson. I was concerned to avoid, through my research, provoking such unhelpful levels of awareness about the process students and teachers are going through (and whether it was, for example, unusual) that may get in the way of their own engagement in the task of the lesson or teacher meeting.

My resolution to this dilemma was to aim to drip-feed conversations about the research when a recording was done. I did gain written permission from parents before any recordings began; in addition, at times throughout the project teachers can be seen, on the video recordings, returning to the issue of what data was being collected and why. No parent questioned whether their child could be involved or refused permission. No student took the opportunity to sit out of camera shot, something that was offered by all teachers. I suspect that part of the reason for students’ and parents’ apparent ease with involvement in the research process was that the camera, being fixed at the back of the classroom and facing the front, made

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it fairly clear that the focus was on the teacher. In general, only the backs of students’ heads are visible on the video recordings.

The University of Bristol’s ethical procedure involves engaging in a conversation about the ethics of any research proposal with a fellow researcher. My conversation was with Richard Barwell, and we were invited to write up our discussion for publication (Coles and Barwell, 2007). This conversation helped me uncover some of my implicit ethical beliefs about discussion, by reflecting on my work with pre-service teachers who would come to our school for teaching practice. I was aware, in lessons, of trying to notice any judgments that arose in me (taking judgment to mean something that carries a negative evaluation). I then used these judgements to track back to the ‘differences’ compared to what I might have been expecting (as I believe it is always possible to do, following a negative emotional reaction). A difference, I find, can then be offered to the pre-service teacher without attachment to the negative evaluation. If I am able to unhook myself in this way from an unthinking reaction, I am able to share openly and professionally in a dialogue about possibilities for action.

Avoiding judgment is, for me, about an ethical stance of trying to be, as far as possible, alongside the prospective teacher. I bring the same beliefs, to some extent, to my work with students. If someone gives an answer in a class that could be judged as ‘wrong’, I try and avoid the use of that word in my response. I believe that all ‘wrong’ means is that, either a slip has been made (e.g., in a calculation) or, more usually, the student has been operating under a different set of assumptions. So, for example, in a first lesson with a class I will often play a name game and get students to try and work out how many names have been said. For the game I play, this involves working out 1+2+3+ … up to, say, 28 if there are that many in the class. Someone almost always offers 28x28 as a quick way of calculating this total. I am aware of usually responding to this by asking what would have had to happen if 28x28 was the total number of names said (i.e., everyone would have had to say everyone else’s name). So, 28x28 is a helpful answer that, in fact, solves a slightly different problem. ‘Wrong’ comes down to ‘different’ again, and I believe that working together as a class to get to where this difference lies both provokes important mathematical thinking and supports classroom interactions in which students value and respect what each other have to say.

Researching in an ethical manner therefore goes beyond the issue of informed consent. I find it to be unethical when I read research that is basically damning of what has been observed. Researchers have to make distinctions about what they interpret but these do not have to be detached and judgmental. Varela (1999) writes about how, ‘praxis is what ethical learning is all about … if we do not practice transformation, we will never attain the highest degree of ethical expertise’ (p.63). Being open to transformation and change in myself, places me out of a judgmental and evaluative stance towards my research.

I know, for myself, that if I audio- or video-record meetings/lessons I am usually able to operate with full attention on my role as discussion facilitator or teacher.

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The recordings allow me to interrogate and reflect after the event. In terms of my approach to managing the classroom or teacher talk, it sometimes feels as though I became more aware of avoiding my own evaluative stance when there is a recording. In other words, engaging in research feels like it supports my developing ethical know-how. To summarise this section, in addition to the ethics of a discussion about the research (e.g., in gaining ‘informed consent’) the whole research project can be seen, in one light, to be about the ethics of discussions themselves.

COLLECTING DATA

In year 1, I took data from the year 7 (age 11–12) classroom of Teacher A. At this school year 7 are the youngest students, so I was able to investigate how ways of working mathematically arose in a new setting. In year 2, I studied the same class as they moved into year 8, still taught by Teacher A. As part of a wider study (not reported here), I took video recordings of three teachers. I report here just on Teacher A, who was acknowledged by both school and external agencies as an expert practitioner. I have come to see her classroom as a ‘paradigmatic case’ (Freudenthal, 1981, p.135) of a very particular practice that I report on in the third section of this book.

As I have described already (Chapter 2), a routine I had established amongst staff in the department, from before embarking on this study, was the recording and joint viewing of videos of each other teaching. Given all the other constraints on time in meetings, I was able to devote time to using video roughly once every six months. I took audio recordings of these teacher discussions. I was interested in exploring in more detail what was happening within them. Over the two-year period of classroom data collection I took audio recordings of three video-discussion meetings. As with the number of video recordings, I deemed this would be sufficient to allow me to look at patterns across the recordings. I also conducted a final interview, towards the end of year 2, with the group of teachers in the department and invited them to talk about anything we had done together that they recognised as having been significant for their own learning.

The enactivist principle of seeking multiple views led me to seek other forms of data, linked to these core video and audio recordings, that might offer different ‘lightings’ (Gattegno, 1987, p.44) on what I observed. Lesh and Lehrer (2000) use video to analyse conceptual change, and, for that purpose, recommend supplementing video with interview data. Along similar lines, Clarke (1997) developed what he called the ‘Complementary accounts methodology’. Central to this research procedure was the use of videotaped classroom lessons and video-stimulated recall techniques within an interview protocol that sought to obtain the following:

1. Students’ perceptions of their own constructed meanings in the course of a lesson and the associated memories and existing meanings employed in the constructive process

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2. Students’ sources of conviction for the construction of their mathematical meanings3. The individuals, experiences, arguments, or actions in which students believed mathematical (academic content) authority to reside (Clarke, 1997, p.100).

My purposes are different, I am not looking at student learning or conceptual change, but more at teaching strategies or the role of the teacher in influencing classroom interactions. I therefore piloted video-stimulated discussion within an interview protocol, seeking the following (roughly equivalent) information:

1. Teachers’ perceptions of their own constructed meanings in the course of a lesson and the associated memories and existing meanings employed in the constructive process2. Teachers’ awareness of strategies employed for specific purposes3. The student contributions that seemed particularly significant in terms of a class discussion.

In order to gain a perspective on these issues, in these interviews, I played the video-tape and invited Teacher A to say ‘stop’ at any moment she wanted to say something about what she saw, i.e., about what she had done, or not done, what students were doing, or anything else that arose from the viewing. We would then talk about what she noticed and I took notes of our discussion. I also occasionally allowed myself to stop the tape and invite comment or discussion. I found I did gain new perspectives on the classroom videos through such a process and conducted interviews of this type with Teacher A as well as, on one occasion, presenting transcript data to a group of teachers from the mathematics department, for discussion of teaching strategies.

In keeping with the enactivist principles of the study, there was a cyclical process of using methods to collect data, working with that data and altering the methods for further data collection as a result. Looking back, the clearest example of this cyclical process is that, having studied the data from year 1, it became apparent that interviews with students might offer something useful in terms of thinking about patterns of classroom interaction, to further support multiple views on those patterns. To support my interpretation of students’ mathematical thinking, I wanted them to articulate as much as possible and judged that this would be facilitated by having them work in pairs. I did not want to work with larger groups as I wanted the work to be collaborative (to force articulation) and was concerned that in threes or fours some students may either not engage, or begin working on their own. Enactivist principles entail beginning with a task for students to engage in together, the important thing is to get students doing something first, to give access to their knowing (as doing) as well as anything they might say about what they know and do. I worked on some mathematics with students to give us a common experience on which to base our conversation. I chose activities that offered scope for them to collaborate, spot patterns, ask questions,

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make predictions and use algebra. I let them work on the task largely unaided (I responded to questions and, if students answered the initial prompt and then stopped, as though they had finished, I decided I would invite them to explore the situation further). I then had some simple questions to ask along the lines of what they think it is to work mathematically (for a full interview schedule see, Appendix 3). With around thirty students in the class I could have interviewed them all, in fifteen pairs. However I did not want to unnecessarily disrupt the usual working of the classroom. Given what I wanted to do in the interviews, I knew (from experience of a similar schedule in my master’s dissertation) they would take around twenty minutes each. I decided that working with three pairs of students (20% of the class) would give me a sufficient range of perspectives to be able to look at patterns across their responses, and would mean that my interviews could all take place in a single lesson (one hour) hence minimizing disruption. I wanted to see the students more than once to be able to track patterns and changes in their responses and so collected two sets of interviews (in October and July of Year 2) with three pairs of students from Teacher A’s class. As a way of trying to ensure a range of perspectives across the class, I asked Teacher A to nominate one pair of with relatively high levels of prior attainment, one roughly in the middle of the class and one at the low end. I also asked for a gender balance across the six students (not necessarily within each pair).

MY APPROACH TO ANALYSIS

In keeping with the enactivist idea of equifinality, my analysis of the video recordings of Teacher A began with the final lesson in year 1. I initially split the videos of lessons into sections, distinguishing periods of ‘whole-class discussion’ from all other activities. These were unproblematic distinctions, in other words, there were generally clear demarcations of times in the lesson when there was a conversation to which all participants were attending, or were meant to be attending (as cued for example by the teacher asking for students to stop talking or put their pens down or turn around to face the front).

To aid the search for patterns within sections of whole-class discussion, which were the parts of the lesson I wanted to investigate, I needed to split these sections further. Simply looking at the order of speech, it became apparent that (of the lessons I analysed) the most common pattern in whole-class discussion was for the talk to proceed: Teacher–Student–Teacher–Student, etc. In some cases, the T-S-T parts of whole-class discussion seemed well characterised by the description of turn taking labelled the Initiation-Response-Feedback (IRF) pattern identified by several authors (e.g., Sinclair and Coultard 1975; Mehan 1979; Mercer 1995). An initiation is typically a question, or a turn in a dialogue that introduces a new issue that is not a response to, or evaluation of what has just been said. I was interested, in part, at looking for times when this patterning is disrupted, for example by the student taking on the initiation role, e.g., by asking a question, or if the turns in discussion go Teacher–Student–Student. I therefore segmented further the lesson sections of

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whole-class discussions, into what I label ‘episodes’, according to what patterns I observe in the discussion.

I began by then characterizing episodes to support my noticing of patterns within and across lessons. I tried to think about episodes in terms of low inference aspects, such as the use of specific words. I used descriptive phrases, linked to my interest in the role of the teacher and patterns of interaction, to characterise each episode. In the final lesson of year 1, I distinguished the following episodes:

– Teacher asks for explanation – Teacher asks for comments – Student question – Student use of ‘because’ – Student explaining an idea at the board.

These labels are significant only for the purpose of supporting my noticing of pattern. I looked at patterns of interaction both within and across episodes, and patterns in the transition points. The patterns identified in the final lesson (from year 1) then became a ‘lighting’ (Gattegno, 1987, p.44) on the rest of the data from year 1.

I adopted a similar process of splitting into episodes, to analyse the audio recordings of teacher discussions. For example, if a teacher raised an issue, I defined an episode to last for as long as that issue is discussed. Any move that shifted the focus (e.g., a teacher asking a question about something new) defined a different episode.

The decision about where an episode begins and ends is sometimes clear cut (for example, a student using the word ‘because’) and sometimes fuzzy (e.g., was this question introducing a new issue, or was it a continuation of the issue just raised?). I imagine other researchers might mark some of these distinctions in different places (an issue of reliability), however, as stated above, the splitting is only a mechanism to support me seeing more in the data and noticing patterns. It is the patterns that I report on and analyse, and it is in relation to these patterns that I am concerned about validity.

VALIDITY

Validity concerns the extent to which any research conclusions or inferences can be shown to follow from the data collected. This is sometimes seen as an issue of the trustworthiness of the inferences, or indeed the researcher (Freeman, deMarrais, Preissle, Roulston and St. Pierre, 2007).

Lather (1986) suggests, ‘[s]pecific techniques of validity are tied to paradigmatic assumptions’ (p.66). Freeman et al (2007) make a similar point when they cite Wilson’s (1994) criteria that evidence of validity should be, ‘consistent with a researcher’s chosen epistemology or perspective’ (Wilson, 1994, p.26 cited in Freeman et al, 2007, p.28). I set out, in chapter 2, my own epistemological stance and while I am not aware of literature specifically dealing with the issue of validity

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within enactivist research, enactivism is commensurable with a range of post-positivist epistemologies that acknowledge the link between knowledge and action or practice (and hence its uncertainty) and the need for researchers to be reflexive about their own assumptions and prejudices.

Wilson’s (1994) criteria for validity within qualitative research suggest that evidence should be, ‘observable … gathered through systematic procedures … shared and made public … compelling’ (1994, p.26–30 cited in Freeman et al, 2007, p.28). These criteria fit well with enactivist principles, particularly the idea that evidence be observable. However, perhaps as important as the criteria, is the issue of how to go about trying to ensure they are met. Creswell and Miller (2000) list nine ‘commonly cited’ (p.25) procedures for validity within qualitative research, ‘triangulation, disconfirming evidence, researcher reflexivity, member checking, prolonged engagement in the field, collaboration, the audit trail, thick rich description, peer debriefing’ (p.26). In agreement with Lather (1986), Creswell and Miller see the choice of procedure as linked to a researcher’s worldview, as well as the emphasis of the project and the audience for any writing. Seven of the nine procedures are ones I used. As mentioned above, researcher reflexivity is a central element of enactivist research, and by taking data over two years (in a school where I worked for over a decade before embarking on this project) I have a ‘prolonged engagement in the field’.

There are further procedures to support validity built in to my research design. By co-watching videos with Teacher A, I engaged in member checking (which ‘consists of taking data and interpretations back to the participants in the study so that they can confirm the credibility of the information and narrative account’ (Cresswell and Miller, 2000, p.127)). This process was also built in to the process of watching excerpts from some videos with the wider mathematics department. These co-watchings both supported my analysis of the classroom data and became part of my data themselves. The discussions of video recordings had the potential to occasion new teacher actions that might be seen in subsequent video recordings of lessons, which is a consequence of the cycling process of data collection, interpretation and further data collection built into the research design.

By collecting evidence from students of the classrooms in this study, there was an element of triangulation of data (i.e., considering multiple sources), for example, in whether student descriptions of thinking mathematically are consistent with processes observable in the video recordings. In focusing on patterns of interaction in classroom talk, I firstly transcribed the whole of the final lesson from year 1 and then tracked patterns back through earlier recordings, transcribing as I went. My analysis was, therefore, based on a large part of the whole corpus of data.

However, rather than assuming that I have dealt with validity through this set of procedures, I agree with Freeman et al that:

There is no single marker of validity in qualitative inquiry … validity cannot be defined in advance by a certain procedure but must be attended to at all times as the study shifts and turns. (2007, p.28–9)

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This conclusion mirrors my thoughts above, relating to the need to be alert at all times to the ethics of the discussions that take place amongst teachers and in classrooms. As a researcher, I need to be similarly alert to what I am seeing in the data I analyse and the extent to which what I think I see is indeed observable.

METACOMMENTARY

I have set out in this chapter my research methods in collecting and analysing video and audio data about classroom and teacher interaction in a way that recognises, to repeat the quotation from the start of this chapter, that ‘[i]t is, after all, an observer who observes the observing’ (Maturana and Poerksen, 2004, p.37). The enactivist ethical approach entails an on-going attention to discussion of potential issues for participants, beyond the gaining of ‘informed consent’. Similarly, there needs to be on-going attention to the validity of research, as well as the design of mechanisms, such as triangulation, to support this validity.

To summarise my data collection, I took three audio recordings of teacher discussions of video, ten video recordings of lessons of Teacher A, two sets of student interviews and interviews with Teacher A about the videos. I audio recorded a final interview with all the teachers who had been involved in watching video recordings (including Teacher A). I transcribed the recordings from year 1 and, in the search for patterns, began my analysis with the last of these transcripts. The first stage of analysis was to identify times of whole class discussion and then, within those times, identify episodes, defined by differences in patterns of interaction.

This chapter concludes the first part of the book, which has set up the background and plan of the study. In the second part, I present findings from my analysis of data on the use of video for teacher learning, after first setting out some of the literature in that area. In the third section of the book, I treat, in a similar way, the classroom video data, setting out some key literature (Chapter 7) and using it to frame my analysis (Chapter 8).

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CHAPTER 4

ON USING VIDEO

Trouble is the engine of narrative and the justification for going public with a story. It is the whiff of trouble that leads us to search out the relevant or responsible constituents in the narrative. (Bruner, 1996, p.99)

VIDEO AS A TOOL FOR TEACHER LEARNING

In this chapter, I review research into the use of video with teachers. I then consider, more widely, how discussion can provoke or facilitate learning of teachers. Past research has not looked into the detail of the role of the discussion facilitator when using video. This is one research gap I address in the next two chapters.

Before setting out the relevant literature, I need to clarify the distinction between what I am calling teacher learning and the common label of professional development. The use of video that I will be analysing took place in a school where I was head of mathematics and so partly responsible for the (in-house) professional development of mathematics teaching staff. I introduced the use of video in the department and saw it as a key element of how I could support staff in developing their teaching. My aim was always to provoke some learning and the possibility for new or different responses. I assume this is the aim of any professional development, and so draw on the literature in this area. I use the phrase ‘teacher learning’ to indicate that the use of video in this school was not a separate course or programme, but integrated into standard departmental meetings, and that my interest is in any learning that occurred.

In reviewing the history of the use of video for teacher learning, Sherin (2007) cites reports going back to 1966 that found mixed results in terms of effectiveness (Fuller and Manning, 1973; McIntyre, Byrd and Foxx, 1966). Sherin (2007) suggested more empirical work was needed to gain appreciation of possible roles for video, work that she herself has continued to conduct (e.g., Sherin and van Es, 2009).

The National Centre for Excellence in Teaching Mathematics (NCETM) commissioned a review of UK research into video use for professional development which found, ‘[u]sing video clips for the professional development of in-service teachers is under researched’ (Hall and Wright, 2007, p.9). Nonetheless, the report was able to draw some tentative conclusions, which are mirrored in a book (Brophy, 2007) on the issue. The NCETM study suggested that effective use of video required videos produced by teachers for use in their own schools, as opposed to ‘universal’ or standardised recordings. In similar vein, Brophy (2007) concluded that ‘ideal’ videos were of ‘teachers with whom viewers can identify implementing a curriculum

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similar to the one they use … in a classroom similar to the classroom in which they teach’ (Brophy, 2007, p.289).

However, a different and earlier study found, ‘[v]ideo clips from classroom settings that are too familiar may be less effective in changing teachers’ thinking’ (Clarke and Hollingsworth, 2000, p.40). I take these conflicting findings to lend further support to the notion that more research is needed about using video for teacher development or learning. They also indicate that perhaps the choice of video is not as important as the use made of it. Despite such doubts about ways of using video, Santagata and Guarino (2011) suggest that at the current time, ‘[v]ideo is commonly used in teacher preparation programmes’ (p.133), which perhaps lends even more weight to the suggestion that further research is needed into what constitutes effective practice and the role of the discussion facilitator.

A COMMON PROBLEM OF EVALUATIVE TALK AND THE OPEN UNIVERSITY SOLUTION

In the conclusion to the edited book referred to above (Brophy, 2007), the author states an essential practice is, ‘establishing norms to ensure discussion is reflective and constructively critical’ (p.298), something echoed in van Es and Sherin (2002). However, establishing such norms can be problematic. van Es and Sherin (2008) report on the workings of a ‘video club’ which included seven teachers who met for ten meetings in 2001–2, at each meeting watching and discussing video clips (taken by the researchers on the project) of each other teaching. van Es and Sherin (2008, p.264) report that some members of the club evaluated or questioned the approaches they saw on video clips. These members would offer advice of what should have been done differently and, in general, draw more from their own experiences as a basis for comment than on any close look at the events on the video.

For two teachers in particular, it was not until a tenth and last meeting of the video club that there was an observable shift in the way they discussed the clips. van Es and Sherin’s description of the evaluative comments of some teachers about video is mirrored in Nemirovsky, Dimattia, Ribeiro, and Lara-Meloy (2005). Drawing on an empirical study of teacher talk linked to a multimedia version of a videotaped classroom episode, Nemirovsky et al distinguish ‘Grounded Narrative whose aim is to articulate descriptions of classroom events’ from ‘Evaluative Discourse’ which ‘centers (sic) on the values, virtues and commitments in play’ (2005, p.365). They conclude, ‘Evaluative Discourse is in our experience, by far, the most prevalent mode used in conversation about videotaped teaching episodes’ (Nemirovsky et al, 2005, p.388).

Strikingly similar findings were reported in Jaworski (1990), many years earlier, drawing on her work at the Open University (OU). Jaworski (1990) cites the following statements as common reactions to watching video, and ones that constitute barriers to valuable discussion (and which I read as ‘evaluative discourse’):

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‘He was railroading them — they didn’t have a chance to think for themselves.’

‘I couldn’t do that with my pupils, they can’t work quietly enough;’ ‘they’re not intelligent enough;’ ‘they don’t wear uniform;’ ‘we can’t arrange the classroom like that;’ etc.

‘I could never do that — I just don’t have the right sort of personality.’ (p.63)

As Jaworski went on to analyse, the problem with such comments as the ones above is that they are interpretations or judgments that (as van Es and Sherin observed in their video club) seem to be based more on viewers’ own experiences than the events on the video. Jaworski, importantly, noted that if such comments were made by the first speaker in discussion of a clip, then those comments can influence the rest of the discussion and result in little of value taking place for the participants.

A particular practice was developed at the OU in order to overcome the difficulty of such evaluative talk. This practice was rooted in Mason’s (1996, 2002) distinction between ‘accounts of’ and ‘accounts for’ data. Mason himself (personal communication) drew inspiration for this method from the pioneering work with video and imagery for the learning of mathematics of Gattegno (1965) and Dick Tahta (1980). Accounts of phenomena aim to report on them as directly as possible, avoiding interpretations, judgments or evaluations. Accounts for phenomena aim to explain what is perceived or interpret it, for example by classifying. I interpret Nemirovsky et al’s ‘grounded narrative’ as essentially accounts of, and their ‘evaluative discourse’ as one type of account for. Jaworski (1990) reports on the practice of using video with teachers in a two-part process, in order to avoid judgmental and unhelpful comments from teachers, particularly at the start of discussion. I quote in full from her description of this practice, as it has been so influential on my own work with video.

… the first step, on switching off the machine is to invite everyone to spend a minute or more silently replaying what they have seen, trying to reconstruct for themselves the most significant parts of it. Participants are then asked to join together in pairs, and try to agree on what they have seen, if possible without overtly entering into interpretation at this stage. This might be described as giving an account of what was seen. It is often surprising to the members of a group that what they notice in a video excerpt varies very considerably from one to another — how what is significant for one might go unnoticed by another — and this draws attention to what emphasis they put on what they see … If there has been disagreement about what was seen, then it may be appropriate to replay the excerpt … Discussion can then move into interpretation, and it is now more possible to back up any interpretation which is made by reference to what happened in the video excerpt. This stage might be called accounting for what was seen — trying out possible meanings and explanations. People are less likely now to jump in with unjustified interpretations, and at this stage it is likely that personal feelings about the teacher viewed have been deflected.

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Experience shows that extremely profitable discussion can result, that issues are raised which are important to the participants, and that the constructive atmosphere can lead to genuine consideration of classroom consequences. (p.63–4)

In the first part, then, the facilitator begins by asking for accounts of what was observed in an attempt to reconstruct the precise words or actions and their chronology, before moving to accounting for what was seen.

Although the OU videotapes generally had a focus (e.g., ‘practical work’), Jaworski is explicit that discussion of the tapes was not limited to this focus: ‘rarely would any particular issue of concern be the only possible focus of any excerpt on video-tape’ (1990, p.62).

In the creation of the tapes themselves, Jaworski wrote of her ‘desire to constrain the viewer as little as possible’ (ibid). In other words, although the videos had been created with some ideas in mind there was an open agenda in terms of where teachers took their discussions and what they noticed. Jaworski also recommends using short sections of lessons for observation.

OTHER MODELS OF USING VIDEO

Several packages of multimedia and interactive resources (involving video recordings of lessons) for mathematics teachers have been developed since the 1990s. Studies on the effect of such resources seem to all report positive results in terms of teacher learning (Lampert and Ball, 1990; Bitter and Hatfield, 1994; Sullivan and Mousley, 1996; Herrington, Herrington, Sparrow and Oliver, 1998; Kim, Sharp and Thompson, 1998; Goldman, 2001; Skiera and Stirling, 2004 and Stirling, Williams and Padgett, 2004). However, like Sherin (2007), who found a much more mixed picture from reviewing literature on video use, I am interested in findings about the specifics of ways of working using video, and none of these studies report on the detail of discussions amongst teachers and role of the discussion facilitator, for example, in some cases the teachers involved made private use of the materials.

I am aware of three alternative frameworks that have been articulated, since Jaworski (1990), for working directly with teachers on video, within mathematics education:

– the work of Santagata and others (e.g., Santagata and Angelici, 2010) in articulating a ‘Lesson Analysis Framework’ (LAF)

– the ‘Learning to Notice Framework’ (LNF) of van Es and Sherin (2002, 2008, 2010)

– the work of Maher and colleagues (e.g., Maher, Landis and Palius, 2010; Maher, 2008) in using ‘videos as tools’ (VAT) for professional development.

In the next sections, I describe these frameworks and compare and contrast them with the OU model.

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Lesson Analysis Framework

In Santagata and Guarino’s (2011) description of the LAF, pre-service teachers are given a series of questions to support their analysis of video clips of lessons. The aim is to encourage the pre-service teachers to use similar analytical skills in the analysis of their own teaching, and hence be able to develop as teachers. The question prompts used are:

What are the main ideas that students are supposed to understand through this lesson?

Did the students make progress toward the learning goals? What evidence do we have that the students made progress? What evidence do we have that students did not make progress? What evidence are we missing?

Which instructional strategies supported students’ progress toward the learning goals and which did not?

What alternative strategies could the teacher use? How do you expect these strategies to impact on students’ progress toward the lesson learning goals? If any evidence of student learning was missing, how could the teacher collect such evidence? (Santagata and Guarino, 2011, p.133)

The approach is markedly different from the one described in Jaworski (1990) and, in Mason’s language, begins with ‘accounts for’ what is seen on the video clips, e.g., judgments about student understanding and progress. Given the findings of Nemirovsky et al (2005), I might expect this model to lead to an evaluative discourse among the teachers. When Santagata and Guarino describe in more detail the workings of their model there are some (what I see as) vital pre-viewing activities. The teachers will work on the same mathematics as the students do in the video, and the facilitators use some of the same teaching strategies used in the video, e.g., getting ‘students’ to come to the board to explain a method of solution. From an enactivist standpoint, such activities are likely to make the teachers far more attuned to notice what occurs on the video, and more likely to be able to see detail that might otherwise have been missed.

Santagata and Guarino (2011) hypothesise a set of pre-requisites teachers need in order to be able to ‘analyse lessons effectively’ (p.136), by which they mean in order to be able use the LAF. These pre-requisites include: appreciation for disciplined analysis of teaching; appreciation for student thinking and ideas; knowledge of strategies that make student thinking visible; evidence-based reasoning about the effectiveness of teaching (Santagata and Guarino, 2011, p.136).

This is again in direct contrast to Jaworski (1990) – anyone can engage in the first part of the OU model, which is to try and agree the fine detail of what was said and done. I link the need for this long list of pre-requisites of the LAF to the fact that the LAF begins at a sophisticated level of analysis. Whereas anyone can begin a conversation about what they heard and saw, it seems to be asking a lot

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of participants to begin by analysing the goals of a lesson. Furthermore, from an enactivist standpoint, I can never know, for example, the teacher’s goals of a lesson. I cannot get inside the teacher’s head. Indeed, Jaworski explicitly states, ‘[a] rule of watching video excerpts has to be that you know only what you see’ (1990, p.64).

Even if a teacher on a video clip states to the students the goal of a task or lesson, it is not clear that we can take that as the goal. It seems highly likely that there will be other, un-verbalised, goals in play for any teacher than those communicated to students, e.g., linked to specific process skills to be developed or, say, linked to a particular mode of classroom organisation to address behaviour management issues.

Learning to Notice Framework

An alternative way of working on video is articulated in van Es and Sherin (2008), where the authors summarise the LNF, which grew out of their research within video clubs. In the LNF, the skill of ‘noticing for teaching’ consists of three main aspects, ‘(a) identifying what is important in a teaching situation; (b) using what one knows about the context to reason about a situation; and (c) making connections between specific events and broader principles of teaching and learning’ (van Es and Sherin, 2008, p.245).

Although Mason’s paradigm of noticing, which is linked to the ideas of accounts of/for, is not referenced in this paper, his influence on the LNF is acknowledged in other descriptions (e.g., van Es and Sherin, 2010). The first dimension of the LNF is the, ‘ability to hone in on what is important in a very complex situation’ (p.245). In van Es and Sherin (2010), this aspect of noticing is linked to Mason’s (2002) notion of ‘marking’.

The idea behind wanting to develop the second aspect, of reasoning about a situation, seems to be that this is what experienced teachers do (van Es and Sherin, 2008. I interpret van Es and Sherin as believing that if more novice teachers are trained in some of the skills of experienced teachers, then the novice teachers may, more quickly than they might otherwise, become expert in their classrooms. Varela (1999, p.32) writes about how beginners can learn in the same way as experts, something that Laurinda and I wrote about in relation to learning to teach mathematics (Brown and Coles, 2011). The third aspect of the framework also draws on observations of the practice of experienced teachers, and their capacity to make connections between specific events and broader principles.

In explaining the use they make of the LNF, van Es and Sherin state, in words that link to Jaworski (1990):

While teaching certainly involves making judgments about what went well or poorly in a lesson, we believe it is critical for teachers to first notice what is significant in a classroom interaction, then interpret that event, and then use those interpretations to inform their pedagogical decisions. (van Es and Sherin, 2008, p.247)

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There is a move here, similar to that described in the OU method, from noticing to interpreting. However, I read a difference between this statement and the OU practice. In using the phrase ‘what is significant’, van Es and Sherin appear to have a pre-existing conception of what counts as important. This weighting of significance is also indicated by the fact that for the facilitators of video clubs their, ‘goal was to help teachers learn to notice and interpret students’ mathematical thinking’ (van Es and Sherin, 2008, p.247). So, whereas the task at the OU for the facilitator was to work with whatever teachers noticed, for van Es and Sherin there are particular aspects of the video (students’ mathematical thinking) to which they want teachers to attend and hence which they (pre-) judge to be particularly significant. van Es and Sherin comment on the fact that some of the teachers in their study had different reasons (than exploring students’ mathematical thinking) for wanting to join the video club (2008, p.249). They report on dimensions of the role of the facilitator and suggest, ‘facilitation methods … likely had a strong influence on teachers coming to focus their comments on interpreting students’ mathematical thinking’ (2008, p.263). Again, what is referred to here is the facilitator moving discussion towards a specific and pre-determined focus.

Star and Strickland (2008) used part of the LNF in a study on the use of video with pre-service teachers (which again reports positive gains in the ability to notice). They choose to focus solely on the first component of the LNF (identifying what is important about a classroom situation) since they see this as the most crucial aspect of noticing. They appear to have a different stance, to van Es and Sherin, on where the focus of the teachers’ attention should be and are explicit that they do not want to prescribe what aspects of a classroom their teachers should focus on, acknowledging that what we notice depends on what we are looking for. Star and Strickland had no equivalent to van Es and Sherin’s ‘exploring students’ mathematical thinking’ that they wanted to privilege.

In their conclusion, Star and Strickland appear to arrive, independently, at something that reads almost identically to the OU practice described by Jaworksi:

An effective task that acted as a catalyst for growth in the present study, particularly in terms of classroom environment observations, was to watch a video once, followed by some sort of ‘‘what did you notice’’ questionnaire, followed by an assignment to re-watch the same video. (2008, p.124)

However, Star and Strickland comment that they did not collect data that might allow them to interrogate, in detail, how what they did in running discussions supported teachers in learning to notice.

Videos as Tools for Professional Development

The model of using ‘videos as tools’ (VAT) for professional development has come out of almost a quarter century of research and video recordings of lessons at Rutgers

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University in New Jersey (USA). In this approach, on a year long professional development programme with teachers:

There are three intervention cycles that occur over several months, and each cycle has four components: (1) teachers doing mathematics, (2) teachers studying videos of children doing mathematics, (3) teachers implementing in their classrooms, and (4) teachers analysing their students’ work. (Maher et al, 2010, p.4)

The focus of the work is explicitly on improving mathematical reasoning, firstly of the teachers on the courses, and then of their students. There is a similarity in this model to the LAF, of a pre-watching activity of working on the same mathematics as the students are engaged in on the video recording. My specific interest in is component (2) of the VAT model, what happens when using the video with teachers. In describing this aspect of the model, Maher et al (2010) again stress the focus of their course, which is on the development of mathematical reasoning, e.g., ‘[e]mphasis is placed on analysing the forms of reasoning exhibited by the students in the videos’ (p.5). When describing an example of the use of the model, Maher et al (2010) make little mention of the role of the facilitator and little detail is offered. For example:

During the workshop, the facilitator initiates a discussion about the third graders’ problem solving and that of the teachers. The teachers are thus enabled to point to comparisons between heuristics and strategies that have been used by both the teachers and the children’ (p.6)

The focus of Maher et al (2010) is on the effects of the VAT model, so little space is given to the role of the facilitator.

DISCUSSION

In considering the four frameworks for using video with teachers, there appears to be a strong connection between the OU practice and the LNF. Both models cite Mason (e.g., 2002) as an influence and, particularly in the version of LNF used by Star and Strickland (2008), both seem designed to avoid the evaluative discourse that is widely reported as a common way teachers will talk about video. In contrast, the LAF appears to prioritise aspects of practice (their focus questions) that the other literature suggests are not essential. Successful outcomes from using LAF have been reported (Santagata and Guarino, 2011). From thinking about the OU method and LNF, I conjecture that it may have been some of the pre-watching tasks that were crucial to teachers’ effective discussions, rather than the LAF focus questions. The pre-viewing tasks were at least a vital aspect to being able to begin with the question prompts straight after viewing a video. In other words, I suspect that Santagata and Guarino may be privileging the role of their questions unduly, and downplaying the role of the context within which these questions were asked. The tasks before the

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watching of video on the VAT model are seen as essential, as are the post-discussion activities where teachers try out activities in their classrooms. A commonality across the VAT model and the LNF, as used by van Es and Sherin, is the explicit focus on one aspect of the video recordings (mathematical thinking for van Es and Sherin, mathematical reasoning for Maher, Landis and Palius).

The more general issue I want to draw out from these research reports is what they have to say about how to facilitate discussion that provokes, or at least does not hinder, teacher learning. Again, there were differences across the four frameworks in the attention paid to the role of the facilitator in using video. The LAF description in Santagata and Guarino (2011) and VAT model in Maher et al, (2010) largely ignore the role of facilitator, or else explain it in unproblematic terms. van Es and Sherin (2008) acknowledge the importance of the facilitating role, and see it in terms of guiding discussion towards pre-set aims. Star and Strickland (2008) also acknowledge the importance of the facilitator role but did not collect data to allow them to report in detail on this aspect of the LNF.

In a different study, also linked to the use of technology in teacher learning, Goldman (2001) acknowledges the complexity involved in facilitating productive discussion (online or face-to-face). Goldman cites three studies (Franke, Carpenter, Levi and Fennema, 1998; Rosebery and Warren, 1998; Wilson, Lubienski and Mattson, 1996) supporting the idea that to make discussion productive in terms of learning, teachers need to talk in ways that are most likely unfamiliar, in particular the need to, ‘articulate their knowledge of content and pedagogy in greater detail’ (Goldman, 2001, p.36). There is a strong echo here of Jaworski (1990) and the LNF, and their practice of starting discussion with ‘accounts of’ video clips. The work of Goldman helps me articulate further my sense of unease at any work where it can read as though the role of the facilitator were transparent. Yet, as Goldman argues:

For facilitators, it is no easy task to decide what to say when, and how much information to provide … Knowing when to insert a question, reframe the issue, redirect the conversation toward more productive conjectures, or offer the “accepted” disciplinary view requires that facilitators have sensitivity, patience, faith in the group-learning process, and deep content knowledge. Furthermore, facilitators can model the inquiry process by sharing their own puzzlements and points of confusion with the group. This helps establish a collaborative learning community by softening the often held “infallible expert” view of facilitators and other professional development staff. (2001, p.37)

From my own experience of facilitating groups of teachers, I recognise the complexities alluded to here (e.g., what to say and when?) and also the need for faith in the group-learning process. These are issues not addressed explicitly in the studies reviewed in this chapter. Yet, as Goldman (2001) notes:

The avenues by which facilitators can acquire this knowledge are by no means clear. This is a serious issue of capacity building … there are scant

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opportunities to develop content and pedagogical content knowledge for purposes of mentoring adults. (p.36)

Of course, in some sense, to learn about facilitating discussion, you have to be in the process of facilitating discussions; I am interested, however, in whether it is possible to articulate anything useful about this role. As Varela (1999, p.32) challenges us, can beginners act like experts from the start? In the next section I turn to this wider issue of ways of using discussion to support teacher learning.

USING DISCUSSION TO PROVOKE LEARNING – PURPOSES

Rather than review a range of approaches, in what follows, I have chosen to articulate in detail the work of Laurinda Brown and her concept of ‘purposes’. The reason I have taken this one body of work is partly that it is the model that has strongly influenced my practice in working with teachers, but also that (as stated by Goldman, above) there are few other articulations of ways of using discussion.

Over a series of papers and book chapters (Brown and Dobson, 1996; Brown (with Coles), 1997; Brown and Coles, 2000; Brown, 2004; Brown 2005) Laurinda articulated her emerging practice as a teacher educator. An initial awareness was that, in discussions with her students who were pre-service (beginning) teachers, talk at a philosophical level (for example about the purposes of education) did not support the development of practical classroom skills. However, discussion at too detailed a level (for example in the form of ‘tips for teachers’) while potentially helpful for a particular lesson, did not support longer-term development.

What did seem to be effective was guiding discussion not in terms of content, but in terms of form, or structure, to a place between the abstract and the detail. Laurinda used ‘the word ‘purposes’ as an emergent description of the sorts of guiding principles that student teachers found energising when learning from their own experience’ (Brown, 2005, p.1). One practice she developed, working in groups of up to sixteen pre-service teachers, was to begin discussion by getting someone to give a ‘brief-but-vivid’ (Mason, 1996, pp.25–6) account of an incident in a lesson. A brief-but-vivid account is a ‘description which enables re-entry into the moment: to be, as it were, ‘back in the situation’ at some later time’ (ibid). As in the literature on using video, this is not necessarily easy for the participants. As humans we are not always used to describing the detail of events and it is easy to slip into generalisations or interpretations. Laurinda might have to work hard with a teacher (for example getting them to slow down and go moment by moment) to get them articulating the detail of their experience. The invitation is then for another teacher to offer an incident that is sparked off by the initial story (because of a similarity or difference). This process continues until some common themes emerge, at which point Laurinda moves discussion to the identification of these themes. Examples of themes that re-occur are: “how do I know what students know?”, or “dealing with

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emotional reactions”. These are labels that are in between the abstract and the detail and are examples of ‘purposes’. The prospective teachers can then discuss a range of possible strategies linked to these purposes.

As I began collaborating with Laurinda in 1995, the first ‘purpose’ that emerged from our discussions was ‘using silence’, as I explain in the story below, taken from Brown and Coles (2008), written in my voice.

Story 4: Finding a Purpose

Reflecting back on my first year of teaching had produced a feeling of inadequacy akin to despair. No lesson really seemed to match up to my ideal image of what seemed possible and there was a strong sense of a gap between where my philosophy lay and the day-to-day practice of what was actually happening in the classroom.

I remember reading articles in Mathematics Teaching [the journal of the UK Association of Teachers of Mathematics] in my first and second years of teaching and becoming frustrated. They were often about what sounded like wonderful lessons, where the students were being creative and investigative – but I wanted to know how I could get my students to even listen to each other, let alone be creative, and I did not find I got much access to what the teachers were doing or thinking.

When I met Laurinda towards the end of my first year of teaching, travelling in a car, with my attention partly taken up by driving, she asked whether I could bring to mind particular moments or times during a part or parts of lessons which had felt closest to my ideal.

This provoked two ‘brief-but-vivid’ (Mason, 1996, pp.25–6) accounts:

Account 1: During an A-Level lesson (17–18 year old students) on partial fractions I was going through an example on the board, trying to prompt suggestions for what I should write. Some discussion ensued amongst the students, which ended in disagreement about what the next line should be. I said I would not write anything until there was a unanimous opinion. This started further talk and a resolution amongst themselves of the disagreement. I then continued with the rule of waiting for agreement before writing the next line on the board.

Account 2: Doing significant figures with a year 9 class (13–14 year old students) I wrote up a list of numbers and got the class to round them to the nearest hundred or tenth, ... Keeping silent, I wrote, next to their answers, how many significant figures they had used in their rounding. Different explanations for what I was doing were quickly formed and a discussion followed about what significant figures were.

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Whilst driving and without any prompting I said, with energy: ‘It’s silence, isn’t it? It’s silence.’ This process felt like somehow staying with the detail of the accounts and seeing the pattern that was there.

We had found an initial focus for our work together. We jointly planned lessons that would begin with my own silence. I had something in mind (i.e., my own deliberate silence), which could inform my planning and actions. There was a markedly more engaged reaction from some of my classes. For almost the first time, I had a real conviction about my actions in a classroom, linked to my ideals about what I thought was important in teaching. (Brown and Coles, 2008, p.28)

This story illustrates key elements of my finding the ‘purpose’ of ‘using silence’. I had identified that a common feature of the most positive moments from my first year of teaching had been my own silence (which seemed to force students to talk in a different way to usual). Articulating this observation, using a succinct label (‘It’s silence’), supported my keeping this idea in mind, e.g., in planning for lessons. In other words, I began experimenting with a more deliberate use of silence (e.g., as a lesson start), and began accruing strategies I could use in other contexts linked to silence. I have worked on many different ‘purposes’ since then (e.g., exploring different ‘ways of sharing responses’ from the students) but ‘silence’ is still an important idea in my teaching. It is as though, having worked consciously on strategies linked to silence, these are now available to me as a teacher and I will use them as needed, without having to put attention into bringing them to mind beforehand.

Given my own experiences of how identifying ‘purposes’ has supported my own learning, it is perhaps unsurprising that I see my work with teachers on video in essentially the same manner. In other words, my own conscious intention, in working with teachers on watching video, is to support the identification of succinct labels for common issues that are being discussed, i.e., to identify ‘purposes’ arising from the discussion. In chapter 1, I introduced the word ‘metacomment’ for a verbal metacommunication. Laurinda and I use the term ‘metacomment’, taken from Bateson (1972), for any comment made in a group that identifies and labels a purpose. Such comments (an example might be, ‘what several people in the group seem to be talking about is ‘how do we know what the students know’’) are metacomments because they are about what has just been said, identifying a statement or series of statements as being of a certain kind.

THOUGHTS ON PRACTICE

In arriving at the conceptualisation of ‘purpose’, Laurinda distinguished purposes from philosophical ideas (which are more abstract) and specific behaviours (more detailed). Purposes therefore lie in the middle layer of a three-part structure. Enactivism carries a commitment to a view of categories that fits well with this idea of a three-part distinction. I now believe there are some theoretical reasons why both Laurinda’s practice with pre-service teachers and the OU method of using

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videos are likely to be effective (i.e., provoke teachers into making new distinctions in a classroom, rather than engage in evaluative judgments). One key point is that, reflecting on the detail of a lesson or reconstructing the events in a video clip (to give an account of what is observed) is something every teacher can engage in and have something to say about, in a way that beginning with interpretations or analysis of events may not. An aspect of starting with accounts of, therefore, may be avoidance of the negativity Jaworski (1990, p.63) identifies. However, I believe there are more fundamental forces at play that I make sense of via the work of Rosch on categories and concepts (e.g., see Rosch, 1999).

Rosch was co-author of one of the foundational enactivist texts (Varela, Thompson and Rosch, 1991). As alluded to in Chapter 2, enactivist thinking places significance on the role of categorising in cognition, and the enactive view of categories is largely derived from Rosch. Rosch (1999) identifies three separate layers in our use of categories which range from the most detailed to the most abstract, which I summarise in table 4.1 (a similar table was used in Brown and Coles, 2000, p.170; with acknowledgment to David Reid for the animal examples).

Table 4.1. Summary of Rosch’s (1999) three layers of categories

Layer of abstraction

Example Example in the context of working on video with teachers

Detail/behaviour layer

my pet ‘Chino’

An account of; a detailed description of an observation e.g., ‘I think Teacher A said ... then a student said …’.

Basic-level category

dog An easily stated label, linked to action. This could be a familiar category linked to familiar ways of acting (e.g., ‘the students were disengaged’), or potentially a new label/issue, arising from detailed observation (e.g., ‘using silence’).

Superordinate category

animal A ‘philosophical’ concept or interpretation, not directly observable e.g., ‘The teacher seems to respect the students’ autonomy’. Discussion at this level is unlikely to impinge on practice.

Words at the basic level are always linked to action. Indeed, one way Rosch identifies basic-level categories is that these are the most abstract words that yet define similar sets of actions in relation to them. “Pen” is something that provokes in me similar actions, i.e., it is a writing-with-object. The more abstract “stationary” does not link directly with the act of writing. The more detailed “this roller-ball in my hand” provokes similar actions in me, but “pen” is a more abstract description and hence “pen” is the basic-level category.

What I believe Jaworski (1990, p.63), Nemirovsky et al (2005, p.388) and van Es and Sherin (2008, p.264) may be reporting, in the difficulties of working with teachers on video, is that by default as humans we will pitch our discourse at this

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basic-level category layer. These are the easiest words to use, but if we begin at this layer the words we use are strongly linked to ways of seeing and actions we already perform in the classroom, and the possibilities for learning are limited. Examples of basic-level comments about a lesson might be: ‘the students seemed lively’; or ‘there was a lot of confusion’. Labels such as ‘lively’ and ‘confusion’ will be linked, for experienced teachers, to well established patterns of behaviour in a classroom (e.g., perhaps to try and reduce these qualities in their students!). By interpreting events at this level of generality, there is little scope for noticing anything that may allow an extension to such established patterns of acting. This is exactly the kind of behaviour reported by Jaworski (1990), when teachers ‘invest all of their energy in interpretation and judgment of the acts and intentions of the particular teacher of the video’ (1990, p.63).

What I am suggesting is that such ‘interpretations’ or ‘judgments’ are at Rosch’s basic level; the descriptions are ‘easy’ and general, invoking concepts such as, for example, railroading, intelligence or personality. These concepts cannot be observed directly on video, they are interpretations that perhaps tell us more about the speaker than anything they observed.

The discipline of beginning with accounts of forces discourse into the detail layer (whether reflecting on a lesson, or working with a video clip) as does supporting pre-service teachers to articulate the detailed descriptions of specific incidents. From these descriptions, generalisations at the basic level have the possibility of identifying new labels, concepts or categories that can act as purposes. There is the possibility that these purposes can become associated with new actions in the classroom. It is the arising of possibilities for new or different ways of acting that I take as the mark of learning.

An implication of this view is that one aspect of running discussion is the need to be sensitive to when talk is at a detail layer and when it slips into more basic-level categories. At the start of a discussion there may be a need to cut short basic-level descriptions or to question what was actually observed. Then, later in the discussion, when the idea is to identify purposes, there is a need to try and shift talk into the basic-level. If discussion remains at the level of details, teachers may have new strategies to try out in the same or similar contexts to the incidents that have been shared, but perhaps not beyond. The identification of purposes allows strategies or behaviours to be discussed that are relevant to a range of contexts. A purpose such as “how do I know what students know?” is potentially relevant to every lesson and strategies linked to this purpose are therefore likely to be used often.

METACOMMENTARY

In my own work with teachers on video, I consciously adopt aspects of the OU method and the idea of trying to identify purposes. A discipline I impose is beginning with accounts of any clip before moving to any more general or interpretive discussion. I am interested, however, in exploring in more detail my role as discussion facilitator

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and how teachers talk. Is there evidence of the identification of purposes? or metacommunication? What occasions such moments?

My analytic focus is on talk. I cannot know what is going on inside someone else’s head, however the enactivist linking of knowing and doing means that, through studying talk, I can get access to what a person is doing and hence to their knowing (in that moment). Enactivism impels an approach to studying interaction that is in the detail of what is said or done, bringing a minimum of conceptual ‘baggage’ to analysis.

In writing this chapter, I am aware of links between my commitment to staying with the detail of events, my use of story in this book and the quotation at the start of this chapter, ‘[t]rouble is the engine of narrative and the justification for going public with a story. It is the whiff of trouble that leads us to search out the relevant or responsible constituents in the narrative’ (Bruner, 1996, p.99). Narrative (or a Bateson story) is needed in response to something unresolved – ‘trouble’, or in Bateson’s language, ‘difference’. In working on video with teachers, I am aware of being alert to differences in the detail of what teachers have heard or seen. Difference can provoke a shared need to search for an account of the ‘relevant or responsible constituents’ (which, in working on video, will be what was said or done on the recording). Similarly, in a classroom, I am aware of wanting to set up contexts that are likely to generate differences in student responses and hopefully a motivation to resolve them.

There are no stories in the next four chapters of this book. The first person voice has felt less appropriate when the focus is so firmly on viewing the data in a principled manner, consistent with my writing to this point. To some extent, I have already set out how my own perspective will play a part in what I see, the next four chapters are focused on the results of this seeing. Instead, it is the chapters themselves that are in some sense (Bruner) stories, containing my (partial) resolution of the predicament of making sense of the data.

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CHAPTER 5

ON TEACHER LEARNING

I get more out of it when I watch a video with all of you (Teacher R)

PATTERNS IN TEACHER DISCUSSION

My work as head of the mathematics department ended in July 2009. In one of the last meetings I ran, I asked mathematics staff to reflect on what had been most useful in terms of their own development, across all meetings, of which watching video made up a small proportion. The following dialogue ensued, without my prompting, with nods of agreement from others.

TR: yeah I’d like to see more videos of people teaching (.) it’s different to when you’re sat in a classroom just observing someone by yourself I get more out of it when I watch a video with all of you

TA: the other people and focusing on a small part/TR: /yeah/ TA: rather than watching a whole lesson where you/TR: /you forget a lot of it/TA: don’t get very muchTR: then you get involved in the lesson

because the kids have started working you start working and you forget about why you’re there (.) you rarely sit and watch a lesson

These two teachers seem to be concurring that an observation of a small clip of a video recording, with others, is more useful to them than sitting in on a whole lesson observation of another teacher. These are striking comments, given the mixed findings of previous research on video use reported in the last chapter. Further evidence for the effectiveness of our use of video in terms of avoiding evaluative discourse and identifying purposes will emerge as I analyse the discussion data.

Having transcribed three teacher meetings, I became immediately aware of some strong similarities. The three recordings appeared more like three instantiations of a similar practice. So, despite the enactivist idea of starting with the final piece of data, it made more sense to consider all three together, beginning my analysis by splitting the three transcripts into episodes and looking at similarities and differences. Looking at the transcripts with the idea of characterising sections, there were broadly ten

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different types of episode (see Appendix 2). From looking across these patterns and considering the literature reviewed in Chapter 4, I have analysed five aspects of the role of the discussion facilitator, which I present in turn.

1. Selecting a Video Clip

Teachers in the department, who had lessons recorded, were always volunteers. The member of staff would select a class and a time. Before the lesson I would set the camera up on a tripod at the back of the room, set it to record and then leave, returning at the end of the lesson to retrieve the equipment. Students and parents were told in advance of these recordings and the students given the option to sit out of shot, not be involved, or have any other concerns discussed. Having taken the video recording, I would transfer it to DVD and watch it, looking for any sections that might be suitable for discussion with other staff. I was looking for small sections of the recording where something ‘interesting’ (as I thought about it at the time) was occurring. The department had agreed on a shared focus for development around the running of class discussions and ‘managing pupil talk’. It was therefore only the times of whole-class discussion that I was selecting from. Indeed the nature of the recordings, and the way the camera was set up, meant the sound was only clear during these periods. By an ‘interesting’ section of video, I meant a time when, for example, students were responding to each other in a whole-class discussion (i.e., the dialogue fell out of a pattern of Teacher-Student-Teacher), or when there was some ambiguity that was discussed as a class (e.g., two or more students expressing conflicting ideas), or if there was a section where several different students were contributing ideas to discussion. I set myself a rule to choose one continuous section from the lesson that did not last more than five minutes, and, in general, I aimed to make the clip around three minutes long, in line with Jaworski (1990).

2. Starting with Reconstruction and Moving to Interpretation

In all three discussions, there is a general move from reconstructing the video clip at the start of the discussion, to identifying teaching strategies towards the end. This is precisely in line with the practice described in Jaworski (1990). Having had a period of time with ‘accounts of’ what was watched, it is possible to move to ‘accounts for’ and avoid evaluative or judgmental comments.

Having shown the video, the first thing that happens on all three recordings is that I make a comment inviting others to ‘reconstruct’ what happened.

AC: so would anyone like to suggest or remind us where it began what the sequence of things that happened there and we can maybe go back and look at things again if we needsee if we can reconstruct as much as possible TD/TB/4-4-08

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This first phase of discussion lasted for 20, 11 and 17 minutes in the three discussions of video recordings. Talk was focused on the detail of what people saw in the video and there was a marked absence of evaluative discourse. Examples of the type of thing said in this phase are as follows.

TB: right and then S2 said so you could take the whole area and just take off the two little bits the one at the top left hand corner and the one at the bottom right hand corner TD/TB/4-4-08

TB here is sharing what she heard a student say on the video clip.

TA: and then he said yeah it goes uphe started trying to describe a pattern [ ]it was a girl who said it was like such and such’s one that went three three TD/AC/19-5-09

TA describes, above, a sequence of two students talking.

TA: that was the second one I’m not sure what the first one was what was the first one? TD/TA/12-11-07

In this turn, TA was disagreeing with what another teacher had just said, about the order in which students on the video spoke. In all three examples, which are typical of this first phase of discussion, teachers are talking about the detail of events and sharing what they heard or asking questions. The absence of evaluative discourse is evidence for the effectiveness of starting with accounts of events, in working with video.

At the end of the first phase, there is then a move to interpreting the video clip. This was signalled by me, with a similar form of words each time, one example is below.

AC: okay fantastic so this is the bit nowwhere TB doesn’t say very much um where I guess so we’ve got [ ] clearly a [ ] okay so clearly an interesting section of dialogue and some interesting things going on there so what was TB doing what were the teaching strategies anyone observed in setting this [ ] (12) TD/TB/4-4-08

The phrase ‘teaching strategy’ was commonly used in the department to indicate the identification of a type of teacher action, i.e., something we observed in the lesson clip we had just watched, but which could be repeated in other contexts. Examples of ‘identifying teaching strategies’ are below.

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TD: … I think you said we need to sort this out um and if it was me I would have chosen one of the options for them um and that’s sorting it out for themI think in some ways you sorted it out for them as well but gave them the option of putting conditions ... TD/TA/12-11-07

In this turn of TD, he makes a connection to this own practice, ‘I would have chosen one …’. For most of this turn, however, (which continues) he grapples with a description and interpretation of what TA did, linked to the general issue of how, as teachers, we ‘sort out’ student difficulties.

Examples of identifying teaching strategies from the other two discussions are below.

TE: on a couple of occasions TB replies to students to comment on what other students have said (10) TD/TB/4-4-08

TD: I don’t (5) it feels a bit like (.) you’ve got the confidence to let them take the lesson (.) but there’s more to it than confidence andthere is subtle guidance there’s (8) they seem to be used to (6) TD/AC/19-5-09

These two examples give a sense of the range of comments that I see as ‘identifying teaching strategies’; from the relatively clear and succinct articulation by Teacher E of how TB responds to students (which is to get them ‘to comment on what other students have said’), to the turn of TD, where he seems to be trying to link ideas of confidence, who ‘takes’ the lesson, guidance and what students are used to, in relation to his observations of my teaching. On several occasions over the three discussions someone offers a label to sum up the strategy or strategies that are being talked about. These labels I interpret as examples of ‘purposes’ (see Chapter 4). These examples are from two different discussions:

TF: there was one part where (1) the boy was talking about something else (1) he was talking about (.) he was referring to the triangles [ ] he was saying if you do the same with the triangle then the area will be three [ ] but you didn’t pick up on that (.) because it wasn’t really what they were talking about at that point (.) so it didn’t interfere with anyone else’s thinking (1) but he still said it (.) he was quite happy to say it

AC: so there’s a strategy there (.) what’s the strategy there thenTA: when to stress and when to ignore (.) when to stress things and when to ignore

things TD/TB/4-4-08

TB: … it’s the image I have when Gattegno does game-like activities which are highly structured and imposedyes this is highly structured and imposed as in what they’re doing TD/AC/19-5-09

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I read Teacher F’s comment as having some similarities to when Teacher D was talking about confidence and guidance, in that there appears to be a grappling with an interpretation of what Teacher B did in the video, which is reported as having the effect that ‘it didn’t interfere with anyone else’s thinking’. I ask for what the strategy is, and Teacher A supplies the label, ‘when to stress and when to ignore’ (which is in fact language borrowed from Gattegno, e.g., 1971, p.16). In the second example above, Teacher B makes a link to some reading and offers the label that what was observed on the video was ‘highly structured and imposed’.

The progression of the three discussions, then, is consistent. Broadly speaking there is a period of reconstruction and then a transition to a period of focusing on identifying teaching strategies that leads to the articulation of a purpose (e.g., ‘when to stress and when to ignore’). There is a marked absence of evaluative talk, across all three recordings of video discussions, in the reconstruction phase.

3. Setting Up Discussion Norms

It is evident from the teacher discussion transcripts that there were some rules or norms in play about how to discuss that were more or less explicit. This section and the previous one overlap, in that starting with reconstruction and moving to interpretation is itself a discussion norm. In the last section, I gave examples of how I set up the beginning of discussions and the move to interpretation. However, it is not enough simply to state that you want discussion to focus on the detail of what was said in a video clip for people to do that. Focusing on detail is, in my experience, not always comfortable for people and can come across as an unusual request. Therefore, particularly in early sessions with a new group, there is a need, as facilitator, to ‘police’ this beginning. I am aware, as a facilitator, of wanting to cut short, in this initial phase, any comment that strays into interpretation or judgment or evaluative talk.

In the discussion below, there had been talk of a student, S2, who came to the board to show a method she had of finding the area of a shape that was being considered by the whole class. S2’s method was different to the one that another student had just demonstrated. TB makes a comment about what she imagines is going on for the class as they see these different methods on the board.

TB: … if you just show other ways then the kids are noticing I guess um the different waysthat would be what they’re attending to I guess

AC: yes but I don’t know whether S2 would have wanted to then (.) because what did she actually saydid she say [ ] TD/AC/19-5-09

TB makes a comment about what students are ‘attending to’. I seem to begin to entertain this idea and then, perhaps catching myself engaging in an issue that we

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can never know and one that is certainly not part of ‘reconstruction’, I return to ask ‘what did she actually say’ and offer one idea (that is indecipherable on the recording) that other teachers pick up on.

In this small section of talk I can be seen, without being explicit about it, to be imposing a discussion norm of focusing talk on what is observable rather than (in this instance) what a student may or may not be attending to.

Since completing this study, I have continued to work with teachers on video and I am aware, almost always, of needing to stop people from moving to interpretation during the initial phase, when the task is to work on describing the detail of what was said or done. The three discussions that I audio recorded were not the first times we had worked on video as a department and several teachers in the group would have been used to the discipline of starting with a reconstruction of events hence there was not so much need to set up this norm. What I do notice, in the three discussions, are opportunities when I think I would intervene now, but did not do at the time. For example, I would not now let the following pair of turns take place without comment.

TN: in the first one (.) S1 doesn’t mention finding out the area of the extra bit but I think she’s got in her mind this whole area and then she says you work out the area of the extra bits [several voices]

TM: she didn’t hear the words whole area (.) that’s what threw her TD/TB/4-4-08

Teacher N refers to something that a student has ‘got in her mind’ and Teacher M mentions what the student did not hear. We simply cannot know what was in S1’s mind, nor what she ‘didn’t hear’. I am surprised that, in this discussion, I did not make a comment to point out the unknowability of what was said here, and then reinforce the norm of speaking from our experience and focusing on what we can observe.

There is evidence in all three discussions of my questioning the meaning of what teachers say. The example below comes from the second half of one discussion, when we were discussing teaching strategies. I interrupt TG just after he has referred to ‘that strategy’. Words like ‘this’ or ‘it’ or ‘that’ I know are ones I am sensitive to, both in written text and verbally. It is easy to assume that the meaning of ‘these’ terms is clear when often it is ambiguous.

TG: there’s also something about the students listening to each other and I think one of the reasons we adopt that strategy is /

AC: /sorry which strategy

This example feels a familiar part of my practice in facilitating discussion. I try to be aware of my own sense making and will interrupt speakers if I feel there is an unhelpful ambiguity.

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4. Re-watching the Video

In running a session looking at video, I am aware of crucial decisions about when, and how often, to replay any clips. There is a subtle mixture of intentions. I do not want the task to be just about remembering. Equally, I do not want to get bogged down in the detail of every moment of the three minutes. I am aware of being alert to any disagreements about the reconstruction. I know I am looking for an excuse to slow down the reconstruction and disagreements about what is said can provide an opportunity to do this, often prolonging discussion, the issue finally to be resolved by re-watching(s). The purpose of the slowing down is to let the complexity of what is happening come to the surface and to allow a focus on the fine detail of a small section of the clip. I find, in a teacher meeting, it is not possible, or even desirable, to do this for every section of the clip due to the time it takes.

Across the audio data of the three meetings, re-playing of video always occurred in the first phase of discussion. I know, from running these discussions, there is a delicate balance linked to the decision to replay. On the one hand, replaying too soon takes away the chance for people to engage in the ambiguity and difficulty of hearing what is said. On the other hand, leaving it too long before replaying means engagement in the reconstruction can diminish.

Typically, re-watching occurred to sort out a specific issue or question that had arisen, and I illustrate some of my decision making in the transcript below. In the video clip we are discussing, TB had shown the class four images of different but linked shapes and students had discussed finding the areas. Student A suggested finding the area of a parallelogram by drawing a rectangle around it. TB drew over the parallelogram on the board (ending up with Figure 5.1). Student A spoke of taking off the ‘extra bits’ at the top left and bottom right.

Figure 5.1. Parallelogram and rectangle.

In the teacher discussion of this sequence, Teacher B commented there was a ‘weird thing’ on the video where she thought she had repeated what S1 had said to her, and S1 paused and responded ‘no’. Discussion below focuses on this moment in the video clip and who heard a difference in what S1 said compared to what TB repeated. This transcript captures, for me, the point of engaging in detailed reconstruction of a video clip.

1 TK: there was [ ] a very subtle difference in the words/ 2 AC: /oh wow/3 TK: /which she seems to pick up on4 AC: oh do you think that’s why she said no 5 TA: yeah

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6 TK: I think because what I understood of what TB had said was exactly what [ ] she had saidbut the words were slightly different

7 AC: oh wow (.) has anybody got that (.) anybody got that detail8 TA: because I didn’t quite understand what TB said

but I did understand what she said9 AC: because I wasn’t sure what the difference was

I was with TK I couldn’t hear the differenceI was surprised when S1 came back and said nobecause it just seemed identical

10 TK: but there was a difference in wordsshe called it the extra bits

11 TA: I think you called it the extra bits firstso didn’t she use the extra bits then

12 AC: okay lets look at that13 TK: or did you say the bits you don’t need14 TB: oh maybe that’s it15 AC: the bits16 TB: you don’t need17 AC: you said the bits you don’t need18 TK: somebody said that 19 AC: ah okay and you think she was just saying the extra bits

Having made this last comment, I make the decision to replay the video clip. At the start of this transcript, Teachers A and K report (lines 1 and 5) having heard a difference in what S1 said, compared to what Teacher B repeated. Teacher B and I appear not to have heard this distinction (e.g., line 9: ‘it just seemed identical’). TK, TB, TA and I all contribute (lines 10, 11, 13, 14, 15, 16, 17) in arriving, by the end of the clip, at a possible reconstruction that S1 talked about ‘extra bits’ and Teacher B talked about ‘the bits you don’t need’. With this conjecture in mind, we then re-watched the video clip. It is in keeping with enactivist beliefs about perception that having some question in mind, or something you are looking for (in this case two forms of words) can support seeing in more detail. One of my aims, in beginning with a reconstruction of events is exactly to get in to the kind of detailed talk above. We are in the fine detail of teacher decision-making, but engaging in the text of what occurred before coming to judgments or interpretations. There is an absence of evaluative discourse (with the possible exception of line 8) as we dwell in the detail of description. The teachers do not discuss values, nor comment on their own experiences in describing what they observe. I know, as a facilitator, I am conscious of looking out for disagreements in what people have seen or heard in this phase of discussion. Given the time constraints of meetings (at this school, a maximum forty-five minutes working on video) it is not possible to engage in this kind of detailed discussion of every part of, say, a three-minute video clip. I aim for at least one sequence that we dissect in detail. There is then a decision point of when to replay the video. I suggest a

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re-playing at line 12, but seeing the transcript I am much happier that it was delayed to line 19. The point of this phase of discussion is not about memory, so I do not want to delay the re-watching beyond the point where there is a clear articulation of two possible sequences of events. This articulation was there at line 19, but was not at line 12 (when I first suggest the replay). In my experience, difference can generate a commitment to further detailed watching of the video and in line 7, I encourage anyone to articulate a difference they have noticed. Becoming ‘carried away’ by discussion of detail seems to function as a means of making evaluative discourse impossible.

This analysis adds detail to what might be involved, as a facilitator, in engaging in the OU method of starting with accounts of video. In particular, I see part of the role as supporting the articulation of difference, leading to making a decision about when to replay a section of video. Jaworski (1990) reported beginning the first phase of discussion by getting teachers to discuss in pairs before moving to a whole group conversation. I am aware I do not do this because I want to be able to ‘police’ this first stage and, for example, interrupt if any teacher begins to engage in an evaluative discourse and re-focus them on the task of describing what they observed. In paired discussions I would be unable to do this and I am wary, because of Jaworski’s report, of how the way that discussions begin can set a pattern that is hard to shift.

5. Metacommenting

In this section, I focus on decision points from the second phase of discussion, when we work on interpreting what the teacher on the clip was doing.

The focus in this second phase is on seeing through the particular actions we have just agreed we observed, to a more general description. In all three video discussions, there is an episode in the second phase either of ‘metacommenting’ or ‘naming a strategy’. I offer three examples below.

In the clip of Teacher A’s classroom, the class had to pick a three-digit number, reverse it and subtract, take their answer, reverse that and add (see Figure 5.2).

325 501 685 685- 523 105 586 586

198 396 099 99+ 891 693 990 99

1089 1089 1089 198

Figure 5.2. The task ‘1089’.

Starting with, say, 685, there are two possible interpretations of how the task should be carried out, resulting in different final answers (1089 or 198).

During the clip we watched, a student articulated precisely this dilemma, which she had come across in her own work. Teacher A does not enforce one interpretation on the class. Instead, she makes a comment that, as mathematicians, they need to

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be clear and consistent about the rules under which they are operating and that these rules need to be articulated when, for example, describing a pattern (or in the language of this classroom, ‘conjecture’) they have noticed. This comment, being about the students’ discussion, is itself a metacomment. However, the issue in this section is metacomments made in the teacher discussion. TD refers to TA’s metacomment about rules in the transcript below. (The first part of this turn was referred to earlier in the chapter.)

TD: because there was this issue that belonged to the classand I think you said we need to sort this out (.) um (.) and if it was me I would have chosen one of the options for them (.) um (.) and that’s sorting it out for them (.) I think in some ways you sorted it out for them as well (.)but gave them the option of putting conditions (.) so you didn’t make the choice but you did sort out the issue (.)I don’t think (.) they didn’t sort out the issue (.) they were wondering what can we do with this (.) we need to sort this out as well (.)and I don’t think it was sorted out by the pupils (.) I think you sorted it out in putting conditions

AC: that’s lovely (.) and it’s a much more enabling sorting outbecause if we just sort it out by answering the issue then the next time pupils come up with this issue they have no (.) they’re in no better position to decide (.) the only resource they’ve got is to ask TA but if you sort it out by making them aware this is an issue and making them aware there are consequences for each one and whatever that is offering them a tool for next time they get in to that situationso yeah I love that TD/TA/12-11-07

I metacomment here on a description from TD about the way TA has sorted out a student dilemma (which in fact TA did by metacommenting herself on the dilemma). It is a metacomment because it is a communication about TD’s contribution, classifying the kind of sorting out he has described. The phrase ‘enabling sorting out’ has the potential to act as a purpose for the teachers in this discussion. In other words, it is the kind of label, easily stated, which can be ‘kept before the mind’ to inform a teacher’s interactions with students. Another possible purpose that could be taken from my comment is ‘offering them a tool’, again as an idea to inform interaction with students. It is this kind of label that has the potential to allow different actions in the classroom. By metacommenting, I am not prescribing what teachers in the discussion can or should work on in their own teaching. But I do see a role, for the facilitator, in raising to awareness possibilities in the form of purposes. Purposes, being at the basic-level, have the potential to become associated with a range of actions. For example, a teacher working on ‘offering them a tool’ might choose to

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make a variety of resources available around the room and direct students to them before asking for help. The fact that my metacomment above is a direct response to the offer from TD, suggests that I do not have a pre-conceived notion of what we will discuss. My attention is on getting to the point of labelling a purpose. I am not concerned about what that purpose is.

Another example of metacommenting to identify a purpose is from later on in the discussion of the video of TB (lines 1 and 2 were discussed earlier). There had been talk of how TB not pursuing one student’s suggestion had helped discussion build and develop.

1 AC: so there’s a strategy there (.) what’s the strategy there then2 TA: when to stress and when to ignore (.) when to stress things and when to

ignore things3 TR: yeah yeah it’s having [ ] knowing what your focus is4 TA: it’s not simply taking everybody’s comments further (.)

because one way of thinking about it might be that [ ] you have to take every student’s comment further to have (.) you know (.) a fair (.) whatever that means (.) discussion (.) but I don’t think that’s (1) that’s not the case here and it’s not (7) that’s helped with the richness rather than made this person feel their comment is [useless]

In this section, I do not do the naming. In line 1, I offer a (meta)comment about the discussion, that I have a sense people have been talking about a strategy, without labelling it. In line 2, TA provides the label ‘when to stress and when to ignore’, which TR confirms and states in her own words, ‘knowing what your focus is’. TA then elaborates further, there is a striking seven second pause before TA comments that what TB did ‘helped with the richness’; TB ignoring one student has helped her focus discussion and stress the mathematics she wants. Enactivist category theory suggests the act of labelling a purpose, creating a new basic-level category (in this case ‘when to stress and when to ignore’), is significant for supporting teachers in making use of ideas raised in discussion. If the new label is basic-level it will be applicable to a range of potential contexts but still linked to specific actions.

In the same discussion, TW names another strategy (that the image TB offered students already contained the answer to the question she wanted the class to work on). I comment:

AC: that’s a lovely thing to think about (.) to try and come up with some images which have the answer (.) and the students’ job is then to talk about it (.) interpret it … has anybody got any others (.) or maybe this is something we can do this afternoon (.)

I am explicit that the label has given us a potential focus for further work. There is an on-going decision, as a facilitator, of when to offer a metacomment or suggest

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a purpose. Having worked with purposes myself I am perhaps sensitised to notice when what is being articulated by a teacher potentially has more general application and so could be used as a purpose – and it is at these moments that I try to offer a metacomment or label.

In my language, a discernable pattern is a recurrence of what might be seen as (positively) evaluative comments: ‘that’s lovely’; ‘so yeah I love that’; and ‘that’s a lovely thing to think about’. These are comments I do not make in awareness and, being evaluative, they might be thought to lead to some of the problems highlighted by Nemirovsky et al (2005) or Jaworski (1990). However, I see a distinction in that these comments are not evaluating the video clip that has been observed (and hence likely to lead to further evaluative judgements of the clip) but rather they are evaluating the discussion about the video, highlighting positive aspects of the nature of the teachers’ engagement.

METACOMMENTARY

In Chapter 2, I stated that one part of my research agenda was, ‘how can I use video to support teacher learning? What patterns of interaction support teacher learning? What is the role of the discussion facilitator?’ The five elements of practice synthesised above are my first pass at addressing these questions. The elements of a way of working with video that I have set out seem to me essential if discussion of video is to move beyond the rehearsal of previously thought-out ideas. They do not guarantee productive discussion but, rather, allow the possibility of the arising of something new. I reported on a metacomment of mine where I offered a label of ‘a much more enabling sorting out’. This comment of mine was made in reference to a teacher’s description of what he had observed on a video clip of Teacher A, where he had described the way Teacher A sorted out a student difficulty without simply giving them an answer. The label ‘an enabling sorting out’ emerged from detailed accounts of what was observed and then trying to account for them. The label represents a shift in discussion from Rosch’s detail or behaviour layer, to the basic-level. It is an easily stated label and has the potential to act as a ‘purpose’ (Brown, 2005) around which teachers could accrue new and different actions in their classrooms.

In the next chapter, I look again at some of the discussion audio data and, in keeping with enactivist methodology, approach it from different perspectives. These alternative perspectives serve to trouble the five aspects of the role of the facilitator I have just offered.

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CHAPTER 6

RE-LOOKING AT TEACHER DISCUSSIONS

In any culture, the individuals acquire quite extraordinary skill in handling not only the flat identification of what sort of message a message is but in dealing in multiple identifications of what sort of a message a message is. When we meet these multiple identifications we laugh, and we make new psychological discoveries about what goes on inside ourselves, which is perhaps the reward of real humour. (Bateson, 1972, p.169)

SHARING PERSPECTIVES ON TWO TRANSCRIPTS

Within enactivist methodology, there are two key ways to gain multiple perspectives on data. The first is to work on data with different people, the second is to consciously adopt a different theoretical frame for viewing. In this project, I engaged in both methods and report, in this chapter, on how re-viewing of data on teacher discussion served to trouble, or at least complexify, aspects of the role of the facilitator that I reported in Chapter 5. I begin by reporting on the outcomes of sharing two transcripts of lessons with the department of teachers and then report on what arose from my analysis of laughter during discussions of video extracts.

In this book I report on a selection of data from a wider project in which I tracked video recordings of three teachers’ classrooms. In September 2008, at the start of the new school year, I looked at video recordings of two lessons taken in that month and was struck by the almost identical words said by both teachers. This seemed an opportunity to engage in a triangulation process by working with the department on the two transcripts to discuss similarities and differences. The transcripts are given on the next page, starting from the use of the similar phrase.

To give a little context to the transcripts, in the first one, just before line 1, a student (S4) had offered a proof of a conjecture the class had been working on for several lessons. The task was ‘7, 11, 13’ (see Appendix 1 for a fuller account, but briefly, you take a 3 digit number, multiply by 7, 11, then 13 and see what you notice about the result). This lesson took place on 12th September 2008, taught by Teacher A. The second transcript is from a lesson taught by Teacher B on 15th September 2008. Just before line 1, S1 had offered an idea. After a pause of a few seconds, another student (S2) made a guttural noise (‘huh’), which I took to indicate confusion. Teacher B had been explicit to the class in this lesson about the need for them to listen to and comment on each other’s ideas rather than to contribute only their own new points. The task was ‘1089’ (see Appendix 1 or Figure 5.2, where students choose a starting number, perform a process and get a surprisingly limited set of possible finishing numbers).

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TRANSCRIPT 1 (TA)1 TA: oh amazing (.) can anyone else explain what she’s done there (4)

can anyone else explain what she’s done there (.) S12 S1: [ ]3 TA: [will you] explain this first?4 S1: yeah (.)5 TA: okay (.) try and do that and then somebody else (.) I want you all to listen to this6 S1: what she’s done [is that she’s timesed]7 TA: go on go on in your own words8 S1: a b c is any number a b c if you times that by one thousand its going to be a b c

with three zeros on it /T: lovely yeah/ and you added another a b c9 TA: if you added another a b c and you get a b c a b c (.) S210 S2: it’s pretty simple11 TA: it is pretty simple12 S2: it’s basically one two three times one thousand and one (equals) one two three one

two three13 TA: does it matter what the one two three is (.) if it was five six seven or five eight nine14 S2: yeah (.) like you said (.) it could be any three-digit number (.)

you times it by a thousand (.)15 TA: and then times it by one16 S2: yeah and then times it by one17 TA: so (.) this is quite key isn’t it (.) the fact that it’s a thousand and one (.) we

talked in other lessons whether multiplication was commutative or not and whether it mattered what order we did it in (.) so would that be the same as doing it in another order (2) I think we convinced ourselves that multiplying was commutative, S3

18 S3: ( ) four digits ( )19 TA: right we’re going to have to be really quick about this because we haven’t got

much time to do it in your books … what I like about S4’s proof is its simplicity (2) and proofs don’t have to be complex (.) so for me I feel convinced by this (.) S4 has convinced me that when she takes any three digit numbers and timeses it by seven eleven thirteen she will get a b c a b c

20 S3: I’ve got a theorem as wellTRANSCRIPT 2 (TB)

1 TB: okay (S2) is a bit confusedcan anybody say what S1 has just said (5) S1 can you go through it again

2 S1: [if there are] all the starting numbers up there have got a nine in on the3 TB: these are finishing numbers (.) these are starting numbers4 S1: yeah (.) er finishing 5 TB: all the finishing numbers6 S3: not one eight one eight7 S1: er (.) don’t know about that one (.) when you put (.)

if you switch them round the nines’ll be together so it’ll be like nine and nine will make eighteen so you’ve got eightand then you add the one onto (.) so the nines go together

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8 TB: okay so why is that important (.) how does that help us9 S1: in case you have numbers with just one and nine and eight in10 TB: you think it’s something to do with (.) okay um (4) S411 S4: I just tried four one six three and it came up with eight nine nine eight

In analysing the two transcripts above with the mathematics teachers (including TA and TB) in the department at a meeting, a distinction was made by the group between different uses of the strategy, ‘can someone say what student x just said’. For TA, this strategy was used at a point when the teacher had made sense of a student utterance, and wanted more people in the class to engage with it. The teaching strategy in line 1 is followed by three contributions from Teacher A (lines 3, 5, 7) supporting students to articulate their ideas. The teacher’s actions slow down the discussion at the key point of a student having offered a proof of a conjecture. The invitation for other students to explain S4’s idea results in discussion dwelling in this proof (lines 6 to 16). At line 17, Teacher A offers a metacomment about multiplication (and the issue of whether it is commutative). In line 19, Teacher A then metacomments about the process of proof, and the fact that they can be simple (picking up on S2’s language in line 10). At the end of the sequence dealing with proof, another student says they have a different theorem and wants to share their proof, which is an example of the kind of mathematical behaviour Teacher A is trying to foster.

In the case of TB, the use of the strategy ‘can someone say what student x just said’ was used at a point when she had not made sense of what a student had said, and when another student had also expressed incomprehension. S1 explains again (lines 2 and 4) and, similar to TA’s transcript, TB supports him in articulating his idea (TB asks for clarification in line 3 and repeats S1’s words in line 5). At line 6, another student (S3) then does respond with a comment about S1’s idea, ‘not one eight one eight’. I take S3 to be offering a counter-example to S1’s idea; S1 had said all the finishing numbers have a 9 in them and S3 appears to be pointing out one that does not have a 9. S1 appears to recognise this as a counter-example and says (line 7), ‘er (.) don’t know about that one’. This is the first time in the lesson when the Teacher-Student-Teacher pattern is disrupted, so the strategy of asking for another’s interpretation (in line 1), despite not getting an immediate response, is effective in terms of TB’s explicit aim for the lesson, of getting students listening to and commenting on each other’s ideas. In line 8, TB asks ‘how does that help us’ directed at S1, with stress on ‘that’, which I know I heard when I first listened to the recording as indicating TB did not think it did help; my enactivist stance cautions, however, against such attributions.

The distinction that was made by the teachers in the department, when comparing these transcripts, was between different motivations in the use of the same strategy. With Teacher A, it was used at a point when she wanted to focus the whole class on a particularly important (to her) idea and with Teacher B, the strategy was used at a point when she (and at least one other student) had not understood what another student had said.

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As I described in Chapter 5, in working on video with teachers, after a period of time focused on reconstructing the detail of what was said and done, I move the discussion to identifying teaching strategies, and getting to some form of words that can potentially act as a label, or purpose, to support the development of new actions in a classroom. The analysis of the two transcripts above suggests that considering the motivation for the use of any given teaching strategy is also important. So, as well as identifying strategies which may support teachers in new or different possibilities for responding to students, discussion of the context for the use of these strategies is important. The analysis suggests a complexity to classroom talk that is not immediately observable.

In my final meeting with teachers (in July 2009), ten months after the teacher discussion in which this distinction was made, I invited staff to reflect on what, of all that we had done as a department, had supported their own learning. TB mentioned the distinction drawn above:

TB: we did one observation I think of me teaching … and it was that I was allowing something to go and it was [ ] how is that relevant to anything else (.) and from that it’s made me a bit more no I know what we want to talk about (.) … if someone says something and no one really knows what he’s talking about [ ] then it’s not helping the class even if you spent ten minutes trying to discuss what he’s saying (.) you have to be thinking oh I know what he’s talking about and it’s really important to spend the ten minutes in the class discussing

I see here some evidence that the process of analysing recordings of lessons was significant, at least for TB, and that she did take on as a ‘purpose’ in her own teaching something that arose from discussion. The purpose I interpret here is around ‘I know what we want to talk about’. TB seems to be articulating a conviction that she needs to take responsibility for the focus of class discussion, linked to what she feels would be important for others to hear. This precisely mirrors the distinction that was made by the teachers re-viewing the two transcripts. There is a seeming paradox here, in that I have been advocating an approach to running discussions with teachers where the facilitator is not in the position of ‘knowing what we will talk about’. One consequence of such a stance, as a faciltator, is that there is no difficulty, however, in a teacher taking as a purpose something seemingly in opposition to how the discussion operates. In fact, I think there is less of a paradox than the words may imply; the facilitator stance I have been advocating does specify in advance ‘how’ teachers will talk (e.g., starting with reconstruction) but leaves the content open.

ON LAUGHTER

An implication of the cyclical method of enactivist enquiry is the need to re-look at data, seeking new and ever finer distinctions and searching out multiple views. As I was transcribing the three teacher discussions, I was struck by the recurrence

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of laughter. I did not pursue this at the time, as my focus was on characterising episodes and looking for patterns linked to the role of the facilitator. Around this time, I was reading Sacks (1995), including a lecture titled, ‘On Suicide Threats Getting Laughed Off’ (pp.12–20). Sack’s approach to analysing talk often seems to shed light on what might be seen as trivial aspects of talk, and this emboldened me to look into an aspect of discussion I would usually dismiss as irrelevant to my concerns. So, in seeking multiple views of data, I decided to return to the audio- recordings of teacher discussions, transcribe episodes that contained laughter and, in keeping with enactivist ideas, look for patterns and differences in the occurrence of laughter. I was also influenced by Sack’s ideas about the role of laughter, which I sketch briefly before turning to the data.

Sacks’ analysis of laughter as a response to suicide threats makes use of the idea of ‘ceremonials’, which are standard or ritualised interactions between people generally lasting two turns in conversation. One example of a ceremonial is:

“How are you feeling?” to which you return “Fine”. If one person, then, uses a ceremonial, the other properly returns with a ceremonial (Sacks, 1995, p.14).

The return, ‘Fine’, ends the ceremonial sequence. ‘How are you feeling?’ is a phrase that can be used with relative strangers, but there is likely to be a circle of people more intimately known with whom it may be taken more directly, e.g., as an invitation to speak of their troubles. Sacks is not saying here that the ‘proper’ return always happens.

Sometimes … a person may take that “How are you feeling?” and attempt to use it to present their troubles. And one sort of thing that happens in that case is that persons who listen when somebody begins to tell them their troubles, talk about themselves routinely as ‘softhearted’, ‘fools’, and that sort of thing … They listen, then they find themselves ‘involved’ … but not knowing what to do. And not knowing how to get out, either, because they ‘know too much’ (Sacks, 1995, p.15).

In other words, ceremonials do not have to be treated as such, but when they are not (particularly with relative strangers) this can lead to difficulties for the person who has avoided the ceremonial response (and can get into a conversation they perhaps rather wished they had not).

Another example of a ceremonial is saying ‘Hello’ on the phone, which can also be used with strangers. The proper return is ‘Hello’, ending the ceremonial, after which the conversation can switch to, e.g., identifying who is speaking and why they have called. Sacks analyses the telling of a joke as a ceremonial, where the ‘proper’ return of laughter ends the sequence. Using these ideas, Sacks then offers an analysis of what is happening when a suicide threat gets laughed off:

When somebody says “I’m going to kill myself,” if the other can cast it into one of the ceremonial forms, then that can end the interchange. One wouldn’t

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have heard the ‘cry for help.’ One would have heard a joke. And one would have behaved properly with respect to a joke … Somebody laughs, or they say “Nice performance, or “Quit playing.” And that would provide, then, for closing that thing off without however, having been in the situation of refusing help in the sense of saying “no,” or other such things. (p.16)

I have chosen to focus on one or two sections from each of the three teacher discussion where there was laughter, to see what emerged.

LAUGHTER IN TEACHER DISCUSSION (TA/14-9-07)

The transcript below begins directly after the first viewing of a video clip of a section of a lesson by Teacher A. The seven-second pause, at the start, I counted from the moment of turning off the video-recording.

1 AC: (7) that was quite long but I think it’s still worth doing (.) so would anyone like to suggest or remind us where it began what the sequence of things that happened there and we can maybe go back and look at things again if we need (.) see if we can re-construct as much as possible

2 TA: (unclear) I started writing it down but then I gave up3 -: [laughter]4 TG: I don’t know if it’s the first thing

but the first thing I remember is the discussion you had about the things you should do if you don’t get one of these answersand I wrote down three suggestionsthe kids came up with three things

At the end of line 1, I offer an invitation to the group to move to reconstructing the video clip we had just seen (this is a standard part of these meetings, see chapter 5). Most of the teachers in the group at this meeting had worked with me on video before in this way, and so would have had experience of the process of starting with a detailed account of what was observed. The invitation I offer is slightly different, however, to other occasions. The new elements are my observation, ‘that was quite long’ (in reference to the clip we had just seen) and then, ‘but I think it’s still worth doing’. The clip had been 5–6 minutes long (compared to a standard of 3–4 minutes) and I interpret some concern, in line 1, about how possible reconstruction is going to be. ‘I think it’s still worth doing’ could be read as hedging, or expressing doubt about whether it is worth doing. This doubt is perhaps picked up in the next turn. Rather than respond to this invitation, TA (line 2) offers a comment about her experience of watching the video (of herself teaching), ‘I started writing it down but then I gave up’, perhaps echoing concerns I voice in line 1 (that the task of reconstruction is too difficult because I have chosen too long a clip). At this point there is laughter from the group.

Line 2 is not a response to the task I offer; it is a comment about that task, and in some sense could be seen as a challenge to the task. If TA’s comment in line 2

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had been responded to with, for example, someone else agreeing, this would have opened up a dialogue about the task rather than an engagement with the task. As I re-work and re-look at this transcript, I see more and more potential dangers in line 2, in terms of possible diversions to my expressed intention to reconstruct the video clip.

The laughter at line 3, I see as operating in a similar way to Sacks’ analysis of laughter after a suicide threat. The laughter, at line 3, turns line 2 into a ceremonial, a joke, and as a ceremonial response it ends that sequence without any further response to TA being necessary. Instead of engaging in discussion about the task, as might have been prompted by line 2, at line 4 (after the laughter at line 3) TG returns to the task of reconstructing the video clip, without any further comment about the issue of whether the clip was too long or not. In other teacher discussion recordings, line 4 is precisely the kind of comment that would come directly after my offer of the task (i.e., a comment that is in the fine detail of words spoken or actions taken on the clip). One interpretation of the laughter, using Sacks’ ideas, is that the group closed off TA’s discussion about the task, without the awkwardness of explicitly rejecting her comment.

METACOMMENTARY

The analysis above is an example of why, from an enactivist perspective, there is a wariness about coding data. In my distillation of using video for teacher learning, I labelled episodes such as the one above ‘reconstruction’, since that appears to be the primary task being undertaken. What the focus on laughter has exposed here is how much more complex the situation turns out to be. The task I offer is one of reconstruction but lines 2 and 3 constitute a micro-episode that is not about reconstruction. There is a subtle shift and then shift back.

I am reminded of Bateson’s (1972) writing about ‘frames’ in thinking about what might have shifted in these lines. Bateson conceived of all communication as occurring within frames that define the kinds of communications that are expected, or allowable. One example Bateson cites is the frame, ‘this is play’, e.g., in animal interaction which might be communicated by a puppy ‘nipping’ rather than ‘biting’ its sibling. Successful interpretation of such messages is vital to animal survival and no less to human communication. Awareness of the frame of an exchange is essential to the interpretation of messages within it. Hence, a frame is a kind of metacommunication. Bateson also reverses this logical implication to assert that ‘every metacommunicative message is or defines a psychological frame’ (1972, p.188)

In the transcript above (TA/14-9-07), line 2, being metacommunicative, would therefore define a psychological frame (we are now communicating about the task, not engaging in the task). This idea, the framing of communication, was taken up by Goffman (1975, 1981). He linked it to the word ‘footing’. A footing describes the alignment we, as speakers, take up in relation to ourselves and others in a

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communication. Goffman analysed the detail of everyday conversations and showed how speakers ‘constantly change their footing, these changes being a persistent feature of natural talk’ (Goffman, 1975, p.128).

Whereas a frame, for Bateson, is psychological and hence perhaps in relation to a ‘thinker’, Goffman’s ‘footing’ is more closely linked to how two or more people align, i.e., is linked more to the relationship between speakers. Goffman goes on to analyse the different types of production and reception roles that speakers can take up. I continue to use the word ‘frame’ but want to invoke the relational aspect of Goffman’s footings. So, another way of analysing what happens in the transcript above would be to say that line 2, being metacommunicative, constitutes a change in frame (we are discussing the task now, not doing it) and that line 3 serves to return the frame to one of engagement in the task of reconstruction, a framing that continues into line 4.

What the focus on laughter makes clear is that I can give someone a list of features of how I use video in teacher learning, but this does not mean they would be able simply to transfer such a way of working into their own context. There is even stronger evidence here for the role of context and history. It is the group of teachers (all used to this way of working) who keep the discussion in its original frame after the change at line 2. This history of reconstructing video footage cannot be transferred to different contexts in the way that a list of ways of working might be seen to suggest.

There is an issue in focusing, as I am, on the role of the discussion facilitator because a ‘role’ is only ever one side of a relationship and, from an enactivist perspective, it is the relationship that is key. A focus on role may appear to reveal aspects of practice but it must be remembered that these aspects arise in interaction and cannot be divorced from the context in which they emerged.

LAUGHTER IN TEACHER DISCUSSION (TB/4-4-08)

The section of transcript below occurs in the ‘reconstruction’ phase of working on the video of a lesson in TB’s classroom. Discussion prior to and in this section of talk focused on a sequence where a student had offered an idea, TB had repeated the idea using slightly different words, and the student responded ‘no’ (this transcript was considered in Chapter 5).

1 TA: you said something slightly different to her/ 2 TB: /did I say/3 TA: /what you said wasn’t as clear as what she said4 -: [laughter]5 TA: the second time she then said it really really clearly (.) when she said no she said

no you draw a box around it and take off the extra bits (.)6 TB: I thought that was what I said7 TK: there was [ ] a very subtle difference in the words /AC: oh wow/ which

she seems to pick up on

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At line 1, TA suggests (about TB on the video recording) ‘you said something slightly different to her’ (i.e., TB repeated something different compared to the student’s original comment). At line 2, TB interrupts with, ‘did I say’, but TA does not hear or acknowledge this turn and, at line 3, I see a shift in framing. Line 1 was a comment about the detail of what occurred in the video clip. Line 3 shifts the frame to more of an evaluation or judgment, ‘what you said wasn’t as clear’. It is after this judgment or change in framing that, at line 4, laughter occurs. In the previous section, I observed that laughter served to return a frame to a previous position, the laughter resisted the change in frame. It does not appear, in this instance, that the laughter acts to end a ceremonial, since TA continues talking in line 5. However, TA does return immediately to the task of reconstruction, ending line 5 with an offer of the detail of what the student had said. There is an intriguing difference between lines 2 and 6, both said by TB. At line 2, before the shift to a more evaluative framing, TB is possibly about to offer a suggestion of what she said. At line 6, I can either read TB as engaging in the detailed dissection of words said (i.e., I heard myself saying exactly the words you just attributed to the student) or as giving a more defensive retort to line 3 (i.e., I was as clear!). In either case, line 7 confirms the return to the frame at the start of the transcript, teasing out the distinction between what the student and TB had said. The word ‘subtle’ in line 7 perhaps operates to smooth any lingering disagreement between TA and TB. If the difference is ‘subtle’, then both TA and TB’s positions are understandable. My interjection of ‘oh wow’ also perhaps serves to confirm this smoothing, i.e., I had not picked up the difference.

In both the transcripts, then, laughter appears following a shift in the framing of the discussion and perhaps acts to repulse this shift. The next transcript, from later in the same discussion of TB’s lesson, reads differently. The transcript begins during a point in the discussion where we had turned to identifying the teaching strategies we had observed being used by TB, and linking what we had observed to our own practice as teachers.

1 TG: there’s also something about the students listening to each other and I think one of the reasons we adopt that strategy is/

2 AC: /sorry which strategy3 TG: the strategy of repeating what students say4 AC: and interpreting it or not interpreting it 5 TG: well (1) right okay I think um (.) we (.) are used [ ] but I think one of

the reasons we have that almost knee jerk reaction to repeat (.) whether it is interpreting or otherwise is because the students will speak quietly (1) and we want everyone else to hear (.) so therefore that’s why we then say it again (.) so that everybody’s heard that comment (.) but it’s so much better if then they listen to each other

6 TD: in some ways by repeating it more clearly students then don’t have to listen to the first

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7 TG: well I I know that’s that’s (.) in a way that’s that’s (.) that’s (.) that’s what I was not saying quite as clearly as you’ve just said

8 -: [laughter] 9 TG: thank you for repeating it 10 -: [louder laughter]

The laughter here feels quite different to other sequences. At line 7, TG was effectively offering a metacomment about line 6, i.e., that TD in line 6 had repeated and interpreted what he had been trying to say himself. The laughter here does not seem to be about rejecting the shift in frame signified by this metacommunication. During the time in the discussion for identifying teaching strategies exactly this kind of metacommunication is warranted. The task is precisely to identify labels for what has been observed. The laughter here seems more to be about the play between the metacommunication TG offers (you have just interpreted what I said more clearly) and the content of the communication (i.e., about whether as teachers we should interpret and ‘clarify’ student comments or not). At line 9, TG then offers another metacommunication. In line 9, TG provides a label, highlighting that what TD had done (in line 6) was an instance of repeating. Again, the laughter seems to come from the play between the levels of communication and the contrast that in the discussion we had seemingly been moving to a position of seeing repetition by the teacher as undesirable, and that interpretation by the teacher is a bad thing. Yet here was an instance of interpretation and repetition being valued by TG, the original speaker. There is therefore a beautiful problematising of an issue that was in danger of being seen more simply (i.e., interpretation and repetition are bad). This is perhaps the ‘real humour’ and the consequent ‘psychological discoveries’ alluded to by Bateson (1972, p.169) in the quotation at the start of this chapter. From my perspective, facilitating this discussion, the shifts in frame denoted by TG’s metacommunications were in fact part of the overall framing for this part of the discussion, which was explicitly about a discussion of teaching strategies. In other words, part of the framing of this section of the teacher discussion (as seen in chapter 5) is a shift in frame to naming or generalising in relation to what has been observed.

LAUGHTER IN TEACHER DISCUSSION (AC/19-5-09)

The final transcript I consider in this chapter is from the teacher discussion of one of my lessons (with a year 7 class, age 11–12). During part of the video clip we viewed, a girl came to the board to explain her idea about how to find the area of a shape being discussed (see bold triangle in Figure 6.1). She began by splitting the shape up horizontally and vertically (which created some squares but also a lot of differently sized triangles, the fainter lines in Figure 6.1). Before she had finished, there were comments from the rest of the class that it was too complicated

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and I took the pen off her. The transcript begins during the phase where we have moved to a discussion of teaching strategies (i.e., we had already finished the task of reconstruction).

Figure 6.1. A girl’s ‘chopping up’ of a shape.

1 TA: well you’re not letting it be a free for allthere are certain thingsyou’re drawing attention toand you’re focusing in certain directions (2)

2 AC: say more3 TA: well (2) you didn’t for example

let the girl who was chopping it up into bits have much time

4 -: [laughter]/5 AC: /right okay/ 6 -: /[laughter]7 TA: but it wasn’t just you who wasn’t letting her have much time

it was sort of everybody really saying how oh that takes too long [1]

At line 1, Teacher A offers a phrase ‘not letting it be a free for all’ that could be taken as a ‘purpose’. At line 2, there is again evidence of my focus on meaning making when I invite TA to ‘say more’. It is evidently not clear to me from the previous discussion what TA is referring to, or what she is focusing her attention on. TA then responds (line 3) with a specific example. The laughter here does not act as a ceremonial ending the sequence. At line 5, I respond to line 3 and the continuing laughter at line 6 does not end the sequence either since, at line 7, TA further develops her interpretation. Line 3 could be read as evaluative, i.e., TA saying that I did not give the girl ‘much time’ could be heard as meaning I should have given her more. There is, therefore, a possible shift in framing at line 3 and the laughter at line 4 could have operated in a similar way to discussion of TA’s lesson, rejecting

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this shift. My comment at line 5 takes line 3 not as a joke but as making a serious point. The laughter at line 6, if it had ended the sequence, would have likewise turned line 5 into a joke, but, in line 7, TA continues with the point she was making, adding another layer of analysis and detail (i.e., recognising the role of the class). It is almost as though TA and I recognise there is something significant being said here about taking responsibility for what is given time in whole-class discussion and we ignore the laughter to maintain our thread.

METACOMMENTARY

Again, I am struck by the complexity of the situation. My analysis above indicates that laughter occurs following a shift in frame and in some cases constitutes itself a shift in framing. In some cases, the laughter acts to return the framing to one I had intended. In others, however, the laughter might potentially have blocked a desired shift in framing (i.e., a deeper analysis of what it means if you are ‘not letting it be a free for all’). So, nothing general can be said about how to deal with laughter. However, this analysis suggests that laughter is linked to shifts in framing, which are key moments in discussion. One aspect of the role of the discussion facilitator is to be sensitive to whether the shift is a desired one or not.

This chapter has served to problematise the five aspects of the role of the discussion facilitator described in Chapter 5. The views of teachers in the department, comparing lesson transcripts, raised for me the awareness of the hidden role of motivation, context and history in making sense of interactions. The analysis of laughter exposed some of this hidden context, in the way the group of teachers laughed off the challenge to the task of reconstructing the video clip. The analysis of the last transcript above adds further complexity to the role of the discussion facilitator, by introducing another dimension to ‘setting up discussion norms’, i.e., a sensitivity to shifts in the framing of talk, and whether these shifts are desired or not. The return to the same research data, each time seeing it differently, is something at the heart of the cycling processes of enactivist research. Rather than be worried that earlier conclusions are troubled, I see this revision as a mark of learning and (Bruner) story-making.

This chapter completes the second part of the book. In the next part, I turn to an analysis of the classroom video- recordings themselves, focusing on Teacher A.

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CHAPTER 7

ON METACOGNITION

D[aughter]: I did an experiment once.F[ather]: Yes?D: I wanted to find out if I could think two thoughts at the same time. So I thought “It’s summer” and I thought “It’s winter.” And then I tried to think the two thoughts together.F: Yes?D: But I found I wasn’t having two thoughts. I was only having one thought about having two thoughts. (Bateson, 1972, p.25)

A STORY OF STUDENT LEARNING

This chapter begins the third section of the book, in which I turn to analyse the video-recordings of Teacher A over a two-year period working with the same group of children.

Teacher A, in interview, was explicit that she saw herself as wanting to develop in students the capacity to notice patterns, make conjectures, find counter-examples and begin working towards proving some of the things they noticed. At the end of year 2, I interviewed three pairs of her students (a high, middle and low attaining pair) and was immediately struck by the clarity with which students articulated how they worked in mathematics, and how closely their descriptions matched Teacher A’s expressed intentions. Teacher A’s description of ‘how mathematicians work’ fitted almost exactly with students’ responses in the interviews. It was clear that students were introduced in the early lessons of the academic year (September) to a new way of working on mathematics compared to their primary schools, where the focus (according to them) was far more on working through text books. Teacher A was getting students to act in unfamiliar ways. All six students in interview spoke of now enjoying mathematics. I offer a selection of comments from the student interviews (for an interview schedule, see Appendix 3).

High prior attainment

AC: … what about thinking mathematically …S1: she (.) we talk about mathematicians a lot and you think it’s quite a hard thing and

like numbers but it’s not (.) it’s really quite easy [ ] it’s just thinking about what you’re doing and trying to find different ways

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Middle prior attainment

AC: is there anything different about how you work in maths compared to other subjectsS1: um (.) in other subjects we don’t really use numbersAC: but more how you work rather than what you work onS1: it’s sort of (1) quite a creative (.) um (.) it depends on what sort of subject you’re

working on in that area (.) um but it’s (.) you can (.) um look at it from different perspectives and um find different ways of working things out and um and we can be working on um hypotheses that you’ve been trying to find and there’s no answer but you can just keep trying and trying and trying (.) everyone can sometimes come up with completely different results [ ] you can all have different answers that can all still be right

AC: I’m really interested in the word creative … can you say a bit more about thatS2: it is (.) cos you’re like making up (.) well when you answer a question that’s being

creative as well cos you’re like actually like saying something S1: yeah you’ve got to bring up certain questions (.) certain things to work on S2: yeah and you’re being creative when you’re trying to figure out like the different

theorems or something and like you make it creative like it’s like making music cos that’s being creative and like painting cos that’s like being creative so when you’re like figuring out a theorem it’s creative but you don’t really think that would you

S1: no (.) everyone thinks the creative stuff is always like art and music and drama and all that and that is (.) but then you’ve got lessons like science and maths that are (.) that you can do a lot more in it if you have the right sort of opportunities to come up with it (.) and I’ve always said I’m quite lucky to have the teacher that I’ve got because we do quite a lot of hands on stuff (.) it’s really good

Low prior attainment

AC: what have you learnt about thinking mathematically or how mathematicians workS1: they always like come up with conjecturesAC: what’s a conjectureS1: like a theory (.) and then they see if they can prove themselves wrongS2: or try and prove it rightS1: and then they find out other people’s conjectures and like test themS2: test ‘em yeah they test ‘em and from other people’s conjectures they also make

new ones

In the high attaining group, thinking mathematically is linked to ‘thinking about what you’re doing’. In the middle attaining group, working mathematically is linked to being creative, which for the students is connected to asking questions and ‘figuring out’ theorems. In the low attaining group, the word ‘conjecture’ has come to symbolize a complex pattern of ways of working in mathematics: looking for connections or patterns, then trying to prove yourself wrong; trying to adapt other people’s ideas to make your own connections; and trying also to prove yourself right. I recognise a strong connection between these descriptions of thinking mathematically from the students and TA’s intentions, and also a connection between

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the student descriptions above and what I observed in the video-recordings of TA’s lesson. It seems apparent that the students in TA’s classroom, in communicating about their actions in mathematics lessons, demonstrated knowledge of their own knowing. What did Teacher A do? What aspects of Teacher A’s role are effective in terms of students being able to display such capacity?

Knowing about your own knowings is one definition for what is labelled ‘metacognition’. From an enactivist perspective, knowing is doing and doing is knowing, hence I take metacognition to mean any kind of meta-awareness about your own actions, or any action that implies a meta-awareness (such as a metacommunication). Supporting mathematical thinking in students therefore entails developing metacognition. It is evident from the student interviews that Teacher A was effective in terms of inducting students into new metacognitive practices. Therefore, to support my analysis of patterns in the video and to try and say something about the role of the teacher, I decided to delve into the literature on metacognition and its instruction.

In the rest of this chapter, I firstly highlight two gaps in the literature on metacognitive instruction; one gap is a scarcity of detailed accounts of the role of the teacher; and a second gap is a lack of consideration of the metacognitive or self-regulatory demands on teachers wanting to instruct their students into metacognitive practices. I take self-regulated learning (SRL) to refer to ‘self-generated thoughts, feelings and actions for attaining academic goals’ (Zimmerman and Schunk, 2004, p.323) and hence see SRL as an aspect of metacognition. I draw on Prawat (1998) and Zimmerman (1999) in arguing that part of the problem is that there continue to be hidden epistemological commitments to outdated information-processing views of cognition in much of the field, and these commitments lead to a dilemma in metacognitive instruction. This chapter ends with these dilemmas, which I aim to resolve in Chapters 8 and 9, through an analysis of Teacher A’s classroom. I believe her classroom practice functions as a ‘paradigmatic case’ (Freudenthal, 1981, p.135) of a method of metacognitive instruction that is not reported in the literature.

TWO LITERATURE GAPS

A number of research studies have linked students’ effective problem-solving with a high frequency of observed metacognitive behaviour both in mathematics and more widely (e.g., Schoenfeld 1992; Carlson 2000; Kramarski and Mevarech 2001; Mevarech and Fridkin 2006; Zohar and Ben David, 2008; Vrugt and Oort, 2008). It has also been reported (Bakracevic Vukman and Licardo, 2010) that the extent of students’ metacognitive self-regulating behaviour is strongly correlated with examination success. It has therefore been seen as worthwhile to develop an understanding of efficient ways of inducting students in metacognitive ways of acting and being.

As an initial point of concern, reading the ‘negative’ of these results, there appears to be an assumption that students who are not successful are lacking in the requisite metacognition. I worry (not least from an ethical point of view) about

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any perspective that starts off seeing students or teachers as somehow in deficit, especially since, in countless aspects of their lives, the very students who are deemed ineffective problem-solvers will negotiate with expertise practical difficulties or changes in framing of a conversation (for example, in displaying effective behaviour on recognising a wallet has been lost or helping a friend ‘save face’ by making a joke of a naïve comment).

In a review of research in metacognition, Brown commented that the concept, ‘is not only a monster of obscure parentage, but a many-headed monster at that’ (1987, p. 105). A similar judgment was reached nineteen years later by Veenman, van Hout-Wolters and Afflerbach (2006) who introduce the new journal, Metacognition and Learning, with terms that have been developed in the field:

[m]etacognitive beliefs, metacognitive awareness, metacognitive experiences, metacognitive knowledge, feeling of knowing, judgment of learning, theory of mind, metamemory, metacognitive skills, executive skills, higher-order skills, metacomponents, comprehension monitoring, learning strategies, heuristic strategies, and self-regulation (2006, p.4).

This proliferation of terms led Veenman et al to conclude, ‘while there is consistent acknowledgement of the importance of metacognition, inconsistency marks the conceptualization of the construct’ (2006, p.4). They give the example that self-regulation is sometimes viewed as a component of metacognition, and sometimes metacognition is viewed as a component of self-regulation.

Looking across research on instruction in metacognition and SRL, I have been struck by the scarcity of reports on the detail of the role of the teacher during instruction, something on which I have data. When the teacher’s role is considered, it tends to be in the context of teacher beliefs (e.g., Maggioni and Parkinson, 2008). The gap I am alluding to is well represented in writing about the IMPROVE model (Mevarech and Kramarski, 1997) for developing student metacognition (the acronym IMPROVE represents all the teaching steps that constitute the method: Introducing the new concepts; Metacognitive questioning; Practising; Reviewing and reducing difficulties; Obtaining mastery; Verification; and Enrichment). In reports on the success of this programme, there are detailed descriptions of the theoretical approach and the student response, but little said about the role of the teacher. For example, in Mevarech and Fridkin (2006)’s report on IMPROVE they explain the role of the teacher as follows:

the teacher first introduces the new concepts, theorems, formula etc. to the whole-class by modelling the meta-cognitive questioning technique (p.87)

At the end of the lesson, the teacher reviews the main ideas and reduces difficulties by modelling the use of the self-addressed meta-cognitive questioning (p.88)

the teacher evaluates students’ progress and provides feedback followed by enrichment and remedial materials, as needed (p.88).

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These statements give the impression that the task for teachers is largely unproblematic. Such an impression is strengthened by conclusions that refer to the method (of IMPROVE) without mention of the teacher implementing it, for example:

students exposed to IMPROVE are better able to construct mathematical models then their counterparts in the control group (Mevarech and Fridkin, 2006, p.88).

The teacher has completely disappeared in this statement as a factor in student development. In another example of this phenomenon, Kramarski and Mevarich (2003) compared the effect of four different methods of instruction on students’ mathematical and metacognitive reasoning – an underlying assumption again appears to be that teachers can un-problematically implement different methods and, as such, are not a significant factor in student learning. The complexity that was apparent in the last section of this book, in working on video with teachers, has made me believe that unproblematic implementation of a set of procedures for working with teachers or students is highly unlikely.

Mevarech and Fridkin (2006) implicitly acknowledge the gap in research on the role of the teacher. While reporting on the success of IMPROVE (in terms of developing metacognitive knowledge and skill; improving students’ mathematical reasoning; and student attainment on standard tests), they suggest that future research involving direct observation of metacognitive instruction is needed to throw light on its benefits. I have such direct observations in my data.

A second gap in the literature relates to one aspect of the role of the teacher – a consideration of the metacognitive demands. Perry, Hutchinson and Thauberger (2008) conducted research into the levels of support needed for beginning teachers to be able to implement SRL strategies in their classrooms suggesting, for example, that discussion of SRL needs to be a prominent feature of lesson observation feedback. However, I am yet to find any study looking at the metacognitive demands on the teacher who works to develop metacognitive skill or knowledge in their students. I find this gap surprising, particularly since there have been reports that how teachers conceptualize mathematics and the nature of student learning has an influence on classroom discourse (Maggioni and Parkinson, 2008, p.446).

In order to probe the issue of how teachers’ metacognition might be conceptualised, I have found it illuminating to look further into the source of current views of student metacognition. Veenman et al (2006) see the ‘most common distinction in metacognition’ as the separation of ‘metacognitive knowledge from skills’ (p.4). This distinction derived from Nelson and Narens (1994) and their model of metacognitive monitoring and control (reproduced below, see figure 7.1). Schwartz and Perfect (2002, p.4) commented that this model took metacognition ‘into the “modern” era’.

Nelson and Narens (1994) conceived of metacognition as resting on the assumption that the brain creates models of the environment in order to be successful

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and efficient (p.10). If these models are thought of as an ‘object-level’, Nelson and Narens suggested the human brain is able to control and monitor these models through a ‘meta-level’ which ‘contains an imperfect model of the object-level’ (p.11).

Model

ControlMonitoring

META-LEVEL

Flow of

information

OBJECT-LEVEL

Figure 7.1. Meta-level/object-level theoretical mechanism (Nelson and Narens, 1994).

Metacognitive control describes the way that the meta-level is able to influence and alter operations at the object-level. For example, part of my ‘model’ of the environment may be that, when tackling a new mathematics problem, it is useful to look at simple cases first. If I choose to employ such a strategy and then make a decision about what, in this instance, constitutes a simple case, this would be the exercise of metacognitive skill. Metacognitive monitoring is where the meta-level is informed and altered by changes at the object-level. For example, I may become aware that whatever strategy I am using to solve a problem is not working and that I am stuck. This is metacognitive knowledge, which generally is in awareness, i.e., we are generally conscious of changes at this meta-level.

My focus on the role of the teacher led me to question, using this model, what metacognitive knowledge/skill do teachers need and how this is similar or different to that of their students? The Nelson and Narens model seems to imply that ‘control’ and ‘monitoring’ can be isolated or viewed as separate from both the learner and the context in which they are learnt. The metacognitive practices of successful problem solvers (e.g., cited by Zohar and Ben David, 2008; Vrugt and Oort, 2008) are presumably ones that the teacher ideally possesses, and that students will learn from instruction. Hence, using Figure 7.1 to conceptualise how students gain the same metacognitive knowledge as possessed by the teacher suggests an image of learning where information or skill gets transferred from teacher to student, independent of context. Yet, if such independence were really the case, it would imply that knowledge (or skill), once learnt, should be able to be used or transferred relatively un-problematically in different contexts. Precisely the opposite has been reported consistently. Haskell (2001) comments, ‘research findings over the past nine decades clearly show that as individuals, and as educational institutions, we have failed to achieve transfer of learning at any significant level’ (p. xiii).

Within the Nelson and Narens model, a way the teacher could be conceptualized as having a different kind of metacognitive skill/knowledge to students, would be

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to add a third ‘meta-meta’ layer, containing an ‘imperfect’ model of the metalayer (which contains an ‘imperfect’ model of the object-layer). However, the complexity of a middle, metalayer, both monitoring and controlling the object layer, and at the same time being monitored and controlled by the meta-meta layer, does not resonate with my experience of teaching.

The problems of how to conceptualise teacher metacognition and of beginning with a deficit model of students are not the only issues. There is also a reported dilemma in metacognitive instruction that I see as tied to the Nelson and Narens model.

A DILEMMA IN METACOGNITIVE INSTRUCTION

As stated above, several studies suggest that, metacognition is displayed by effective problem-solvers and mathematicians. There is, therefore, interest in effective ways of teaching metacognition. However, there is a problem, reported by Schoenfeld (1992), linked to teaching problem solving heuristics (an example of metacognitive instruction):

the critique of the strategies listed in How to Solve It [Polya, 1957] and its successors is that the characterizations of them were descriptive rather than prescriptive. That is, the characterizations allowed one to recognise the strategies when they were being used ... [but] did not provide the amount of detail that would enable people who were not already familiar with the strategies to be able to implement them (p.353).

Schoenfeld seems to be suggesting that a prescriptive approach is preferable to a descriptive one; yet the difficulty then, as he goes on to point out, is that any single description of a strategy must be translated into a dozen or more prescriptive routines and before long, a list of, say, six heuristics turns into close to one hundred prescriptions. In other words, as a teacher, I can choose a small set of heuristics that seem important (‘trying out simple cases’ might be one) and I might emphasise their importance to students. This approach may allow my students to recognize such a strategy when they see it being used but, according to Schoenfeld, is unlikely to mean they are in a position to know when would be a good time to use it, nor how to use it and it is such control that is the mark of metacognitive skill. So, taking a descriptive approach, students would gain metacognitive knowledge but not skill. If I split the heuristic into more prescriptive routines (for example, ‘if you have an algebraic function, look at when x = 0’; ‘if you have a number sequence, look at when n = 1’; etc) then my students may develop metacognitive skills; the “if” clauses imply some control is taking place. As a teacher, however, I end up with an unmanageable list of scenarios to cover and for students to memorise. It appears that the splitting of metacognition into the dichotomy of skills and knowledge leads to a teaching dilemma. Do I focus on “knowledge” and risk students not developing skill, or do I focus on “skill” with the risk of overwhelming students with lists of specific strategies?.

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Schoenfeld (1992) reported on training his students to use metacognitive self-questioning, on a problem solving course he ran, as a way through this dilemma. Mevarech and Kramarski (1997) drew on Schoenfeld’s work to develop their IMPROVE model. In this method, students are trained to ask four kinds of metacognitive, self-addressed questions: comprehension questions; connection questions; strategic questions; and reflection questions. The asking of questions such as ‘what is this problem all about?’ (a comprehension question) is an example of a procedure or metacognitive skill. The approach taken is therefore on the ‘prescriptive’ side of Schoenfeld’s dilemma and there are indeed several questions that students are trained to use within each of the four categories. Given the success reported in using IMPROVE, Mevarech and Fridkin (2006) suggest future research involving direct observation of metacognitive instruction is needed, to throw light on its benefits. This section of the book offers such direct observation and, in the process, I suggest an alternative view of metacognition, removed from the knowledge/skills dichotomy and the descriptive/prescriptive teaching dilemma.

In the next section, I suggest a source of the dilemmas and difficulties above is that the metacognitive skill/knowledge division (and by implication any study that draws on it) arises from commitment, within the Nelson and Narens model, to an information-processing view of cognition that is now out dated.

EPISTEMOLOGICAL COMMITMENTS

Prawat (1998) suggested that the dualities of mind/body or process/content, which he linked to the view of learning as gaining objects, were inherent in much of the literature at that time on metacognition and self-regulated learning.

Nelson and Narens (1994) state their alignment with just such an ‘object-based’ view by endorsing the conclusions of early (i.e., 1970s) cognitive science:

the living brain, so far as it is to be successful and efficient as a regulator for survival must proceed, in learning, by the formation of a model (or models) of its environment (p.89, their italics)

Indeed, in labelling the lower layer of their model the ‘object-level’, Nelson and Narens are explicit in conceptualizing experience in terms of objects in the mind (e.g., representations of the world) that provide information to the metalevel. This assumption of an ‘object-based’ and information-processing view of the mind is therefore inherent within all metacognitive research that draws on the knowledge/skills distinction which, to reiterate, is the ‘most common distinction in metacognition’ (Veenman et al, 2006, p.4).

Schunk (2005) traces the history of SRL, showing it was seen as an information- control mechanism, much in the same way that Nelson and Narens view metacognition as controlling and monitoring the flow of information within a system. As with metacognition, within the self-regulated learning literature, there is a proliferation of terms and approaches. Zimmerman and Shunk (2001) describe seven distinct

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approaches to SRL, each one stressing different key processes. In a special issue on SRL, Azevedo (2009) calls for researchers to be more explicit about the theoretical model or framework informing their studies.

A year later than Prawat’s (1998) criticism of a hidden dualism in much metacognitive theorizing, Zimmerman (1999) sees a similar assumption in a special issue of the International Journal of Educational Research on self-regulation and SRL. In particular, Zimmerman notes one problem with the object-based, dualistic view of self-regulation (and by extension cognition) – the social cannot be viewed as a resource, echoing the absence of a consideration of ‘context’ in theorisation about metacognition. Zimmerman and Schunk (2001) articulate an alternative, socio-cognitive version of SRL, yet the underlying cognitive model is still one of information processing (Schunk, 2005, p.86). Winnie and Hadwin (2008) put forward a model of SRL that makes this information-processing commitment explicit. If the mind is seen as an information-processing device, it is not surprising that metacognition is seen as a way of managing this information flow (metacognitive skill) and the possession of a different class of information to normal (metacognitive knowledge).

The enactive view is explicitly opposed to seeing cognition as linked to information exchange. However, the rejection of this view of cognition is not limited to enactivist researchers. In a review of conceptualizations of learning, Hager and Hodkinson (2009) critique the object-based view of knowing and learning while also highlighting problems with the implied metaphor of learning as ‘transfer’ (of objects). Hager and Hodkinson (2009) analyse commonly held ‘conceptual lenses’ through which to view learning. The first two of which ‘the propositional learning lens’ (p.622) and ‘the skill learning lens’ (p.624) mirror the concepts of metacognitive knowledge and skill. Hager and Hodkinson (2009) see these two lenses as sharing the following assumptions:

(a) What is learnt is a product, a thing or substance that is independent of the learner … (b) Learning involves movement of this thing or substance from place to place … (c) What is learnt is independent of and separate from the context in which it is learnt. (p.622–3)

Whether what is learnt is thought of as propositional knowledge (as in the case of metacognitive knowledge) or skill (metacognitive, or otherwise), the assumptions above can be seen in the ways we talk. Assumption (a) is hidden in discussions about ‘gaining’, ‘losing’, ‘passing on’ knowledge or skills, as though these are objects we possess in our mind or bodies. Assumption (b) can be read into the language of learning transfer in which we commonly talk as if knowledge is a ‘commodity or substance being literally moved from one location to another’ (Hager and Hodkinson, p.622). The situation becomes more confusing when we think of knowledge being transferred from a teacher to a learner, ‘since the teacher also retains what was transferred to the learner’ (p.623). The metaphors become misleading.

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If the object-based view of mind and learning is rejected, for leading into confusion, where does that leave conceptualisations of metacognition?

METACOMMENTARY

What I have argued in this chapter is that conceiving metacognition as split between knowledge and skills is linked to commitments within the field to information-processing views of the mind and of learning. Such a view of metacognition leads to three problems:

• it is not clear how it would be possible to conceptualise the metacognitive demands of teachers wanting to teach metacognition

• the model of metacognition is a deficit one, students initially are seen to lack metacognitive skill and knowledge, which they then gain from their teachers

• there is a dilemma in metacognitive instruction, of whether to focus on knowledge (and risk not gaining skill) or to focus on skill (and risk overwhelming students with unmanageable lists of things to memorise).

In the next chapter, I move towards a re-framing of metacognition that is not tied to an object-based, information-processing view of cognition and that I believe offers an alternative perspective on the problems raised above.

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CHAPTER 8

THE STORY OF TEACHER A

In the living of living beings living entails knowing, and knowing entails living. (Maturana and Poerksen, 2004, p.67)

PATTERNS IN THE DATA

In this chapter, I offer an alternative perspective on the concept of metacognition arising out of my analysis of the classroom of Teacher A.

Enactivist research entails a cycling between data collection and analysis. I began looking seriously into the role of Teacher A at the end of the first year of data collection. The enactivist principle of equifinality (see Chapter 2) meant that I began my analysis of Teacher A’s role by viewing and re-viewing the last video recording I had, which meant the final recording of year 1 (June 2008). As with the teacher meeting data, I initially split the lesson into episodes characterised by different patterns of interaction (see Appendix 4 for a full transcript) and began looking for patterns within and across those episodes.

A pattern observable at the end of year 1 is the use of the word ‘conjecture’. For example:

TA: I want you to try what has been suggested (.) so can you find a counter-example to the conjecture that all thedifferences will be eight

TA: okay (.) could you try some different starting numbers for your conjecture then

Having identified a pattern in the use of a specific word, I then looked through the rest of the data to trace the emergence of this language. Starting with ‘conjecture’, my analysis of other lessons threw up three more words that were used frequently in the same episode, namely ‘counter-example’, ‘proof’ and ‘theorem’.

At this stage in my analysis, I was not explicitly focusing on metacognition. Looking at ‘conjectures’ had arisen as an issue in the manner described. It has only been later that I have recognized the language of conjecture as being metacognitive, and the way of working with this language (developed by Teacher A) to be different to anything I have found in the research literature, provoking me to look for an alternative conceptualisation of metacognition. I focus here on the emergence of the (metacognitive) language of conjecture in Teacher A’s classroom.

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The first video recording I have of Teacher A with her year 7 class is their second mathematics lesson of the year, and these students’ second mathematics lesson at secondary school (which goes from age 11 to 18). In this lesson, the word ‘conjecture’ is used forty-two times in whole-class discussion, six times by students and thirty-six times by Teacher A. There is one extended whole-class discussion that lasts twenty minutes of the hour lesson, when most of these occurrences take place. I present below four of Teacher A’s contributions to the discussion (numbered for later reference), which represent the range of ways in which she uses the word ‘conjecture’.

This first contribution is early in the lesson as Teacher A sets up an initial task for students:

TA: what I want you to try and do now is while you are doing these sums be thinking about the conjectures we talked about last lesson (.) be thinking about whether you think they are true (.) be thinking about whether you can explain why they are true and why you don’t think they are true and also writing down all of these things (Lesson 14/9/07, #1)

It is clear from this comment that the word conjecture has been introduced in the first lesson of the year, and that some conjectures have already been found by the class. The transcript also suggests a connection between conjectures and writing. Teacher A described (in interviews) the importance she places on students’ writing, linked to developing with them a process of thinking mathematically. Teacher A described writing as a mechanism by which students are forced to consider what they are doing and hence it supports them in making the connections that form conjectures. Later in the first video recording, Teacher A again links conjectures and writing:

TA: you need to have written at least one sentence about what you have noticed (.) it might be something about J’s conjecture or J’s old conjecture or it might be something completely new you’ve discovered yourself (Lesson 14-9-07, #2)

During the one extended class discussion in this lesson, there is a disagreement amongst students. The issue is in fact one where both options could be mathematically correct. Rather than close off this ambiguity and force one interpretation on the students, Teacher A leaves the decision to the students, and uses the opportunity to make a point about working mathematically:

TA: so (.) I think what you need to be clear about as somebody working on mathematics as a mathematician (.) you need to decide what you’re going to do and be consistent about that […] J’s conjecture might not apply to your rules (.) so when we’re making conjectures we need to be clear about the rules we’re using (Lesson 14-9-07, #3)

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There is a further comment later in the lesson, relating conjectures and counter-examples to how mathematicians work, in response to two students who changed their conjectures:

TA: this is what mathematicians do (.) they develop their conjectures so they begin with something they believe to be true and then they might change their minds having got some results (.) so it’s interesting that I’ve got the same two people in the group (.) whose conjectures have actually changed (.) but what you need to be doing as mathematicians is thinking about starting testing these conjectures (.) and it might be that leads you to develop conjectures of your own (.) it might be that it leads you to disprove one of these (.) and what T has given us an example of (.) and what J has given us an example of (.) is where this doesn’t hold to be true (.) called a counter example (.) an example that doesn’t fit the conjecture (Lesson 14-9-07, #4)

In this instance, and the one before, it can be seen that Teacher A uses and expands on the word conjecture at the moment when a student exhibits a behaviour that fits the notion of what it means (in this classroom) to be an effective mathematician. Teacher A (in interview) reported having told students in their first lesson (the one before this recording) that the purpose for the year for all of them was ‘becoming a mathematician’. It is also evident in transcript #4 above that the word conjecture is linked to this purpose – ‘this is what mathematicians do’.

In subsequent video recordings of Teacher A, the word ‘conjecture’ is used less frequently. It seems as though the process of looking for connections rapidly becomes part of what students expect to do in lessons with Teacher A and, as such, the word conjecture perhaps only needs to be mentioned infrequently. In the fifth recording (in January 2008), there are four uses of the word, twice by Teacher A and twice by students; in the sixth recording (June 2008), there is one use of the word by Teacher A. However, in both these lessons (and in fact across all six recordings), there is a common pattern of students making statements of generalities, for example:

Student: with the rhombus or kite (.) whatever one we were doing (.) you could um cut it in half and then on the sides you get a little box and it’s like half (Lesson, 5/1/08)

Student: the two answers in the little circles no matter what you start with will always be the same (.) because if you times five times two it’s like times ten (Lesson, 6/6/08)

In both examples above, students make general statements, in the latter case, with a reason to justify the thinking. Both statements could be called conjectures in Teacher

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A’s classroom, but by January and June in the academic year there is little explicit need to mention the word. Making statements such as these and discussing them have become part of what students do, unprompted, in whole-class discussion.

The decrease in frequency of the use of the word conjecture, with very high occurrences of teacher use in early lessons, mirrors a result reported by Schoenfeld (1992) that he needed to pose metacognitive questions less and less as his courses on problem-solving progressed. It also seems to be significant that the word conjecture is barely defined by Teacher A. It is used in response to student behaviours and, as the year progresses, different aspects of what it means to conjecture are commented on by Teacher A (e.g., considering the ‘rules’ behind a conjecture, looking for counter-examples). The word therefore acts as a placeholder for a variety of actions that students can usefully perform in mathematics lessons.

The word ‘conjecture’ has come to symbolize a complex pattern of ways of working in mathematics: looking for connections or patterns, then trying to prove yourself wrong; trying to adapt other people’s ideas to make your own connections; and trying also to prove yourself right. Considering again the student interviews quoted at the start of Chapter 7, student responses do not translate the word ‘conjecture’ neatly into a set of propositions, or link it directly to specific skills learnt. In other words, their responses link ‘conjecture’ to a complex mix of metacognitive knowledge and skills. The meta-language of conjecture seems to ‘hold’ the process of doing mathematics with Teacher A, linked to the notion of ‘becoming a mathematician’. The language of conjecture and counter-example had been introduced in response to student actions. The words did not need to be memorized by students because they became useful shorthand for these new ways of acting. The language of conjecture, counter-example, and proof is subsumed underneath the purpose of ‘becoming a mathematician’, a purpose that is embodied and enacted in the person of Teacher A. The language gets linked to vivid and common experiences as it is initially introduced and used in response to what students do and say.

RE-FRAMING METACOGNITION

The process by which Teacher A’s students take on metacognitive practices is highly efficient. By the second lesson of the year, some students were using the new (to them) word ‘conjecture’ and engaging in metacognitive activity in lessons (looking for patterns in results; making and testing conjectures; and looking for counter-examples). Teacher A used this meta-language linked to an explicit process of ‘becoming a mathematician’, which was offered to students as their purpose for the year. The activity of writing was emphasized to encourage students to think about and become aware of what they noticed. The words of the meta-language (conjecture, etc) were introduced and used at the moments when students themselves exhibited behaviours that Teacher A judged as fitting with what it means, in her classroom, to be an effective mathematician. It is apparent in Teacher A’s comments in the second lesson of the year (#3 and #4, above) that she does not respond to the

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content of student responses. This is a pattern observable in all her responses to students in the early lessons. For example (in #3), rather than responding to a student dilemma in terms of its mathematical source, she responds from the perspective of how mathematicians resolve such difficulties. From interviews and the video recordings of lessons it is clear that Teacher A has a well established (though not fixed) framework of what it means, to her, for students to work mathematically, which comprises precisely the types of activity I reported students can be seen doing in lessons (such as, asking questions and noticing patterns). Also, from interviews and video recordings at the end of the year, it seems that students had entered into a different relationship to mathematics since coming to secondary school, and that the language of conjecture, though not often needed in class by the end of the year, ‘held’ the process of working mathematically with Teacher A.

While it may be possible to analyse developments in student metacognition in terms of skills and knowledge, the data from this study serves to highlight connections between terms on either side of the distinction. For example, in Teacher A comment #4 (above), she introduces the label ‘counter-example’. It could be argued that she is pointing to the development of metacognitive knowledge (by offering students a vocabulary which could support awareness of their own or others’ work in mathematics). Equally, however, the comment could be seen to be about metacognitive skill, in encouraging students to be aware of when it could be useful to look for counter-examples (which they talk about in interview: having found a conjecture, ‘[mathematicians] see if they can prove themselves wrong’). To take another example, in my interview with students, they linked conjectures to proof (proving a conjecture is true and trying to prove it wrong), testing, and making new conjectures (see Chapter 7). These statements display metacognitive knowledge (they are descriptions of what goes on in lessons), but equally it is clear from the lesson recordings that these are general statements of processes students engage in; these processes involve control and regulation and hence are evidence of metacognitive skill.

The mechanism of Varela’s ‘star’ (Figure 1.1) entails viewing each side of a distinction as arising from the context of the other. Teacher A consistently links specific metacognitive skills (e.g., about conjecturing and proving) to what mathematicians do. These skills (‘the process of becoming it’) arise from the classroom context of students being placed as mathematicians (‘the it’). Equally, what it means to be a mathematician in this classroom (‘the it’) is constituted precisely by the exercise of metacognitive skill (‘the process of becoming it’) and hence ‘the it’ itself is constantly changing. Each side of the distinction arises from the other and each side is dynamic. In Teacher A’s practice, the meta-level which bridges metacognitive knowledge and skills is in the concept of ‘mathematicians’. Students are at the same time placed as already ‘being mathematicians’ (in the way Teacher A draws out aspects of mathematical practice from actions she observes in the classroom) and as still being in a process of ‘becoming mathematicians’, by developing the kinds of skills and awarenesses she highlights. This analysis points to an alternative way of

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viewing metacognition that does not emphasise the knowledge/skill dichotomy and avoids a deficit view of focusing on what students lack.

The data I have re-presented from Teacher A, I see as indicating her practice is a ‘paradigmatic case’ (Freudentahl, 1981, p.135) of what it would mean to teach with a view of metacognition as a star. What I see as different about Teacher A’s practice, to the other models in the literature, is her holding of both the ‘process of becoming it’ side of metacognition and the ‘it’, by consistently linking specific metacognitive skills (e.g., about conjecturing and proving) to what mathematicians do and by linking metacognitive knowledge (about what it means to be a mathematician) to the exercise of these skills. In this sense, I see her teaching as a paradigmatic example of what it might mean in practice to view metacognition as both a process of becoming, emerging from the context of learning specific skills, and as constituted by those skills, as they in turn emerge from the context of the ongoing process of learning about becoming a mathematician.

A SIMILARITY

There is a similarity in practice between the classroom of Teacher A, where metacomments were offered following actions or statements students had made, and my practice of metacommenting in teacher discussions. I quoted above from Teacher A, when she metacomments about a student dilemma that arose in the context of working on the problem ‘1089’ (see Appendix 1):

TA: so (.) I think what you need to be clear about as somebody working on mathematics as a mathematician (.) you need to decide what you’re going to do and be consistent about that [ ] S2’s conjecture might not apply to your rules (.) so when we’re making conjectures we need to be clear about the rules we’re using (Lesson 14/9/07)

This is precisely the section of the lesson that TD refers to in the transcript below, to which I metacomment (as reported in Chapter 5):

TD: because there was this issue that belonged to the classand I think you said we need to sort this out (.) um (.) and if it was me I would have chosen one of the options for them (.) um (.) and that’s sorting it out for them (.) I think in some ways you sorted it out for them as well (.)but gave them the option of putting conditions (.) so you didn’t make the choice but you did sort out the issue (.) I don’t think (.) they didn’t sort out the issue (.) they were wondering what can we do with this (.) we need to sort this out as well (.) and I don’t think it was sorted out by the pupils (.) I think you sorted it out (.) in putting conditions

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AC: that’s lovely (.) and it’s a much more enabling sorting outbecause if we just sort it out by answering the issue then the next time pupils come up with this issue they have no they’re in no better position to decidethe only resource they’ve got is to ask TA but if you sort it out by making them aware this is an issue and making them aware there are consequences for each one and whatever that is offering them a tool for next time they get in to that situationso yeah I love that TD/TA/12-11-07

As with TA metacommenting on student actions or comments that have just occurred, I metacomment here on a description from TD about the way TA has sorted out a student dilemma. What I do not do, that TA does, is offer any equivalent label to that offered students of ‘becoming a mathematician’. However, in both cases we offer a label for something we see as significant that has just occurred (in TA’s case, an example of working as a mathematician and in my case, the identification of a purpose, ‘enabling sorting out’).

There is another similarity in the comments made in the video and the discussion of that video, which is alluded to by Teacher D. A little before the transcript quoted above, Teacher D says (in a turn quoted already in Chapter 5):

TD: it was as if TA was listening to their discussion and then (2) because what struck me while watching it was that this is actually a really sophisticated skill for the pupils to be thinking about and that byI would like to talk about that in my lessons but to talk about it wouldn’t mean anything to themso it’s as if it came from their discussion and you spotted that this was the issue they were having TD/TA/12-11-07

In commenting ‘I would like to talk about that in my lessons but to talk about it wouldn’t mean anything to them’, Teacher D makes what I take to be a profound point, and one that is warranted by the evidence I have in both classrooms and teacher meetings. I interpret what he is saying as touching on the issue of the impossibility of ‘double jumps’ in levels of comment. This interpretation will need some un-packing.

It seems possible in thinking about the lesson to which Teacher D is referring, to see the communications occurring on three levels. By ‘level’, I mean to distinguish communications from metacommunications and meta-metacommunications. A second level indicates that comments in that level are about comments in the level below. So, a student conjecture is about their answers or results, since it is a generalization of them. A teacher’s metacomment about student answers (e.g., the

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way they organized them, or the speed with which they found them) would also be in the same second level.

Teacher metacomments about student conjectures

~

Student conjectures, or

Teacher metacomments about student answers

~

Student answers

Figure 8.1. Levels of classroom communication.

A metacomment from the teacher about student conjectures would be at a different level again, being about the conjectures themselves. I see Teacher D as effectively saying, there cannot be comments at this third level, if there is not activity at the second level on which to base the comment. Of course, Teacher D could go in to his next classroom and start talking about conjectures and rules, but his insight I read as being that this would not be effective. Such talk would no longer be metacommunication that served to highlight something of importance to the class that has just occurred. The evidence from the recordings of TA is that the power of metacommunication comes from it providing ways of talking about (and promoting the recurrence of) certain kinds of action that have just taken place.

In a similar way, I am only able to make a comment about Teacher A offering an ‘enabling sorting out’ compared to actually making a choice for the class, because Teacher D has raised the issue of who sorted out the student dilemma. The levels of communication in this case seem to be something like this (Figure 8.2):

Metacomment about teaching strategies

(e.g., I contrast different types of ‘sorting out’)~

Description of teaching strategies or metacomment about reconstruction

(e.g., TD comment about how TA sorts out a difficulty by giving a choice)~

Reconstruction of events on the video

(e.g., teacher observations of talk and gesture in the lesson)

Figure 8.2. Levels of teacher-meeting communication.

In just the same away that TA’s metacomment about student conjectures became possible as a result of a disagreement about the conjectures, so my metacomment about ways of sorting out student problems became possible as a result of discussion about how the problem was resolved. I suggest (as I read Teacher D doing above) that in this instance a ‘double jump’ in levels would not be effective. If I was to watch the same clip of video with a group of new teachers and, as they were talking

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about the section where Teacher A metacomments to the students, if I were then to make a comment about what an enabling way of sorting out a dilemma it was, then it would surely come across as though I had an ‘answer’ in mind, or a specific interpretation that I thought was ‘correct’ about this video clip.

A distinction seems significant here between metacomments and metamessages. I use metacomment to indicate a specific form of verbal statement that is about the communication occurring. Following Bateson (1972, p.247), I take it that all messages carry metamessages about how the message is to be read and about the relationship implied between the participants. We communicate the ‘frame’ of any message. Such metamessages about framing are carried by facial expressions, body position, tone of voice and no doubt are generally unconscious. Even metacomments therefore carry metamessages. The ‘double jump’ issue alluded to by Teacher D can be read in terms of different types of metamessage. A metacomment about something one level below (as in the examples of TA in the classroom and me in the teacher discussion) in response to what is said, carries metamessages about whose agenda is to the fore. The teacher responds to what is said, and the student is placed as working like a ‘mathematician’. The relationship implied is one in which the teacher takes account of what is said by others. A metacomment that was a ‘double jump’ would carry metamessages that the teacher/facilitator is the one who will control the direction of the conversation, and that the role of the group of students or teachers is to fit in to something fixed. The relationship implied is one in which the leader does not allow what others say to alter his/her ideas.

METACOMMENTARY

I raised, in Chapter 7, three problems with current conceptualisations of metacognition, namely:

1. how to account for the metacognitive demands on a teacher wanting to develop metacognition

2. whether there is an alternative to a deficit view of metacognition3. the teaching dilemma of Schoenfeld (1992).

The re-framing above allows me to respond to problems (2) and (3).To summarise the perspective offered in this chapter, I suggest that metacognition

can be viewed as a (meta)category that encompasses both knowledge and skills and need not be reduced to those components, or their combination. From this perspective, it would be an error of logic to see metacognition as defined by any list of knowledge and skills. If metacognition is a meta-category, or a category of categories of behaviour it cannot be reduced to those categories of behaviours. It is not enough to know, say, a list of skills that can help with problem-solving heuristics. No matter how metacognitive the skills are, I need some criteria for when to use them, when to stop using them and when to search for something new. Such criteria must be about the categories of behaviour, i.e., they constitute a metacategorisation

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or metaframework. In reducing metacognitive instruction to a list of skills or items of knowledge to be learnt, there is an attempt to equate a category of categories of behaviour with the categories of behaviour themselves (lists of skills and knowledge). This kind of equation is an example of what Bateson calls an error of ‘logical type’ (1972, p.301) where categories and members of a category are confused. There is a danger, in this confusion, that attention (for teacher and student) will not be placed in developing a metaframework that seems so vital in, for instance, choosing which strategy to employ or when to give up on it. In contrast, TA’s practice, by explicitly linking student behaviours to a metaframework of what it means to think mathematically, directly supports students in connecting their actions to categories of behaviour, and to ‘mathematicians’ (connections students demonstrate in interview and in lessons).

The metaframework of what mathematicians do, with the associated categories of pattern, conjecture, counter-example and proof appear to ‘hold’ the process of students’ work in mathematics lessons with TA. I see a connection to Gattegno’s (1988) articulation that ‘[e]very arithmetical problem has a background algebra. This represents awareness of the process … algebra should be stressed wherever possible’ (p.6). For example, x + y holds the process of addition. There is, as yet, no resolution or answer – the algebra is a static expression of a dynamic process (of adding x and y) and it ‘holds’ that process. This algebraic expression is a categorization, it represents any addition of two numbers, yet it is also linked to a more abstract framework of algebraic processes in that things can be done to this expression (adding another variable, for example). It is as though TA offers her students an algebra for the process of doing mathematics, with words (such as conjecture) that are intimately linked to both actions in the classroom (such as looking for patterns and making predictions), and a metaframework of working mathematically (students do things with their conjectures, like testing them or proving them). In the same way that Gattegno suggests that wherever possible the algebra in a mathematical context should be stressed, in the early lessons of year 1 TA can be seen stressing the process (the algebra) of thinking mathematically and introducing students to a language to describe that process.

The re-framing of metacognition I have offered, which arose from the analysis of one classroom, suggests a resolution to the teaching dilemma that follows from seeing metacognition as split between knowledge and skills. Metacognitive learning need not be seen as an inner event that is either an item of knowledge or skill, declarative or procedural, knowing-that or know-how. Rather, metacognition can be seen as a way of teaching and learning in the classroom, which encompasses both knowledge and skill through focus on a meta-level that transcends the distinction (in this instance, what ‘mathematicians’ do). There need be no dilemma, as a teacher, about choosing to develop either metacognitive skill or knowledge if attention is placed, instead, in offering students an explicit meta-level language to describe and support their learning. In Teacher A’s practice, there is evidence of how a manageably small list of words can, over the course of a year, accrue a wide complexity of

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associated actions (skills) and awarenesses (knowledge), within a context of students ‘becoming a mathematician’. In the classroom data I have presented, the teacher’s role in focusing on metacognitive aspects of working mathematically supported a transformation in classroom discourse and practice. The data from video recordings of Teacher A in year 2 confirm that the transformation apparent by the last recording of year 1 (students questioning and conjecturing) was sustained. There is evidence that the conjectures students make deepen mathematically, for example, students begin to make conjectures about how negative number arithmetic works. Their attention appears to turn towards aspects of mathematical structure as well as the constraints of any particular task.

Teacher A is explicit to students that their task is ‘working on mathematics as a mathematician’. Students are told in early lessons, ‘what you need to be doing as mathematicians is …’ and Teacher A had given students a purpose for the year of ‘becoming a mathematician’. There is a direct and obvious link here to the side of the metacognitive star of a process of becoming, i.e., Teacher A is offering students the idea that this year will not simply be about learning how to do new things but, to borrow words from Hager and Hodkinson (2009), ‘reconstructing themselves’ (p.633). They will be working as mathematicians. I am struck, re-looking at these statements, that there is nothing in them that suggests students cannot already act as mathematicians, although there is still work to be done. In other words, the aspect of the star that views learning as becoming avoids placing students as lacking or deficient. There are still skills students need to learn (e.g., to do with negative numbers or proof) but by interpreting students as already mathematicians there is no feel of a deficit model of metacognition even from the other side of the star.

Maturana (quoted at the start of this chapter) articulates his ontology in the form of a wheel or circle: living is knowing is living. In the same way, the re-framing of meta-cognition that I offer in this chapter has a circularity to it: ‘becoming’ is ‘being’ is ‘becoming’ is ‘being’ …

In the next chapter I address problem (1), the question of the metacognitive demands on a teacher wanting students to learn metacognitive practices.

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CHAPTER 9

HEIGHTENED LISTENING

[O]ne must listen with as many ears as possible (Maturana and Poerksen, 2004, p.116)

INTRODUCTION

I ended the last chapter with the challenge to address the problem of what the metacognitive demands are on a teacher interested in developing metacognition with their students. I will again draw on data from Teacher A in developing a response to this question. This data bears similarities to data from the teacher-discussion recordings and I also return to the connection between my research issues: ‘What patterns of interaction support teacher learning?; What is the role of the discussion facilitator?; What patterns of interaction get established and alter in a classroom over an academic year?; and What is the role of the teacher?’. In other words, considering the metacognitive demand on the teacher in a classroom (an issue that I have not seen addressed in the metacognitive literature) has led me to seeing similarities between the role of the teacher and the role of the discussion facilitator. In this chapter, I theorise these roles and introduce the notion of ‘heightened listening’ as a label for one common feature I see across this data.

I begin with a story.

Story 6: Extracts from a Public Conversation – ‘Being Alongside’

Laurinda: I was sitting at the back of your classroom recently and you were using what we refer to as the Gattegno chart:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90100 200 300 400 500 600 700 800 900

Your Y7 [age 11–12] group was chanting, “… three point two; three point four; three point six; three point eight; three point ten”.

Your pointer was following their chant by a fraction of a second and you looked around with nowhere to point. I remember laughing – enjoying the moment, “… four; four point two; ...”. It was five next time around …

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Alf: I have images of observing you working with students throughout the years of working in lessons together and being struck by how you somehow ‘get alongside’ the students. It is as though the ‘mathematical’ part of yourself is left in abeyance as you engage with the students’ world (of mathematics) … the issue is where, as teachers, we place our attention. And particularly with very low attaining students, any sense they get of their thinking being judged or evaluated is destructive of their capacity to learn. (Brown and Coles, 2007, p.8)

Dick: ...... three-point-six, three-point-eight, three-point-ten. The rhythmic trip along the signifiers suddenly pauses with a shift into the underlying signified. A surprise! I was moved by the way your accounts noted the reaction of teacher and observer … here was something powerful being said about the “undefined sea of whiteness which surrounds the letters on all sides” … I link the complex situation of the teacher following the chant with that of the psychoanalyst who also notes a surprising shift by attending to that undefined sea … “Like a mother you have to take whatever is slung at you”, writes J O Wisdom in an article on Bion.

And so the issue, neatly described as being ‘alongside’. Teaching involves a waiting stance like that of the psychoanalyst. The surprise has to really be a surprise – unexpected and initiated by the other. You have to take the wanting out of the waiting ... What is the waiting for? In his article, Wisdom suggested that it was for the present, in the sense that the present arises from the intersection of the future characterised by uncertainty and the past whose meaning is repetition. This is how I interpret the crucial notion that to attend to the other’s mathematics you submerge your own (to avoid that repetition?). It’s not, of course, easy … How do you prepare oneself to be able to be alongside? (Tahta and Williams, 2007, p.11)

In Teacher A’s metacomments to her students (see Chapter 8), I interpret her also as being ‘alongside’ her students, attending to their mathematics, not repeating a pre-prepared plan but able to be with the uncertainties of the future. What is she doing in these moments, what are the metacognitive demands?

METACOGNITIVE DEMANDS ON THE TEACHER

From my analysis of the work of Teacher A in year 1, it became clear that she had a focus at the start of the year on (among other things) using the word ‘conjecture’ and associated actions. The transcript below (from TA’s Lesson 1) ends with one metacomment that I have already looked at. I offer it here including more of the lead up to the comment, which was about mathematicians and the rules they use. The students had been discussing a dilemma about whether to write an answer as 099 or 99. In re-presenting this transcript, I am interested this time in how what is said

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may shed light on where TA is placing her attention, and hence on the metacognitive demands she is placing herself under.

S1: could I say what I was going to say about number twoTA: I’d love you to say what you were going to say [on number two]S1: on number two (.) when I done it I put five hundred and thirty four take away four

hundred and thirty five and I come up with oh nine nine /TA: hold on/ [TA writing on board as student in speaking] and then when I done it on the number line I checked it and I got it right (.) and when I had to turn it round and add it together (.) I didn’t know if I had to put ninety nine or oh nine nine (.) yeah oh nine nine and then add nine nine oh (.) so when I done it with oh nine nine I think my answer was ten eighty nine and when I just done it with ninety nine it was one ninety eight (1) [TA gestures towards S3]

S3: you don’t necessarily have to put the zero in front of the ninety nine (.) so if you changed it round [it’d be the same]

TA: go on (.) anybody got any comments (.) S2 S2: um (.) um on that one if no one never left off the zero all the answers would be (.)

um all the answers would be (.) um um one thousand and eighty nine so (.) um that would [ ] our conjecture

TA: go on (.) can you say what your conjecture isS2: so if (.) it doesn’t really matter what number it is so long as (.) so long as a is bigger

than c um (.) you get one oh eight nine TA: if you leave the zero on (3) [TA writing on board] so with three digits a b c if a is

bigger than c you always get one oh eight nine (.) sure? [looking at S3] (1) sure (.) what are we going to do about this (.) because this feels like it might be something we are going to have to sort out (.) so S3’s going (.) if we always put a zero in then we’ll always get one oh eight nine (.) S4

S4: when you actually do the sum first and you come up with oh nine nine (.) I reckon you should actually leave the oh on because that’s the answer you come up with (.) it’s like doing four take away two and saying oh (.) I’m not going to use two I’m going to use three (.) it’s basically changing the sum.

TA: okay any thoughts on that (.) S5S5: no (.) because even if you done something to the zero it’s still the same number (.)

ninety nine and in the sum you’re only using ninety nine and not nine hundred and ninety as well as oh nine nine [TA gestures to S6]

S6: I was going to say the same thing as S5TA: so (.) I think what you need to be clear about as (.) as somebody working on

mathematics as a mathematician (.) you need to decide what you’re going to do and be consistent about that (.) now S1’s feeling strongly you should leave the zero in (.) you might not feel strongly you should leave the zero in (.) in which case S2’s conjecture might not apply to your rules (.) so when we’re making conjectures we need to be clear about the rules we’re using I suppose (.) so you might want to add on to your conjecture S2 that (1) this is when you leave the zero in or something like that (Lesson 14/9/07)

I take it that throughout this transcript TA is hearing the content of student contributions, since her comments always seem to be a response to what came before.

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However, it is also evident this is not all that is going on. Rather than responding to a student dilemma in terms of its mathematical source (the issue about 99 or 099), she responds, in the final turn above, from the perspective of how mathematicians resolve such difficulties. I see two levels of categorising in operation here. She has firstly heard or noticed what students are saying as an example of the issue of ‘consistency’ of rules behind conjectures. This is the first layer of categorising. She also appears to be able to place this category within a metaframework of categories linked to becoming a mathematician, ‘as a mathematician (.) you need to decide what you’re going to do and be consistent about that’. So there is the initial hearing of the content of what students say, then hearing this content as being about rules and conjectures, and then the hearing or placing of this categorisation within the context of a third layer of what mathematicians do. This third layer, for Teacher A, is a well-established (though not fixed) framework (I would say metaframework) of what it means, to her, for students to work mathematically (which includes categories of behaviour such as asking questions; noticing patterns; making conjectures; and being consistent about rules). To be able to make the connections I see in TA’s last turn above, between the content of student talk, a categorisation of it, and ideas of becoming a mathematician, seems to me to entail some form of metacognition on the part of the teacher.

At the end of Chapter 8, I considered Teacher A’s metacomment about rules and consistency, in comparison to a metacomment I made in the teacher discussion about that moment in the lesson. This comparison suggested links between the role of the teacher in a classroom and the role of discussion chair, for example, that metacomments are responses to what is said, and are not pre-planned. I offer more of the context leading to my metacomment (in the teacher discussion) below and consider, as with TA in the classroom, how what is said may give a clue as to where I was placing my attention and what I was hearing.

TD: it was as if TA was listening to their discussion and then (2) because what struck me while watching it (.) was that this is actually a really sophisticated skill for the pupils to be thinking about and that by (.) I would like to talk about that in my lessons but to talk about it wouldn’t mean anything to them (.) so it’s as if it came from their discussion and you spotted that this was the issue they were having

AC: the sophisticated skill meaning/TD: /the putting conditions on and having choices (.)

where there isn’t necessarily a right or wrong but being aware of your choices and so putting conditions on conjectures (.) I mean it’s really sophisticated stuff (.) and you didn’t introduce that (.) that came from them but it was your spotting that this was their issue (.) and so it was slight re-direction back to it but (.) it was we need to sort this out and it was still their issue

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TB: because I think I remember focusing on that after a lesson that I had with this and I remembered (.) I think I remembered hearing you say and now we need to make a decision, a mathematician needs to make a decision (.) and I did use that and I said a mathematician needs to make a decision and I (.) but I think that I decided what to choose

TA: so you made the decisionTB: so I think I made the decision (.) I said probably something like

let’s use this but admittedly looking at that maybe one person has made a decision (.) and you wrote it up (.) I like the formality of it (.) it’s all there (.) I think that I just made the decision but didn’t write down that’s what we’d done as well (.) and um I remember the lesson (.) so I’m remembering that not everybody had (.) that not everybody was up to there (.) there’s a kind of yes it was spoken about (.) but I don’t think everybody was [ ]

TD: because there was this issue that belonged to the class and I think you said we need to sort this out um (.) and if it was me I would have chosen one of the options for them um (.) and that’s sorting it out for them (.) I think in some ways you sorted it out for them as well (.) but gave them the option of putting conditions (.) so you didn’t make the choice but you did sort out the issue (.) I don’t think (.) they didn’t sort out the issue (.) they were wondering what can we do with this (.) we need to sort this out as well (.) and I don’t think it was sorted out by the pupils (.) I think you sorted it out (.) in putting conditions

AC: that’s lovely (.) and it’s a much more enabling sorting out (.) because if we just sort it out by answering the issue then the next time pupils come up with this issue they have no (.) they’re in no better position to decide (.) the only resource they’ve got is to ask TA (.) but if you sort it out by making them aware this is an issue and making them aware there are consequences for each one and whatever (.) that is offering them a tool for next time they get in to that situation (.) so yeah (.) I love that TD/TA/12-11-07

In the first turn in this transcript, TD mentions TA’s listening, but then changes track after saying, ‘it was as if TA was listening to their discussion and then …’. My analysis of TA’s comment, and the different levels on which she was hearing, is perhaps a continuation of this line of thought, exploring the ‘and then’!

My contribution, in the final turn of the transcript, is a metacomment about TD’s analysis of TA. As with TA’s metacomment, it is apparent that I am attending to the content of what TD is saying, and also categorising (as an ‘enabling sorting out’). As I commented in Chapter 8, I see a difference compared to the comment of TA in

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that I do not link my categorisation of TD’s observation of TA’s behaviour to any metacategorisation or framework. The kind of comment I am imagining here, that I do not make and that would parallel TA more closely, might have been something like, ‘… and supporting students becoming aware of these issues is part of what it means to be developing mathematical thinking in our classrooms’. That would be if my metaframework had been about teachers in this department supporting mathematical thinking. As I analysed in Chapter 8, what I offer in the labelling of an ‘enabling sorting out’ is a purpose. I could also imagine a metaframework at a more abstract level and hence a comment such as, ‘… and getting to a label like an ‘enabling sorting out’ means this is the kind of thing you could work on consciously in your teaching, this is an example of what reflecting on our teaching means’. It was at this latter level that I thought about my role in running teacher discussions. I wanted to be able to label a ‘purpose’ in these discussions. So, although I did have a metacategorisation, unlike TA I do not share this explicitly.

In thinking further across these two transcripts about where attention is being placed, I am also struck by the fact that in neither case can the metacomments at the end have been planned. My label of an ‘enabling sorting out’ was a new awareness, a connection made in the moment. I am sure TA would have thought about the issue of rules behind conjectures before, but her articulation of this idea in the lesson quoted above is closely bound to the context in which the issue has arisen (i.e., linked to S2’s conjecture) and so in some sense is also created anew.

A commonality, then, across the two transcripts is that listening appears to be taking place on three levels: the content; a categorization of the content; and a link between this categorization and a metaframework (that either is, or is not, made explicit). I label such a facility of listening a ‘heightened listening’. I believe this mode of listening constitutes a metacognitive demand on the teacher.

Previous studies into listening in mathematics education have suggested there is a powerful mode of ‘hermeneutic’ (Davis, 1996, p.153) or ‘transformative’ (Coles, 2001, p.283; Coles, 2009, p.138) listening that a teacher can employ, and which implies an openness to being affected and changed by what is said. The notion of a ‘heightened listening’ is in a slightly different place, since it is only possible in relation to a set of ideas, or a framework that, though open to change, pre-dates the interaction. However, there is a sense from both TA and me that we have been sensitive to what has been said in the discussion and allowed ourselves to be affected by it so there is, perhaps, a connection to hermeneutic listening.

Davis (1996) also theorised about a mode of ‘evaluative listening’, in which student responses are categorised but in a way that feels quite different to the categorisation involved in a heightened listening:

Within the mathematics classroom, [‘evaluative’] listening is manifested in the detached, evaluative stance of the teacher who deviates little from intended plans, in whose classroom student contributions are judged as either right or

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wrong ... and for whom listening is primarily the responsibility of the learner. (Davis, 1996, p.52)

In this evaluative form of listening, the categorisation of student responses is generally into ‘right’ or ‘wrong’. What is different, however, is that (e.g., in Davis’s (1996) examples) there is no linking of this categorisation to any metacategorisation. There is also generally a binary in the case of evaluative categories (essentially, comments are judged as either good or bad) compared to the kinds of categories that might be invoked in a heightened listening. In the cases of heightened listening I have been discussing, the categories of comments are not fixed in advance and are not a closed set (e.g., my category in the teacher discussion was created in that moment). The stance of the teacher is engaged, not detached.

METACOMMENTARY

I am wary about the analysis I have offered above about listening. In the same way that I have suggested metacognition cannot only be seen either as a list of skills and knowledge, or as a process of becoming, I do not want to be read as suggesting that employing a heightened listening can be reduced to a set of procedures (listen to the content; categorise it; and link it to a metaframework) that can be done unproblematically. I know for myself that finding a way of being in a classroom that allows me to listen has been a long and on-going journey. I offer two snapshots on that journey in order to emphasise the complexity of the task. The first story, below, is taken from the beginning of my Master’s dissertation, submitted in 2000.

Story 7: Waiting and Listening

I am a teacher. I taught for some years before I felt able to use that word as a description of myself. What made the difference, I think, was learning to listen and learning to hear. From when I was a child I have always listened – around the family dinner table soaking up adult conversation – but only recently have I recognised that I did not always hear. I listened passively, what was said or done did not affect me.

To listen and to hear I need to be able to wait. I learned to wait in Zimbabwe – during a nine month teaching ‘experience’ in between leaving school and going to university (I was eighteen). There were no bus timetables. ‘We are leaving now!’ could mean one hour, four hours or not at all. I think I learned from observing others to quell my Western urges to ‘find out what’s going on’. All there was to do was wait. I later noticed a pattern – the bus would leave when it was full … I am learning that in my classroom I can also wait for students. And if I wait, I find that I may hear. But even hearing is not enough. I am the teacher in this room and I have to act. I have to find within myself the resources

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to respond to my students. Part of the journey I am on, in my teaching, involves developing this capacity to respond. (Coles, 2000, p.1)

In the first paragraph of this story, I mention whether I allow myself to be affected by what I hear, a link back to the discussion of hermeneutic listening. The capacity to wait seems to me linked to some sense of what I am waiting for. When Teacher A commented, about one student contribution (while watching with me the video of her first lesson), ‘that’s a classic in terms of the kind of shift I want (.) I want this kind of comment in a discussion’, she articulated a sense of what she wants to happen in discussions. I read in her words a belief that it is the students who must make this ‘kind of shift’ in the discussion. So, although she has clarity about what she wants, she has not decided in advance on the content of the shift she wants to occur. In a similar way, in running teacher discussions, I was clear I wanted to get to an articulation of a ‘purpose’, but had no investment in what that purpose would be about. The ideas sketched out above about heightened listening represent, for me, a further development in my capacity to respond, and I have been intrigued to consider the possibility of being explicit (as Teacher A is) about the metaframework of my work with teachers. I am no longer working with a department of teachers but now working at the University of Bristol. I work with prospective teachers and offer, below, a story from this work that gives a sense of what heightened listening might look like in this context.

In my job as a lecturer on a PGCE course, I regularly visit prospective teachers in school and watch them taking lessons. There is an established practice on this particular PGCE course of how these lesson observations and subsequent discussions are run. Before the lesson, I ask the prospective teacher what they would like as their focus for observation. During the lesson, which is always co-observed with a teacher from the school, I write down as much as I can of the detail of what is said, especially in relation to the focus. After the lesson, the prospective teacher always begins the discussion of what occurred, usually prompted by the university tutor with a question such as, ‘What felt comfortable, what felt uncomfortable, anything you would have done differently if you had your time again?’. In a recent round of observations, I was struck by the difference between how these discussions went with consecutive prospective teachers who I visited.

Story 8: Listening at University

Prospective teacher 1:

In response to the prompting question, this prospective teacher responded that he felt more confident in class than he had done previously and also that he felt the pupils had followed the directions and explanations better than before and had got themselves more involved. Following this statement, the co-observer (another teacher at the school) offered some observations and detailed some of

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what they saw as the positive features of the lesson. I then asked the prompting question again and this time the prospective teacher responded that he had picked up a common misconception among three of four pairs of pupils when going around and checking how they were doing on an exercise. The prospective teacher concluded that he would now have changed the starting activity to address this misconception at the beginning of the lesson if he had his time again.

The co-observer continued with some further reflections. At some point, I then mentioned a sequence of interactions in the lesson I had found interesting and, using my notes (for the first time), read out the words I had noted down of a question from the prospective teacher, response from a pupil, and the reply by the prospective teacher. The prospective teacher expressed recognition of this sequence, and remembered feeling uncomfortable that he had not really engaged in the pupil response. A discussion ensued that ended with him articulating an “action-point” for future lessons: ‘don’t just look for the answer you want’ when listening to pupil responses.

Prospective teacher 2:

In response to the same prompting question, this student commented on being aware that there were several moments in the lesson when she got the class quiet and had then not been able to sustain pupils’ attention. I referred to my notes and read out the words I had written down, said by the prospective teacher and pupils at some of the moments I had also recognised. There were four such moments and we were able to discuss what had been the same or different in what had happened. In one case, for example, the prospective teacher had asked for quiet, the class had responded, and the next instruction was for four pupils to come to the board at the front of the class to tell the others what answer they had got to a task. It took several minutes for these pupils to get to the front, during which time many of the others had begun different conversations. On another occasion, the prospective teacher had again successfully got the pupils to stop talking and look to the front. The next instruction had been for volunteers from four groups in the class to give answers they had got to a task. As she invited pupils to speak and wrote their answers on the board, other pupils began talking between themselves.

Having gone through the detail of the interactions during these four incidents, she then generalised the issues as being (a) thinking through the order in which to give instructions (e.g., invite the pupils to the front and then ask the class to be quiet) and (b) communicating what is the purpose of the activity (e.g., the idea had been for all students to compare and contrast the answers from the different groups – if this had been said before collecting the answers there may have been more engagement).

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At the time of these incidents, I was struck by how much smoother the conversation with the second prospective teacher had been. In the discussion, she spoke of specific incidents in the lesson, my notes supported the joint remembering of the detail of these moments and it was then possible to notice similarities and differences and the prospective teacher was able to articulate herself some key issues to think about in future planning. I see the first issue she identifies (thinking through the order of instructions) as something in the detail of her actions as a teacher and the second one ‘what is the purpose of the activity’ as itself a purpose.

With prospective teacher 1, his first comment was about himself and his observations about the students were generalised (e.g., as though the class were a single entity). I repeated the prompting question in order to try and get at more detail of his awareness of what had just happened. This did provoke description of some interactions with pairs of students (and a common misconception). The prospective teacher’s conclusions from this linked to planning. There was no indication of raising to awareness a possibility for alternative actions, to the ones he had taken, in the moment of the lesson. I then offered an incident I had found intriguing and it was during the discussion of this moment that the prospective teacher did remember a moment of discomfort and imagined some alternative actions that may have been available and could be tried in future. Again, he gets to a purpose ‘don’t just look for the answer you want’.

I see, in both these stories, a connection to my practices as both head of mathematics and the role of the discussion facilitator – my focus on the articulation of a purpose arising out of the detail of observations. It is apparent that, as in the teacher discussions and unlike Teacher A, I do not offer any articulation of this metaframework. During my work in school, there was an unstated and perhaps shared metaframework that we were all trying to develop students’ mathematical thinking. In my work at the University, I am clear that my task is to support prospective teachers in finding their own metaframeworks that will support their own and their students’ learning. My framework must be unambiguously about the prospective teacher’s learning hence, in working with teachers, I am not convinced that articulating this metaframework would be helpful in the way it seems to be when working with students.

In summary, in this chapter I have suggested that the mode of listening of a teacher working on student metacognition represents a vital (metacognitive) dimension of their role and one that has been largely neglected in past research in the field of metacognition. There is evidence from the data I analysed that an effective method of occasioning metacommunicative and metacognitive practices entails the teacher or facilitator learning to bring a ‘heightened listening’ to responses.

On this view of metacognition, the difference in a classroom between teacher and student metacognition is one of awareness. For students to become successful mathematicians in Teacher A’s classroom, they must be able to act according to a particular metaframework (i.e., they must conjecture, find counter-examples and so on). Teacher A must be able to do this as well. Teacher A is a mathematician, as are

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her students. What is different is that the students do not need any awareness about this metaframework – they need to use it (as indeed is also the case for the teachers). Teacher A needs to be aware of when certain actions are within the framework and when not. The metaframework needs to be something Teacher A can use to categorise in a way that is not necessary for her students. When students are in mathematics lessons with Teacher A, the framework will support effective action. For Teacher A there is an extra level of awareness, which brings with it a choice about what aspects within the classroom to stress and what to ignore.

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CHAPTER 10

CONCLUSION

In the end, all we have are our stories. (Tahta, 2006a, p.239)

LOOKING BACK

In the first section of this book, I set out my research agenda in the following terms:

– how can discussions of video occasion teacher learning? What patterns of interaction support teacher learning? What is the role of the discussion facilitator?

– how can classroom discussion occasion mathematical thinking? What patterns of interaction get established and alter in a classroom, over an academic year, that support mathematical thinking? What is the role of the teacher?

– how are the roles of discussion facilitator and teacher the same or different?

In Chapter 5, I analysed five elements of the role of the discussion facilitator, in using video to support teacher learning: setting up discussion norms; starting with reconstruction and moving to interpretation; re-watching the video; metacommenting; and selecting a video clip. These aspects (particularly the first, second and fourth points) put constraints on the kinds of patterns of interaction that will take place.

In returning to the audio data on using video with teachers, it became apparent as I looked into the detail of the talk within, say, a period of ‘reconstruction’ that there are subtleties and shifts taking place that cannot be captured in a generic heading. One section of discussion I analysed entailed an implicit challenge to the task of reconstructing the video clip that was effectively laughed ‘off’ by the group. In another context, this challenge might have led to the task of reconstructing never being addressed. The five aspects of video use do not constitute a recipe that can be followed straightforwardly. For example, one aspect of the context of the use of video in this study was the recursive nature of the taking of video recordings, analysis of clips and then further taking of video recordings, as I highlighted in Chapter 6.

In Chapter 6, when I reported on a teacher meeting looking at two lesson transcripts, it became apparent that the context of communication, which is not necessarily visible in the transcript of a lesson, seems to be a crucial element in interpreting how one turn follows another. Different motivations behind almost identical words spoken by Teacher A and Teacher B seemed linked to the different things that occurred next.

My second set of research concerns were around using discussion to support mathematical thinking. In Chapter 7, I analysed the classroom of Teacher A in which

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there was evidence that there may be a significant role for teacher metacommunication about aspects of talk, in terms of setting up new patterns of interaction.

The data from Teacher A’s classroom and my enactivist convictions led me towards a re-framing of metacognition, away from seeing it as a set of knowledge or skills, towards a ‘star’ (Figure 1.1), i.e., both a set of skills and a process of becoming.

I also set out on this book, wanting to look at any connections between the two sets of research concerns. In Chapter 9, I introduced the notion of ‘heightened listening’ that seemed to be one common feature of the way I ran teacher discussions and Teacher A ran discussions in her classroom. The notion of heightened listening addressed a gap, identified in Chapter 8, in the literature on metacognition around accounts of the metacognitive demands on teachers of metacognition. I saw a key role for heightened listening for a teacher or facilitator wanting to occasion metacommunicative and metacognitive practices with others.

Looking back across the last 9 chapters, I am struck by several places in which I have introduced a three-level distinction based on some kind of logical categorisation between events, categories of events and categories of categories of events. The distinction I have used throughout this book between a comment in a discussion and a metacomment (about something in the discussion) is an example of this kind of logical shift from an event, to the placing of the event as a member of a category. I have summarized these different classifications in Table 10.1.

By drawing together these different distinctions, I am not suggesting there is a hierarchy or that one column is better than any other, nor that there is any simple mapping across rows in the table. There are some connections between rows, however and some contexts in which I might want to privilege one column over another. An example of how there is no easy mapping across rows is that my comment about an ‘enabling sorting out’ I classified as a ‘purpose’ (category level) and also a ‘metacomment about teaching strategies’ (meta-category level). Perhaps more important than positioning within this table is the sense of a shift from one column to the next, for example, articulating a metacomment, labelling a purpose, or moving from reconstructing a video clip to discussing teaching strategies.

Yet there are some suggestive links between rows. I am particularly struck by the connection between a listening that categorises what is said and Rosch’s basic level categories. For Rosch (1999), basic-level categories are the easiest distinctions to be made in the world. I wonder whether listening in terms of categories (which include evaluations) may also be easier than either listening to the detail of what is said, or linking what is said to a metacategorisation (such as ‘becoming a mathematician’). There also seems to be an important distinction to be made about different kinds of listening in terms of category. Heightened listening is a listening to the category (and to the content), but in a way that is linked to a metacategorisation. Or, as I wrote about in the last chapter, there seems to be a difference between a listening that categorises what is heard into a binary (such as in evaluative listening) or closed

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set of categories, and a listening that allows what is said to alter the categories themselves (e.g., by creating a new one).

Table 10.1. Different types of logical levels

Categorisation Event level Category level Metacategory levelBrown’s (2005) levels of comments to teachers

Behaviours Purposes Abstract/ philosophical

Rosch’s (1999) levels of categories

Detail/behaviour layer

Basic-level category Superordinate category

Levels of classroom communication

Student answers Teacher metacomment about student answers Student conjecture (about student answers)

Teacher metacomment about student conjecture(s)

Levels of teacher meeting communication

Reconstruction of events in a video clip

Metacomment about reconstruction Description of a teaching strategy (i.e., category of teacher behaviour)

Metacomment about teaching strategies

Metacognition as offered by TA

Student mathematical behaviour

TA’s labels, e.g., making conjectures, finding counter-examples, proof

TA’s labels linked to ‘becoming a mathematician’

Levels of listening Listening to the detail/content of what is said

Listening and categorising what is said

‘Heightened listening’, categorising what is said, in relation to a higher level categorisation

LEARNING TO LISTEN

An aspect of enactivism that has bubbled beneath the writing of this book but never surfaced fully is its spiritual dimension. Varela was a Buddhist. Rosch (2008) has written about the importance of the spiritual notion of ‘beginner’s mind’ (p. 135). The idea of heightened listening is touching on such dimensions of experience. There is a connection here also to a quotation I have lived with for many years (as a purpose informing my own teaching) that comes from Bion and his injunction that a psycho-analyst enter the therapeutic session avoiding ‘memory and desire’ (1970, p.34). I read this not to imply that as a teacher I can enter the classroom with

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no planning but that I need to enter with no attachment to that planning and where we might end up (no desire) and with no limit placed on what I think students can do based on their past (no memory). Employing a heightened listening, as I think of it, is consistent with these principles in that there is necessarily a responsiveness to students that cannot be pre-prepared. Varela (1999) wrote of the Buddhist idea of emptiness which is linked to an absence of the experience of an ‘I’ controlling actions. In an earlier publication, the loss of a sense of ego is linked to the capacity for concern for others:

The possibility for compassionate concern for others, which is present in all humans, is usually mixed with the sense of ego and so becomes confused with the need to satisfy one’s own cravings for recognition and self-evaluation. The spontaneous compassion that arises when one is not caught in the habitual patterns … is called supreme (or transcendental) generosity. (Varela, Thompson and Rosch, 1991, p.249)

I take it from this quotation that, in the spontaneous compassion of supreme generosity, there is no longer the mix with a sense of ego and the need to satisfy one’s own cravings. I wonder whether Teacher A’s placing of all students, from the start of the year, as ‘mathematicians’ helps them avoid a ‘craving for recognition’. There seem to me connections here between the compassionate concern and avoidance of habitual pattern mentioned in this quotation and heightened listening.

The notion of a heightened listening and the link to compassionate ways of being, inevitably brings with it a question of how a facilitator might learn to pay attention in this manner. From this study, I am only able to reflect on my own learning. As I began collaborating with Laurinda, in the 1990s, a significant learning point in my own teaching was recognising how different my classroom could feel when there was something ‘other’ for students to be thinking about or working on, as well as the doing of any task. So, for example, when working on the problem ‘Dice Games’ (see Appendix 1) I might metacomment to the class that the important issue is: how do they know they have found all the routes? Or, when doing the task ‘1089’ (Appendix 1) if several students were having difficulties subtracting numbers, we might as a class have a focus on subtraction methods for a lesson, or part of a lesson. In this instance, my metacomment might be that the key is: try out at least two methods and find one that works for you. These metacomments give students a ‘purpose’ for the activity. There is a reason to engage beyond getting right answers or the completion of a certain number of questions. I would state these purposes explicitly and repeatedly to students. They needed to be easily stated so they could be kept in mind, to inform actions. When there was a purpose that seemed to ‘take hold’ with students, I felt different, with more conviction about what I was offering students, more conviction that this activity was worth their while to do. Having a purpose for the activity meant that my attention began to shift away from whether students were getting answers correct, onto issues such as ‘can they be sure?’, or ‘do they each have a method?’.

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After a few years of working with purposes in my own teaching and trying to find purposes for activities to offer students, Laurinda and I talked about the idea of offering students an over-arching purpose for the year. The first time I tried this, I told students that the task for the year was ‘being a mathematician’ and I began using the word ‘theory’ as consistently as possible to describe students’ ideas. This language shifted the next year to ‘becoming a mathematician’ and the use of the word ‘conjecture’. When I was head of the mathematics department, we agreed this was common language we would use across year 7 (age 11–12) classes. There would still be purposes for individual lessons (e.g., linked to finding out ‘why?’) but underneath was the over-arching purpose of ‘becoming a mathematician’ and this was practice that Teacher A had clearly adopted. My attention began to shift again, not only to be mindful of the individual lesson’s purpose but also the more generic issues of ‘becoming a mathematician’. In the departmental handbook, we had an image of concentric circles to try and get across this idea that there might be specific purposes or objectives for an individual lesson (the inner circle), that might fit into slightly wider purposes over a sequence of lessons, that fit inside the overall purpose of the year.

I remember once sitting at the back of my classroom as a prospective teacher from the University of Bristol took a lesson, working on a task in our scheme of work where year 8 (age 12–13) students had to pick a matrix and investigate the geometric effect of applying that matrix to every point of a shape of their choosing. A student put up their hand and made a comment along the lines of: ‘I’ve noticed that when I try the matrix 2,0,0,2 my shape gets two times bigger’. The prospective teacher acknowledged this comment and invited another student to speak. I remember an almost violent emotional reaction. Everything inside me was wanting to pick up on this student comment, to work with the student or class to turn what they had noticed into a ‘conjecture’, to generalise in order to end up with some predictions (e.g., what might the matrix 3,0,0,3 do to a shape?) and to link this to the idea of becoming a mathematician. Thinking back on this incident, I recognise that it shows how attuned I had become to noticing student comments that could be generalised. In this noticing, I interpret a heightened listening. There is the content of the student comment, there is a classification (it could become a conjecture) and there is a linking to what it means to be a mathematician.

Given these classroom experiences, it is perhaps not surprising that I was interested in the OU video method of work on video, which starts by reconstructing the events. This idea of ‘reconstruction’ acts as a ‘purpose’ for the activity (for the teachers). My attention, as facilitator, shifts from the content of what teachers say, to a monitoring of the ‘category’. Over time I needed to attune myself to notice when a teacher is reconstructing and when there is a move into interpretation or judgement (in order to stop such a move!). This process, of learning to attend to more than the content of what teachers say, is on-going and there are things I see now, in transcripts of teacher discussions, that I did not comment on that I think I would do now.

Whereas a spiritual compassion or mindfulness is a way of being that I recognise coming in an out of, in the living of my life, a heightened listening in a classroom

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or teacher meeting feels more to me like something I am unable to ‘turn off’. Having worked on myself to become aware of different levels in student and teacher comments, these are awarenesses that feel as though they are, in some respects, always available (whilst at the same time, I am always learning to notice more). I may choose to ignore or not act on my noticing and perhaps over time this would mean I ceased to notice, but at the moment it feels as though I cannot not notice when a teacher, for example, slips into judgment and evaluation. I am interested in gathering stories of other people’s learning, such as the brief one I have just offered and continuing to explore the role of facilitators of teacher development.

It feels fitting, however, to conclude this book with a story, ‘[i]n the end, all we have are our stories’ (Tahta, 2006a, p.239). The one below is credited to The Hodja. These tales were quoted in the mathematics education publication I have probably used and thought about more than any other (Banwell, Saunders and Tahta, 1986). The tales are Sufi in origin and have been linked to the ‘crazy wisdom’ of traditions such as Zen Buddhism; I find them invariably salutary. I have been at pains throughout this writing to avoid the impression that I am arriving at somewhere definite, that I have a recipe. The relative unimportance of recipes and a reminder to focus on where the action really takes place, are two pieces of wisdom I interpret in the story below.

Story 9: The Recipe

The Hodja was walking home with a fine p iece of liver when he was met by a friend.

‘How are you going to cook that liver?’ asked the friend.

‘The usual way,’ said the Hodja.

‘That way it has no taste,’ said the other. ‘I have a very special way of preparing a very tasty meal with liver. Listen and I’ll explain.’

‘I am bound to forget it, if you tell me,’ said the Hodja. ‘Write it down on a piece of paper.’

The friend wrote out the instructions, and gave them to the Hodja, who continued on his way home. Before he arrived at his door, however, a large crow swooped down, seized the liver in it claws, and flew high up into the sky with it.

‘It won’t do you any good, you rogue!’ shouted the Hodja triumphantly waving a piece of paper. ‘I’ve got the recipe here!’ (Banwell, Saunders and Tahta, 1986, p.4).

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APPENDIX 1

TASK: 1089

The Beginning of Year 7

We begin by articulating the purpose of year 7 for the students as being about becoming a mathematician and thinking mathematically, i.e.,:

• thinking for yourself and so not asking the teacher if things are right• noticing what you are doing, e.g., patterns• asking why patterns work• writing down everything you notice• being organised• doing things in your head.

We want to establish a purpose for the year that is removed from the content level of what we do in class and is an easily stated label that can gather complexity and meaning for each individual as the year progresses. We believe the purpose of ‘becoming a mathematician’ supports students in becoming aware of what they do when working in a mathematics lesson, by allowing them (and us) to question and reflect on whether something they (and we) do is mathematical or not.

Lesson Extract 1:

I issued the following instructions, at the same time going through an example on the board:

Pick any three digit number with 1st digit bigger than 3rd 7 5 2 Reverse the number and subtract – 2 5 7 4 9 5 Reverse the answer and add + 5 9 4 1 0 8 9

Several comments were made by students that they also got 1089 and the challenge I gave to the class was: ‘Can you find a number that does not end up as 1089?”

There are several reasons behind the choice of this activity as the first one with our year 7 groups. It is an activity we are familiar with but more important for our purposes is the fact that it is self-generative. By that we mean that, having set up the task, we do not need to direct students as to what numbers to try out – they can generate their own examples to try. Also, some students typically become convinced very soon that all answers will end up at 1089, so if a student gets a

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different answer they can check their working with someone who thinks that is impossible, so the task becomes self-checking amongst the students. Both these elements leave our attention free to eg, notice aspects of mathematical thinking that we can highlight to the group and also gives the students an immediate experience of having to ‘think for themselves’ which we had said was part of ‘becoming a mathematician’.

One of the tasks we set ourselves for the year is to comment as much as possible on activity that we consider to be mathematical. Below are two examples, from a first lesson.

Lesson Extract 2:

‘This group noticed something about their answers – it proved not to be 100% correct but it’s an example of what it means to think as a mathematician.’

‘This group had an idea which they wrote down and tested and found it didn’t work so they changed their idea, that’s a great example of what it is to think mathematically.’

Commenting on activity in this way is part of our attempt to set up a classroom culture in which students’ ideas and questions are valued and in which there is an acceptance that it is okay to make mistakes, since everyone can learn from mistakes. The self-checking nature of the first activity supports the development of this culture in which we, as teachers, act as a role model.

We move on to looking at the same problem using 4 digits – where there are several different answers. We set up two ‘common boards’ for students on which to write up their answers.

45519741

1520

10890 109899999

10009271

7164

8304

On the first board, students write the numbers they started with underneath the total it came to. They write their initials next to their number. If another student checks their number and agrees, they come and write a tick next to it. When a number has two ticks it is rubbed off (by the original student) and written on the second board of checked results. The challenge with 4 digits is to predict what total your number will come to. These ‘common boards’ develop the self-checking nature of the activity. The checked results board allows every student access to working on the higher-level questions of looking for patterns, for instance, in which columns do different numbers end up in?

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The following sequence took place in a sixth lesson on this activity, during which time different students had worked on different questions and different numbers of digits.

Lesson Extract 3: Algebraic Proof

With twenty minutes to go I stopped everyone and went through the work shown here, which I introduced as a way of proving what we found out for three-digit numbers. Alongside the algebra, I did a numerical example and at each stage of both the numerical and algebraic example I elicited answers for what to write from the class. I then wiped the proof off the board and set the class the challenge of reproducing it and then extending it to prove things they had found out about the problem with different numbers of digits.

a–1 b–1+10 c+10

–

+

c b a

a–1–c

a–1–c

c+10–a

c+10–a

9

9

9801

1

We are keen that students’ first experience of algebra in secondary school is one that is meaningful. We do not expect everyone (in these mixed ability groups) to follow the proof or be able to reproduce it but at the least they see that algebra can answer a question (‘why is it always 1089?’; ‘why is there always a 9 in the middle after the subtraction?’) that they have been asking and which they had not been able to answer in any other way.

At the end of this activity, we ask the class to write anything they could under the heading ‘What have I learnt?’ since arriving at secondary school – what bits of mathematics they had learnt and also what they had learnt about what it is to think mathematically. Some examples are given below of what students have written:

‘I’ve learnt that you have to think about the problem and not just do the sum. Also you have to maybe carry on thinking about the problem and see if it carries on. You could also have suggestions on why there are problems and how the problem works. It is a lot different to primary school because at primary school we just had to do the sum, we didn’t think about the problem of the sum we had to just do it.’

‘I’ve learnt mathematicians have to think quickly and solve a lot of problems. You’ve got to jot little theories down. On a lot of theories you have to write

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why. You’ve got to correct your mistakes. You’ve got to confirm things … You’ve got to explain your findings.’

‘I’ve learnt its OK to make mistakes … maths is more exciting in secondary school than in primary because … you can write on the board and make suggestions and talk about the work and write in our books.’

TASK: BOTH WAYS

A First Lesson

Drawn on the board the following image:

X3

X3

+2+2

d=

Invite a student to suggest a number to go in the top left-hand circle. Work this through, and calculate d, the difference in answers.

Depending on the classroom culture that exists within the group you may, at this stage, be able to ask for questions from the students such as: will the difference always be the same?; what happens if we x4? Otherwise, work through one more example together and invite comments. At this stage, it is likely someone will predict the difference will always be the same no matter what number you put in. Write this up as a conjecture.

Invite the class to try out big numbers, negatives, fractions, decimals, staying with x3 and +2. There is an opportunity for a common board, e.g., starting numbers where d does not equal 4.

If some students quickly become convinced that d=4 they can be invited to think about proving this, or else multiplying/adding by a different number and seeing if they can predict the difference. It is important that the operations on opposite sides of the square are always the same.

At the point where the majority of the class are convinced about d=4 in the case above, introduce an algebraic proof (or even better get a student to show the class). Students should always write down what a proof shows them. The class are probably now ready for the challenge.

Challenge: If you x and + by any number, can you predict the difference?

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TASK: DICE GAMES

A First Lesson

Draw the following on the board:

A

B

C 1,2,3

4,5,6D

E

F

G

H

I

“Start at the black circle; if you roll a 1,2 or 3 move up, if you roll 4,5 or 6 move down. We are going to roll the dice until we get to the end and I want you to predict on which letter we will finish. (Possible addition: You win 5 points if you guess right, you lose 1 point if you guess wrong.)”

Now get a student to roll and draw the route on the board.

“Okay, it ended up on ___. (Addition: Is the game fair? How many points should you get for winning?) We are going to play again and you can change the letter you are betting on if you want … Has anyone changed their letter? Why? (hear from several students) … Has anyone not changed their letter? Why? (again, hear from several students).”

After several games, students should come up with some conjectures eg, ‘you can’t end on B, D, F and H’; ‘the middle letters are more likely to win’. Push them to give reasons for these conjectures eg, ‘the middle letters are more likely to win because there are more ways of getting to them’.

At the point where there are conjectures to test, get students to play the game in pairs (have grids copied on to card and counters – they can choose more than one letter when playing in pairs). They should keep a tally of which letter wins, and class totals can be collected after, say, 5 minutes.

“Okay, we had several conjectures, in total the class have played the game 100 times (for example). Is there evidence for or against the conjectures?” (Discuss results.)

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It could be at this point, or for homework, students write up their interpretation of the class results. Or, the class may be talking about the different ways of getting to each letter, in which case, in pairs, they should work on finding all the different routes. (The class must do this at some point.) After 5 or so minutes this will need to become a class discussion with the aim of agreeing on all the routes there are. The teacher will then need to show how these can be converted into probability fractions.

Where This Can Go

It is important with the first grid, as a class, to agree on the number of routes to each letter and to derive the associated probabilities. These probabilities should then be compared to the class results from playing the game (or, after getting the probabilities, the class can play again and collect results to test against the theory – try to get a multiple of 22 games in total, e.g., get each pair to play 22 times). This can be a good time to introduce (remind about) zero probability implying something is impossible and probability of 1 or 100% meaning something is certain to happen. There can be rich discussions about how individual results may be far from what is expected but that, as a class, the results are pretty close – why is this so?

Individuals can then choose to look at different size grids, play the game, and try to work out the probability. The probability must then be tested against actual results (they can decide how many times to play).

Probabilities can be collected on a common board going as far as you like. (You may decide not to collect results on B, D, etc):

2by1 4by2 6by3 8by4 10by5 … 2nbynA 1/2 1/4 1/8B 0 0 0C 1/2 2/4 3/8D 0 0E 1/4 3/8F 0G 1/8H…

This board should provide rich material for conjecturing. All theoretical probabilities should be tested against experimental ones and comparisons commented upon (this is important – getting students in to the habit of commenting on results supports their noticing of patterns).

APPENDIX 1

127

TASK: 7, 11, 13

Take any 3-digit number e.g. 830

Repeat them to get a 6-digit number e.g. 830830

Take the answer and divide by 7 using short division (118690)

Take the answer and divide by 11 using short division (10790)

Take the answer and divide by 13 (830)

Compare this with the 3 digits you started with ...

Try this for your own 3 digits. Does it always work? Can you find one that doesn’t work? Why does it work?Can you prove any of your results?

This task can, of course, be done the other way round, starting with a three-digit number and multiplying by 7, 11 and 13.

129

APPENDIX 2

The episodes across all three teacher-discussions are on the next page. I give a brief description of the working characterisations below:

• reconstruction: the collective attempt to agree on the detail or account ‘of’ what was seen and heard in the video clip (what I call the ‘text’ of the video)

• questioning text: doubt or ambiguity expressed about a particular version of what happened in the video clip; this type of episode is close to ‘reconstruction’ in that the focus is on the detail of events in the video

• negotiating process: discussion of what to do next as a group (e.g., to re-watch the video or not)

• identifying teaching strategies: describing what the teacher on the video was doing in more general terms than during a ‘reconstruction’ episode (a ‘strategy’ could include: a form of words the teacher uses when teaching; a gesture; how the lesson has been planned; how the classroom has been laid out physically; or things the teacher has done in previous lessons, such as routines or expectations established from the start of the academic year)

• naming a strategy: giving a succinct label (or ‘purpose’) to a teaching strategy that is being discussed

• linking to own practice / linking to reading: relating what has been seen or talked about to habitual or particular instances of a teacher’s own behaviour, or to previous reading of education literature

• metacomment: a comment about the discussion that has just been taking place• ascribing intention: describing the imagined motivation of someone in the video clip• questioning intention: asking a teacher, who was in the video, why they did a

certain thing; or questioning what might be in the thoughts of individual students in the clip

• questioning meaning: a teacher asking another in the discussion what they mean by a particular word or phrase.

To aid comparison across the three discussions, I have formatted them on the next page so they fit in one column each and I used the abbreviations below for labels of episodes.

Episode AbbreviationIdentifying teaching strategies Identifying t. s.Identifying student strategy Identifying s. s.Identifying an absent teaching strategy Identifying a. t. s.Negotiating process Negotiating p.Link to own practice Link to own p.Questioning meaning Questioning m.

APPENDIX 2

130

Teacher discussion on TA Teacher discussion on TB Teacher discussion on TCReconstructionQuestioning textQuestioning textReconstructionQuestioning textReconstructionIdentifying s. s.Identifying a. t. s.ReconstructionQuestioning textReconstructionQuestioning textReconstructionQuestioning textReconstructionNegotiating p.*re-watchNegotiating p.*re-watchNegotiating p.*re-watchReconstructionNegotiating p.*re-watchQuestioning text*re-watchReconstructionIdentifying t. s.Link to own p.Identifying t. s.Link to own p.Identifying t. s.Identifying t. s.Questioning m.Link to own p.Identifying t. s.Metacomment

ReconstructionNegotiating p.* re-watchReconstruction* re-watchReconstructionReconstructionQuestioning textReconstruction* re-watchReconstructionReconstructionNegotiating p.* re-watchAscribing intentionReconstructionMetacommentIdentifying t. s.Identifying t. s.Identifying t. s.Identifying t. s.Naming a strategyIdentifying t. s.Identifying t. s.Identifying t. s.Ascribing intentionsIdentifying t. s.Link to own p.Questioning m.Link to own p.Link to others’ p.Identifying t. s.Naming a strategyLink to own p.

ReconstructionNegotiating p.ReconstructionNegotiating p.* re-watchReconstructionQuestioning textQuestioning text* re-watchQuestioning intentionReconstructionAscribing intentionIdentifying t. s.Naming a strategyIdentifying t. s.Identifying t. s.Linking to p.Identifying t. s.Questioning intentionIdentifying t. s.Link to readingLink to own p.* re-watchReconstructionLinking to own p.Ascribing intentionIdentifying t. s.Link to own p.Identifying t. s.Questioning intentionIdentifying t. s.[Writing for Mathematics Teaching]Identifying t. s.Identifying t. s.Questioning intentionLink to readingLink to own p.Questioning m.[Year 8 scheme of work]Identifying t. s.

An example of a full transcript of a teacher discussion (on TA) is below. Episodes are coloured alternately in two grey scales.

APPENDIX 2

131

Time Who? Dialogue Episode00.00 AC (7 sec) that was quite long

but I think it’s still worth doing. So, would anyone like to suggest or remind us where it beganwhat the sequence of things that happened there and we can maybe go back and look at things again if we need

RECONSTRUCTION

00.20 AC see if we can re-construct as much as possible00.29 TA (unclear) I started writing it down

but then I gave up[laughter]

00.32 TG I don’t know if it’s the first thingbut the first thing I remember is the discussion you had about the things you should do if you don’t get one of these answersand I wrote down three suggestionsthe kids came up with three thingsI can’t remember whether that was thewhether that was (.) yourinitial comments were about these answersI think you said thatyou might have said that actually [laughter] when you were telling us how the lesson startedand then it came in with TA saying that I think

1.14 AC yes (.) was there something about what are you going to do if you haven’t got /one of

TG /one of these1.18 AC one that’s circled or something

TG yesTR there was some sort of voting about the six sums

1.23 AC that’s right they voted and I think1.29 TG so if you don’t get one of these (.) what can you

do1.32 TR one was use a calculator to check it1.38 TA that was the third one

[various voices]1.42 TG the number line1.43 TW was that the first suggestion QUESTIONING TEXT1.46 TA that was the second one

I’m not sure what the first one waswhat was the first one?

TG the first one/TB /do it again

1.53 TA no, I wrote down / ./I think I said thatTG

APPENDIX 2

132

Time Who? Dialogue Episode1.56 TD oh right1.57 TR wasn’t another check against a number line2.00 AC or use a calculator2.02 TA use a calculator was the third one

TG no, no, that’s right the first one was ‘do it again’, you said, you said

2.07 TA I said, or you could speak to the person next to you

TG2.11 AC which I don’t think had come from them/

TA /noTG /that’s right

2.13 TA so they said; do it again, check on a number line, and use a calculator

2.19 AC and you added/TA /and I added, speak to the person next

to you2.24 TA I don’t know if I repeated all three of those

things or just couldn’t remember; I know I added something, I don’t know if I repeated all three as well

QUESTIONING TEXT

(12 sec)2.48 TB [are they doing] something about on a

conjecture, with you focusing back on a conjecture. because I know after that someone said about zero nine, nineor was that a long time after that

RECONSTRUCTION

3.03 TA I think /TD /started talking about how she did the sumTA she

3.13 AC and did that just come from the pupilor had TA said something

QUESTIONING TEXT

TB no I think that had come from the pupil3.19 TR I think it just came from the pupil3.20 TB but you were talking to another pupil at

the time and then they put their hand up and said when I did this I can’t remember what you were saying

3.30 TG because she said something before she started talking about itshe said can I (.)I couldn’t make it out it was either

APPENDIX 2

133

Time Who? Dialogue EpisodeTA I’ve written can I sayTG can I say

and it was almost as if it was about something elsecan I sayI wasn’t sure if she said can I say something about that group or as if it was some other

AC oh it was the second one did she say something about the second one

TG the second onethe second oneit might have been that yes

AC and you said something likeI’d love you to say something about (.)did you (.) was that

RECONSTRUCTION

TG yes (7)4.10 TA and then she began to say something

AC and then she spoke for a long time didn’t she

TB she was saying it could b: :ewhen I did it I didwhat numbers did she sayfive three four

AC yeahTB yeah

and it changed to four three five[laughter]

TB and they took it away//TG //what was interesting

was that when she took it away she got zero nine nine (.)and (.) and then she saidand if I then reverse that I’m not surewhether I should write ninety nineand then went on to saywhat (.) that was interesting thatyou don’t see students (.)they either write zero nine nine or write ninety nineand it’s not a decision that they usually makethat was what was interesting (2)

IDENTIFYING STUDENT STRATEGY

APPENDIX 2

134

Time Who? Dialogue Episode5.15 AC yes they’re usually not aware there’s a decision

therethat’s right isn’t it (.) yesand that’s something we could flag up isn’t it as teachers if somebody says that kind of thingit’s like a possiblean opportunity for some kind of metacomment thereisn’t there aboutthat’s great to be aware you’ve got a choice thereyeah it’s great to be not be sure

IDENTIFYING AN ABSENT TEACHING STRATEGY

TD it was flagged up [at the time]well it was talked aboutmaybe not the awareness of the choice being therebut (.) there was talking about what happens when there is a choice and we’ve got different people choosing different things

AC yes she did actuallyTD and you went and wrote on the board

that was quite near the endand you added to the conjecture umwhat’s the word

RECONSTRUCTION

6.08 TB modifyingTD you modified it but you

you set (.) what’s the word umTA conditionsTD conditions on it (.) on the conjectureTW when you put a b c on the board

had you already spoken to them about a b c[like number one two three or not or did you just]

TA well I think that didn’t come from me I think QUESTIONING TEXTTW what the a b c didn’t come from you

6.36 TA well originally it didbut I think it that lessonthere was a girl that said

TR yeah it was Jodie RECONSTRUCTIONTA it was JodieTR she just wentTA she said

if a is bigger than c

APPENDIX 2

135

Time Who? Dialogue EpisodeTW she said if a was bigger

because I wasn’t quite able to hear what she saidthat’s why I wasn’t sure

TA so it mustI guess we must have talked about that before handmaybe inI think that was the second lessonso it must have been in the first lesson we talked about that (5)

7.00 TB are youI don’t know when you did itbut when the first person was talking about zero nine nineand nine nine zeroyou wrote something upand I can’t remember if that happensare doing that at the same timeor had they finished and then you put it upbecause I got the senseof either it was you two working together and you put it up at the same time or (.)um you were kind of translating what she was saying

AC it might be an [item]we can go back and possibly look at that clip againI mean has anybody got a sense of thatwhat happened next as she was talkingyeah she was talking about your five three fourand TA was writingin some kind of synchronisationI’m not quite sure whatwhat what’swhere did it go from there (1)does anybody have a sense about TB’s questionwere they in step or was TA translating

(NEGOTIATING PROCESS)QUESTIONING TEXT

TR yeah I felt like Ayesha was kind of dictating to youand you were writing down

RECONSTRUCTION

AC right

APPENDIX 2

136

Time Who? Dialogue EpisodeTB but it’s not

she didn’t write exactly what she saidbecause she was talkingand she was sayingso likebecause it seemed like you were writing zero nine nineas she was sayingI wasn’t sure whether to put in a zero or notand at that point you’d written nine nine zero (1)I don’t knowso she didn’t sayput in nine nine zero

8.20 AC yeahTG she was quite detailed

because she then saidcos when I did nine nine zeroI got one oh eight nineand when I did ninety nine I got

TB yeah yeahTG you know she was actually quite

she was very fluentin her description of what she found (3)

AC and there were some interesting things that I can’t quitewhat happened at the end of what she was sayingbecause I was fascinated by the transitionsgo on why

TR well I justafter she finished you didn’t really commentsomeone else came backshe didn’t really say anythingsomeone just

TB but I could see that you were (.)when someone said something you would not (.)um (.) you would kind of like gosometimes you’d nod but you wouldor just say nothing or say yeah or somebody elseor anybody want to make a comment on that you said sometimes

9.20 TG I think you said any commentsI think you said any comments

TR the first she didn’t say anything the first timeshe just kind of finishedand someone else just [jumped in]

APPENDIX 2

137

Time Who? Dialogue EpisodeTG did they they might have done I [laughter] QUESTIONING TEXTAC has anybody else got a sense of that TD I think remember both happeningTA I think I remember both

I don’t know which order it isTR I think once she’d done it

it was quiet and someone else said somethingthen after that you said any comments

9.50 AC I remember you gesturing at a couple of pointswhich felt like it was with no wordsif somebody had spokenand I just got this sense of you slightly learning back and [ ]I don’t know gesturing to somebody else (.)but I’ve no idea what that second person saidat the end of Alisha’s

TA no I can’t remember who it was or what they said

TD and then I thinkthat almost (.)I don’t remember what that person said and itI remember it as a distraction which is

AC ah that’s interestingTD and then I remember you saying

but what about thisthis seems like something we need to sort outor something like that (.)put the zero nine nine or ninety nine and that//

TG yes because the conjectureTD and I remember it lead in//TG //because the conjecture bit came before didn’t

it (1)the a b c conjecture came before

AC before whatTG before TA then said

this is somethingthat as a mathematicianwe then need to (.) we need to sort outand people came back then withshe might have said any comments then

TD there’s a pupil saying we should really leave the zero because

RECONSTRUCTION

TA that’s rightTG that was lovely wasn’t it

APPENDIX 2

138

Time Who? Dialogue EpisodeTD because if you take off the zero it’s like if you

had fourand then just changed it to a three

TA that was Alisha again wasn’t itTG was itTA wasn’t that her againTD I don’t know who was who

11.15 AC and didn’t some boy come back and saybut if you take a zero off ninety nineit’s not changing the number or something

TG yesTA yeah he was quite detailed about his response

to thatAC yes yesTA yeah that’s rightTB using detailed reasoning

[laughter]AC and you’re right at some point TA then said

well this is something we need to sort out as mathematicians (1)

TD I remember that being at the startbefore this discussionand this was then sorting it outbut I think (1)that it got raised as an issuethen there’s some distraction I rememberand I remember you bringing it back cosbut what about this again we need to sort this outand then that led in to

TA modifying the conjectureAC then the pupils sorting out

well talking about the options12.06 TG yeah that’s interesting

because I think the distraction was the girl saying the conjecture

TA yeahTG about a b c which which

but that then became relevant again becauseyou were then able to say um if if you are happy with it being ninety nineor nine nine zero then that means you can work with this conjectureor I don’t know what your words weredevelop the conjecture add conditions to the conjecture I can’t remember

APPENDIX 2

139

Time Who? Dialogue EpisodeTA modify I thinkTG modify it (2) yes (1) yes (3)AC shall we quickly look at that clip again

just that little bit that we’re talking about againand then I think we canthink about what teaching strategies TA was usingbecause one of the things we talked about last timewas conversations that flip out of the sort ofteacher student teacher student patternand it seemed to bethat was happening today as well

NEGOTIATING PROCESS

13.30 [re-watching clip]AC do we need to go a bit earlier NEGOTATING

PROCESSTA did you want to see how I wrote [ ]

I think you need to go back[continue re-watching]

TA it’s before that NEGOTIATING PROCESS

AC do you want the bit beforeTA do you want to see the synchronisationTB I can’t see the [ ]TG there we are

[continue re-watching]15.20 AC let’s leave it there

we may well want to see for a bit longerbut it seemed to me there were quite a few issues we were talking about there (.)what was your sense of the synchronisation TB there

RECONSTRUCTION

TB that she was talking about itas you were writing itbut she corrected herself seeing what you’d written (.)because she wentum yeahand she read it off what you’d writtenthe zero nine nine [ ]it was what she was sayingbut I think when she said it the first timeshe was getting a but muddled

APPENDIX 2

140

Time Who? Dialogue Episode15.55 AC and you’d already written a thing underneath

ninety nine add ninety nine (.)my sense was you’d written oh nine ninenine nine ohas she was talkingand then you wrote the ninety nine plus ninety nine

TA and then I wrote the answers in after thatshe was saying [ ] (1)

AC and it wasthere was no comment there was thereso then at the end of thatand I think there were even two comments after

TA no there was a comment after that oneAC but not from youTG there was somebodyTA no not from meAC no that’s what I mean sorryTG yes so

did somebody say (.)so you should put the nought in because you get one oh eight nineis that what they said

TA I think they said you should put the nought inbecause it’s a different number if you don’t

TG right okaysomebody said about putting the nought in didn’t they

TA because that’s the number you get or somethingTG right (.) and thenAC I think there may even have been another

studentsaid something as well wasn’t there beforeand then you said at one pointany comments

TG //and that’s when the girl saidTA //I said go onTR //someone was going to sayTA and then they said [ ]AC and then they didn’t

oh okayTA and then I said are there any other commentsAC and then we had the girl with the conjecture

yeah

APPENDIX 2

141

Time Who? Dialogue EpisodeTA she said something like

with that oneand then she didn’t reallyand then she saidshe said something I couldn’t really hearand then she saidand that’s a conjectureand then I think I said can you say what the conjecture isand then she said itand then I wrote itor started to write it down

AC and she did introduce the ashe said a greater than cit did come from her

TG there was something interesting I noticed as wellwhich was when that girl was talkingpossibly because you couldn’t quite hear herbut you movedand what was interesting was watchingthe other pupils in the class[either] they moved with youbut they certainly were listening to that pupilyou could see them turn and listen to that pupil talkand I wasn’t sureI was interested if that was because of you moving

TB you’d be able to tell because she started talking before

TG yeahAC yes I was really struck by some of them turning

round at one pointparticularly the girl who’s conjecture it wasshe turned right round to look back at her didn’t she

TB was that because you’d movedI’d also be interested to see

AC do you want to seejust check that again

NEGOTIATING PROCESS

TB yeahI mean I guess we know that it’s probably more likely

TG but it’s interesting that isn’t it[start playing clip]

TA no I’ve got to be the other side of the board[clip playing]

APPENDIX 2

142

Time Who? Dialogue EpisodeTA you’ve got to make me the other side of the board

[clip playing]TA haven’t I already moved QUESTIONING TEXTTG no noTA no oh okayTG here you go

[re-watching clip]18.55 TB they move because you move

[laughter]TG look how far they’re turning round

[clip continuing]AC yeah that’s fascinating

are they following TA or following the erRECONSTRUCTION

TA yeahAC or following following were it’s coming from

[laughter]TG there’s something interesting about that isn’t

therebecause in order to get the conversationthe trouble is we’re there aren’t weand you talk about this conversationand physically you can see why the conversation goes like that

TA I’m quite often drawn to the person who’s talkingso I’ll often sort of move towards them

IDENTIFYING TEACHING STRATEGIES

TG yeahTB I think in your classroom it’s easier

I’m looking at mineand I think thatbecause you’ve got two sets of threeI don’t think they’re necessarily going to the endmaybe they are on this sidebut on the other side there’s a gapso you can walk roundso it’s like you can get in to the middle of the roomso I think whereas here I can’tif someone’s spoken over thereme going to stand over there would mean that um I’m in front of somebody else to the board or somethingso I can’t stand here and talk to that personbut if there was room on other sideI think I could go and stand

LINK TO OWN PRACTICE

APPENDIX 2

143

Time Who? Dialogue Episode20.14 AC interesting

so we’re moving yeahthis is where TA doesn’t say very much nowwhat were the teaching strategies that she was employing there? There was a lot going on, and we’ve said one about the classroom organisation; what else was TA to, as you say, well, do whatever was happening there; generate a discussion where they were listening to each other, and she was able to metacomment

IDENTIFYING TEACHING STRATEGIES

20.38 TR She didn’t personally get involved in what they were doing ... because I know I would have sort of ‘look at this’, you know I would have commented [unclear] I just know I would have, especially when it got down to the bit where it was 99 or 099. I just know I would have jumped in.

LINK TO OWN PRACTICE

21.02 TB It feels like a battle doesn’t it between wanting to say ‘yes, so what you’re saying is de de de’ ... um, and instead of saying ‘that’s a really good thing’, saying ‘yes, you’re completely correct’; she’s kind of, you’re not saying anything; your offer that you’re writing is on the board; you’re kind of being there more as a support rather than/

LINK TO OWN PRACTICE

21.29 TR /as someone who comments on other people’s work /

IDENTIFYING TEACHING STRATEGIES

21.33 TB /until you got to talking, ‘as mathematicians’; we need to make a decision ... do you decide on what to say, no but you still put somebody else’s comments on the board

IDENTIFYING TEACHING STRATEGIES

21.46 AC Yes, the comment was at a level above, it wasn’t a comment about the actual work, it was a comment about how they’ve got to organise their work and you weren’t saying, she wasn’t saying which one to do just that they need to be clear.

22.02 TD It was as if TA was listening to their discussion and then ... because what struck me while watching it, was that this is actually a really sophisticated skill for the pupils to be thinking about, and that by; I would like to talk about that in my lessons but to talk about it wouldn’t mean anything to them; so it’s as if it came from their discussion and you spotted that this was the issue they were having

APPENDIX 2

144

Time Who? Dialogue Episode22.37 AC The sophisticated skill meaning/ QUESTIONING

MEANING22.39 TD /the putting conditions on and having choices,

where there isn’t necessarily a right or wrong but being aware of your choices and so putting conditions on conjectures; I mean it’s really sophisticated stuff. And you didn’t introduce that, that came from them but it was your spotting that this was their issue, and so it was slight re-direction back to it, but, it was we need to sort this out and it was still their issue

23.28 TB Because I think I remember focusing on that after a lesson that I had with this and I remembered, I think I remembered hearing you say ‘and now we need to make a decision, a mathematician needs to make a decision’; and I did use that and I said, ‘a mathematician needs to make a decision’ and I, but I think that I decided what to choose

LINK TO OWN PRACTICE

23.52 TA So you made the decision23.53 TB So, I think I made the decision; I said, probably

something like ‘let’s use this’ but admittedly looking at that I that maybe one person has made a decision; and you wrote it up, I like the formality of it, it’s all there; I think that I just made the decision but didn’t write down that’s what we’d done as well; and um, I remember the lesson, so I’m remembering that not everybody had that not everybody was up to there, there’s a kind of yes it was spoken about, but I don’t think everybody was [unclear]

24.40 TD Because there was this issue that belonged to the class, and I think you said ‘we need to sort this out’, um, and if it was me I would have chosen one of the options for them, um, and that’s sorting it out for them. I think in some ways you sorted it out for them as well, but gave them the option of putting conditions, so you didn’t make the choice but you did sort out the issue; I don’t think; they didn’t sort out the issue, they were wondering what can we do with this, we need to sort this out as well; and I don’t think it was sorted out by the pupils, I think you sorted it out, in putting conditions

IDENTIFYING TEACHING STRATGIES

APPENDIX 2

145

Time Who? Dialogue Episode25.33 AC That’s lovely, and it’s a much more enabling

sorting out, because if we just sort it out by answering the issue then the next time pupils come up with this issue they have no; they’re in no better position to decide; the only resource they’ve got is to ask TA. But if you sort it out by making them aware this is an issue and making them aware there are consequences for each one and whatever, that is offering them a tool for next time they get in to that situation; so, yeah, I love that.

METACOMMENT

26.02 TG It would have been when they were doing pentominoes and they had to decide whether to include reflections

147

APPENDIX 3

Schedule for interviews with students. The questions remained the same across both sets of interviews (October 2008 and July 2009), the activity changed.

1. Thank students for their time, explain again the context of this interview, the reason for recording, and offer the option not to be involved.

2. Explain there will be an activity for them to work on mathematically and then some questions.

3. Offer activity: “You can work together on this activity. There is a starting point for you to try and then I’d like you to work mathematically on investigating the problem after that.”

4. During activity, offer no help to students unless asked. At the end of the initial task prompt students to “work mathematically” and “investigate” if they do not start doing this themselves.

5. After approximately 10 minutes bring the activity to an end, and ask students “if you had time, what would you do next?”

6. Questions: a. Think back over the lessons you have had with Teacher A – what have you

learnt in maths?b. What have you learnt about thinking mathematically and how mathematicians

work? (try to link their answers to what they have just done on the activity).c. What do you enjoy most about maths or maths lessons?d. Is there anything different about how you work in maths compared to other

subjects?e. How important is class discussion for you in maths lessons?

7. Thank students for their involvement.

149

APPENDIX 4

The five broad categories of episodes in Teacher A lesson 6 (the last one of year 1) are as follows.

1. Teacher asking: for answers/for comments/for thoughts/what students think/for ‘ones that don’t work’/for counter-examples – often Teacher A writes responses on the board; typically the episode continues either as long as the idea is discussed, or until Teacher A has collected (without further discussion) a range of ideas from students.

2. Teacher asking ‘how do we know?’ or ‘why?’ – having received a range of different answers, or perhaps after having received a single comment, Teacher A may ask students to explain or give the reasons behind an answer.

3. Student question – I take a question to be defined to how it is responded to, i.e., a question is any statement that is answered like a question; a student saying ‘oh Miss’ where Teacher A responds ‘go on’, I take as asking a question (implicitly – ‘can I say something?’, since the response is to grant permission to speak); the episode will last for as long as the idea the student puts forward is discussed.

4. Student use of ‘because’ – typically such an episode interrupts one of ‘Teacher asking for comments’, where instead or as well as giving a comment or answer, a student also offers reasons behind their response. I chose not to define a new episode if a student uses ‘because’ following Teacher asking ‘why’.

5. Teacher ends discussion – these episodes are monologues, as Teacher A comments on the discussion that has just occurred, and sets up a different activity for the class.

Having established these categories, the first (twenty-three minute) section split into twenty episodes. The first twelve were as follows:

Episode 1: Teacher asking for answersEpisode 2: Teacher asking ‘how do we know’ Episode 3: Student offers alternative reasonEpisode 4: Teacher asking for answersEpisode 5: Teacher prompting reasons Episode 6: Student questionEpisode 7: Teacher asking why Episode 8: Student questionEpisode 9: Teacher asking what students think Episode 10: Teacher asking for answersEpisode 11: Student use of ‘because’Episode 12: Teacher ends activity.

These twelve episodes were centred around the feedback on answers to the starting activity offered to the class. Teacher A then introduced a new topic (called ‘Both Ways’, see Appendix 1) and there were a further eight episodes before students were given two minutes to work on their own.

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Episode 13: Teacher asking for answersEpisode 14: Student question Episode 15: Teacher asking for answersEpisode 16: Student questionEpisode 17: Teacher asking for commentsEpisode 18: Student use of becauseEpisode 19: Teacher asking for answersEpisode 20: Teacher asking for thoughts

The second of the class discussions split into seven episodes:

Episode 21: Teacher asking for commentsEpisode 22: Student question Episode 23: Teacher asking ‘why’Episode 24: Student questionEpisode 25: Teacher asking for ones that ‘don’t work’ Episode 26: Student questionEpisode 27: Teacher ends discussion.

The third and final discussion, of two minutes, contains just one episode that is ‘Teacher asking for counter-examples’.

The full transcript from Teacher A lesson 6 is below. The clip numbers refer to the video recording; times where there was no whole-class discussion have been edited out. Other lessons were transcribed in similar manner. Episodes are coloured alternately in two grey scales.

Who? Dialogue/event EpisodeFIRST CLIP

TA okay six minutes (.) to write down in the back of your books what you think the answers are (.) back of books

SECOND CLIPTA quite challenging this (2) okay (4) okay (1) put your hands up if

you’ve got an answer to question one (3) question one (1) S1Teacher asking for answers

S1 three hundred (.) three hundred and eighty two point twoTA okay has anybody got anything different to that (1) S11 (.) no (.)

anybody got anything different to that (.) S2S2 three point eight two twoTA anything different to that (3) okay so how do we know which

one it is (.) how do we know which one it is (.) S3Teacher asking ‘how do we know’

S3 it can’t be three point eight two two because that’s smaller than twenty hundred and thirty seven

TA smaller than

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Who? Dialogue/event EpisodeS3 two hundred and thirty sevenTA two hundred and seventy threeS3 oh yeah sorryTA so (.) why can’t the answer be smaller than two hundred and

seventy threeS3 because you’re timesingTA because you’re timesingS4 the number by itS5 and MissTA hold on hold on (.) I’m not sure it’s just because you’re

timesing (.) because I sometimes think I can times and get a smaller answer

S6 but only if it’s a minusTA only if it’s a mi: nusS7 three point eight two two’s wrong because the decimal place

moves back (.) back not forward (1)TA go on what do you meanS7 because it says times and the decimal places move back because

it’s been divided by tenTA so what is it that’s been divided by tenS7 fourteenTA okay so from here to here [draws arrow on board] (1) I’ve

divided by ten (.) from here to here nothing’s happened (.) so from here to here I’ve got to divide by ten (.) a: nd (.) if I divide by ten I move the decimal place back one

Ss yeahTA so it’s got to be that one [circling one answer on board] (2)S8 [can I also say] there’s only one number in the times that’s

got the decimal point in (.) if there’s only one digit behind the decimal point there’s only going to be one digit behind the decimal point // [ ]

Student offers alternative reason

TA //do you mean behind it in that one or that oneS8 the fourTA so there’s only going to be one number there so there’s only

going to be one number there (.) that’s a good way of doing it thank you (1) okay number two (.) different answers to number two e: :er S9

Teacher asking for answers

S9 three eight two two ohTA anything different (.) S11 (2) everybody thinks it’s thatSs yeahTA okay (.) so anybody could tell me why you think it’s that (1)

good lots of hands (.) S10Teacher asking why

S10 because you’re basically doing fourteen times two hundred and seventy three and basically because the hundred and forty has a zero on the end so [you’ve that]’s got to have a zero on the end of it

Student use of because

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Who? Dialogue/event EpisodeTA so you’ve just got that extra zero so you’ve just got that extra

zero (.) any other explanation or shall we move o: :n (.) okay let’s move on (.) number three (.) S11

S11 nought point three eight two two TA okay anything else (.) all agree (.) S12S12 [I got] zero point zero [three eight two two]TA okay that’s similar (.) okay (.) I’ve got two answers there (.)

how do we know which one’s right (1) S13S13 well I think it’s zero point three eight two two because (.)

you’ve got zero point zero already (.) and you just have to take the decimal point back (.) four times

Student use of because

TA from thereS13 no (3) [TA writing] and you’ve got to do it on the answer (.) so

it’s got to go back three times (1) four times so it would be at the beginning of the three

TA one two three four yupS13 and then you put a zero (.) before the decimal point [ ]TA s: :o this oneS13 yeahTA ClaireS14 I think it’s the top oneTA go onS14 because it’s been taken back (3) five spaces (.) so you take it

back [on the other one]Student use of because

TA what’s been taken back five spacesS14 the decimal pointTA so from two hundred and seventy three five spaces (.) one two

three four fiveS? fourTA oh four spaces (.) so I have to do it the same do IS14 and you [take it] on the answer at the topTA one two three four puts it there (2) yup (.) so your extra zero

Alicia (.) because you’ve described exactly the same thing as Claire but you’ve for some reason stuck on an extra zero (.) I think if you decide the point’s there you just need to stick a zero in there (1) S15

S15 not being [ ] but I know why she’s got the zero there Student question [implicitly ‘can I explain’]

TA go onS15 because two hundred and seventy three’s only a three digit

number (1) when it was divided by like [point four] like the decimal point moved four (.) they’d have to put the zero in to make it a four digit [ ]

Student use of because

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Who? Dialogue/event EpisodeTA okay (.) what do you think Teacher

asking what students think

S? I thought like if you times’d (.) the number gets biggerTA e: :r that’s interesting because that’s what S3 was saying (.) so

S3 you said it would only get smaller if it’s a negative (2)S6 I did (1) yeah //Ss: laughter//TA but this is (.) there’s no negative numbers in here (.) and the

number’s got smaller than fourteenS6 it’s a decimalTA oh because it’s a decimalS6 and it’s below oneTA so it’s a number less that one (.) so if we multiply [writing]

by a number less than one it gets smaller (.) so it’s not just multiplying makes it bigger (.) okay (.) I think we’ve sorted it (.) S11

S11 um (.) because you divided two hundred and seventy three by ten thousand

TA oh nice (.) ten thousand okay yeahS11 you move the decimal point five (.) four places backTA five or four places (.) okay thank you (.) so we’re going to go

with that one (1) okay (.) number four (.) S16Teacher asking for answers

S16 er (.) three eight two twoTA three eight two two (.) okay (.) anything different (2) S3S3 um (.) three eight point two twoTA I’ve got three thousand eight hundred and twenty two and

then I’ve got twenty eight point (.) thirty eight point two two (.) S17 (.) no (1) something different (.) something different

S17 three point eight two twoTA three point eight two two (.) anything else (1) okay so (.) S16

why did you say three eight two twoTeacher asking why

S16 um (.) because [on the first one]TA yeahS16 it’s been divided by (.) [the decimal point’s gone] back one two

three spacesTA on the fourteenS16 yeahTA right so I’ve back three numbers on the fourteen (.) back threeS16 and on the other one it’s gone (.) forward threeTA forward one two three (.) forward three yeahS16 and because if you like (1) say you had zero and you take away

three [and add three you’d have the same]TA right so what are you doing by going back three places (.) what

are you actually doing (.) dividing b: :y

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Who? Dialogue/event EpisodeS16 a thousandTA a thousand (.) and then timesing by a thousand (1) so that’s like

doing nothingS16 yeahTA you times by a thousand and then divide by a thousand that’s

like doing nothing (.) s: :o three thousand eight hundred and twenty two (.) last one then S15

S15 um (.) um (.) it’s like (.) I think it’s what Jacob said [ ] but the number (.) the bottom two numbers (.) can’t be right (.) because if you’re timesing it’s got to be lower than two hundred and seventy three thousand (1) oh no

TA even if (.) the number’s less than o: :ne (.) I think all of these numbers are less than that

S16 [ ] because I reckon what it’s done (1) no don’t worry (.) I was going to say cos (.) if you go back three noughts

TA yeahS16 cos there’s [ ] there you might have done one point fourTA okay (.) okay (.) yeah (.) have you got an answer for number

five (.) Grantley have you got an answer to number fiveTeacher asking for answers

S11 [I think it’s] twenty seven point threeTA twenty seven point three (.) anything else (1) er (.) S16S16 two hund (.) two thousand seven hundred and thirtyTA anything else (1) so we’ve got a dividing one here (.) we’ve

got three thousand eight hundred and twenty two divided by one point four (.) so (.) like we’ve got here (.) we’ve got to do something with the two hundred and seventy three (1) go on S1 what were you going to say

S1 cos at the top it’s like roughly the same thing but with two hundred and seventy three swapped with three hundred and eighty

Student use of because

TA yeahS1 [three thousand (.) yeah] so like you (5) um (2) [it’s hard to

explain it] (1) TA do you agree with either of theseS1 yeah I reckon it’s the twenty seven point threeTA you reckon it’s the twenty seven point threeS1 yeah cos on the like (.) two hundred and seventy three it equals

the three hundred and eighty two point twoTA rightS1 and the decimal point’s at the end so it goes point two and then

it goes point three so (1) and (.) that’s about itTA so (1) are you comparing that with that (1) or (.) is something

else going onS1 number five and number one (1) but just swapped around

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Who? Dialogue/event EpisodeTA oh (.) okay (.) so we’ve got one point four one point four that’s

the same (.) the two hundred and seventy three we don’t know (.) and the answer is three eight two point two (.) but here it’s three eight two two (.) so we’ve got to do something to the two seven three (.) and you think it’s put the decimal point there (2) e: :r Jodie go on

S16 um (.) I think it’s two thousand seven hundred and thirty (.) because (.) if the [thing] was like [turned around] it would be two thousand seven hundred and thirty [because it’s] divide instead of times it’s got to go the other way

TA nice (.) okay (.) so (.) does that make sense to you (.) because you’re dividing not times you have to move the decimal point forward not backwards

S1 yeah but like (.) you’re going down (.) so like the decimal point’s like nearer the end (.) so I still reckon it’s twenty seven point three

TA okay (.) anyone elseS17 I think it’s two thousand seven hundred and thirty because if

you did twenty seven point threeTA yeahS17 times one point four TA oh that’s a good way of doing it (.) so the opposite of it //S: it’s//

hold on let’s write that down (.) one point four times twenty seven point three yeah

S17 it’s going to be nowhere near three thousand eight hundred and twenty two

TA right (.) it’s going to be more like one lot of thirty isn’t it (.) it’s going to be nearer (.) nearer to thirty-ish (1) so (.) that’s not thirty-ish it’s three thousand-ish (.) so can we try doing it with that one then (.) so if I do one point four lots of two thousand seven hundred and thirty am I going to get (.) eight (.) three thousand eight hundred and twenty two (.) is that more like it

S17 yeahTA so it feels like that oneS18 yeah but it’s dividedTA so (.) come on S17 explain why we have to do thisS17 because (.) if you put equals and then two seven three at the

end (.) of the sum [ ] (.) you just do it backwards (.) with the opposite sum (.) so the opposite to divide is times

TA so if I got that times that I should get that (.) and that’s not going to give us that (.) but that is

S17 yeahS19 I think because (.) um you’re dividing (.) and (1) because three

thousand eight hundred and twenty stays the same and one point four is divided by ten (.) you divide the two hundred and seven three by ten (.) and that’s how I got twenty seven point three

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Who? Dialogue/event EpisodeTA you got [ ] that one (.) what about what S17 is saying (.) cos

he’s given us a check to do (1) I mean if I do that times that I’m not going to get that I don’t think (.) S15 (.) last comment

S15 I’ve changed my mind from twenty seven point three to two thousand seven hundred and thirty

TA okay go onS15 beca: :use (.) if you do two thousand seven hundred and thirty

(.) times one point four (.) cos it’s two thousand seven hundred and thirty (.) times the one point four by ten to give fourteen (.) so it’s two seven three times fourteen

TA so that’s the same as fourteen times two seven threeS15 yeahTA which is that (1) lovely (.) okay we’re going to stop there (.)

that’s been really useful (.) um I hope there are things you can take from this discussion that you can use next time you do this (.) um (.) and also I quite like (.) I want to highlight what S17’s done there (.) and then what S15’s done (.) and (.) if you can’t do the sum that’s there (.) then often it’s a case of transforming it into a sum you can do (.) and so what S15’s got (.) is a transformation of this sum (.) and saying it’s the same as that one (.) we’ve already got the answer to that one it was there (1) so always be thinking if I (.) if you (.) can transform something (.) to make it easier to do (.) into something you can do (.) okay (.) shut books for a minute

Teacher ends activity

TA put everything aside (.) a: :nd (.) look at what’s on the board (.) no (.) we’re going to concentrate on this (.) put everything aside (2) and (.) on board there (2) we’ve got what I am going to call a number machine (.) right (1) so you put numbers in and you get get numbers out (2) someone give me a number (.) S20

Teacher asking for an answer

S20 seventy fourTA seventy four (.) ohhS21 can it be a decimal Student

questionTA well it could be any number I suppose but we are going to have

to work with seventy four (.) someone tell me what’s going to go in there (4) what goes in that box (7) S22

TA taking answers from students – directing

S22 is it [one four eight] Teacher asking for answers

TA one four eight (.) woah what goes in that box (4) S23S23 forty fiveS huhS5 how did you get

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Who? Dialogue/event EpisodeS23 I don’t knowTA so follow alongS wait (.) how did you get ten // // ohhTA ohh (.) S4S4 three hundred and seventyTA three hundred and seventy (.) I’ll go with that (.) I have no idea

(.) okay (.) um (.) what goes in that circle //Ss: uhhh// don’t blame me (.) blame Sara she said seventy four //Ss: ( ) // S24

S24 seven hundred and fortyTA seven //S: hundred and forty // four oh (.) you happy with that (.)

okay what goes in that box (3) S25S25 seven hundred and fortyTA oh right (.) okayS yeah but why (.) why does it come to that Student

questionS is that meant to happen missS yeah it isS no Students’

unelicited responses

S oh yeah missS because if you do seventy four times [ ]TA um (4) any comments (.) any comments (.) yeah Teacher

asking for comments

S cos it can’t be seven hundred and forty which is the bottom one (.) because seventy four times five is three hundred and seventy and then (1) no (1) no ignore me

Student use of because

S it (.) the two answers in the little circles no matter what you start up there will always be the same (.) because if you start there no matter what (.) if it’s times five times two it’s like times ten

TA so if I’ve got something (.) times five (.) and times two (.) I’ll always get the same as times two times five

S it’s like times tenTA it’ like times ten (.) oh (.) like that times tenS I think it’s eight fortyTA go on whyS I just do cos I worked it outTA okay (.) so (.) at what point did we go wrongS I think it’s a hundred and forty time the [ ]TA did you do that on your calculator Caitlin (.) yeah (.) I think a

hundred and seven forty there (.) S10S10 I didn’t have my hand up

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Who? Dialogue/event EpisodeTA ah alright okay (.) you did originally (.) so what S15’s saying is

times five times two is the same as times ten it doesn’t matter what order you do it in (1) are we happy with that //Ss: yeah// shall we try another number or are we happy (.) so (.) er okay shall we try something easier (.) S12 give me an easier number

Teacher asking for answers

S12 oneTA okay (,) one times fiveSs fiveTA times twoSs tenTA one times twoSs twoTA times fiveSs tenTA so S’s thing (.) times it by ten (.) you get the same thing (2) go onS can we do number seven hundred and twoTA seven hundred and two (.) you know what Jacob (.) you can do

thatS yeah (.) the answer’s seven thousand and something TA okay I’ve changed the number machine (.) you may not have

noticed (.) okay can we have any thoughts at the moment (1) about what’s going to happen in these two circles (1) any thoughts (.) Tommy yeah go on

Teacher asking for thoughts

S might be (.) still going to be the sameTA same (1) anything else (1) any other commentsS I think they’re going to be different because they’re different

order um (4) what’s it called whenTA order of operationsS ( )TA so you think it’s different (.) timesing by two and adding two is

different to adding two and timesing by two (1) TA is gathering ideas on the board

S yeahTA okay so because they’re different operationsS I think it’s going to be the same as well (.) because both numbers

are times five and added by two (.) so it’s kind of the same thing you’re doing

S I reckon they’re going to be different (.) because if you start with one again (.) if you times by five and plus two it’s going to be seven (.) and if you add two first it’ll be three and then you times five which is fifteen (.) so plus-ing on two first will make it a bigger number

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Who? Dialogue/event EpisodeTA okay everybody can you draw that back of your book (.) don’t

worry if it’s a mess (.) just draw circles squares and two more circles (.) put those in and I want you to choose your own starting number (4) okay you’ve got two minutes to do that (.) two minutes and no more

CLIP THREETA okay (1) stop there (.) you should have done at least one (.) any

comments (.) any comments (.) TomTeacher asking for comments

S if you times five plus two the answers will be smaller (.) if you plus two times five the answer will be bigger

TA so that gives you the smaller one (.) add two times five gives you the bigger answer (.) so they’re different Tom

S yeahTA AlexS when I did the bottom twoTA yeahS they were eight apart and then I checked I did it again [ ]

and it was still eight apart (.) TA difference of eight [someone at door] um BenS I reckon they’ll all have a difference of eightTA you reckon they’ll all have a difference of eightSs yeah // yeah they will //TA yeahS what number did Alex start with on there Student

questionTA Alex what numbers did you start with S [seven]TA seven (.) and what was the other oneS [thirty five]TA thirty five (.) no what was the other starting number (.) you did

it twiceS um oneTA seven and one she triedS I think it’s going to be all a difference of eight because they’re

an odd numberTA you think all a difference of eight (.) so Claire you think (1) odd

(.) has anyone done an even number (.) JodieS I done two and ten and they both had a difference of eightTA ahh (.) two and ten (1) okay um (.) does anybody think they’ve

got any reasoning they can do here as to why we’re getting eight coming up (4) any reasoning so why is times five plus two and plus times five (.) why are they bigger and smaller and then particularly (.) why are they eight apart (2) Tom

Teacher asks why

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Who? Dialogue/event EpisodeS I reckon what I should try is (.) it should be five times at the top

and five times on the left side and then it would come out as the same I reckon

TA what (.) so if I made that times five S yeah I reckon that would come out the same [at the bottom]TA so you think if I left that as add two and times fiveS yeahTA would you try thatS Miss I just did (.) her thing’s wrong (.) cos I just did two and

two’s an even number and it still has a difference of eightTA right (.) so (.) Claire (1) are you happy with me to rub it out (.)

are you sure (1) er FranS can I tell you mine (.) what I got Student

questionTA yeahS my starting number is eight thousand four hundred and seventy

oneTA oh my (.) eight thousand four hundred and seventy oneS times it by five is four two three five fiveTA yeahS add two is four two three seven (.) four two three five sevenTA okay (.) I’ll do the add bit because I can add two (.) that’s easy

(.) times fiveS times five is four two three sex fiveTA so is that eight apart //Ss: yeah// so big numbers work (.) are

there any numbers that don’t workTeacher asks for ones that don’t work

S decimals I think (.) I think we should try decimalsS yeah (.) or minus numbersS minus numbersTA okay what I want you to do (.) negatives that would be a good

ideaS I done it Miss and it came out equal (.) the sameTA oka: :y (.) could you try some different starting numbers for

your conjecture then (.) um (.) Jodie last comment S I got lots of things Student

question [implicitly – can I say them]

TA okay go on (.) let’s listenS I got one (.) that the end number ends in five and seven or zero

and two TA so the last digit of the answers (1) the last digit will be five and

seven or zero and two

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Who? Dialogue/event EpisodeS yeah (.) and [what I done] also Miss [ ] cos if you like

times two by five yeah (.) then you get ten and because it’s round the other way on the other one you get eight

TA say that againS if you like (.) um times two by five you get tenTA yeahS and like cos the other one’s round the other way you can take

away two instead then you get your eightTA (1) take away two instead (4) go on ClaireS I don’t really understand it’s like she means [it’s seven] on

one of the numbers or both numbers (.) cos on one on the first number

TA she means the last digits of the answersS cos I got five on one and seven on the other and then five on

one and seven on the other againTA okay (.) that fits (.) what I’m going to do (.) Jodie you need to

think more about what you’re saying and I need to think more about what you’re saying (.) um (.) I want you at the front of your books (1) I want you to try what has been suggested (.) so can you find a counter-example to the conjecture that all the difference will be eight (1) you can try decimals (.) you can try negatives (.) you can try fractions you can try big numbers tiny numbers (.) anything that you want (.) you’ve got to do at least (.) listen (.) try at least six of those (1) and I’m going to give you eight minutes

Teacher ends discussion

FOURTH CLIPTA okay (.) um (.) we’re going to finish in a minute (2) now (.)

we’re going to finish in a minute (2) um (.) well done if you’ve managed to do four of those that’s fine (.) er (.) what I am interested in is some last comments about well what we were doing actually with you trying to find some counter-examples (.) and (.) the difference was eight (.) so has anyone got any starting number that doesn’t give us a difference of eight (.) that’s the first thing (.) and second thing I want to hear about what Alex has tried and what Tommy has tried (.) they were trying something slightly different (.) it’s quite interesting (.) um (.) any counter-examples that did not give us a difference of eight (.) okay (.) Jacob what’s your number

Teacher asks for counter-examples

S nineteen million six hundred and three thousand nine hundred and twenty one

TA nine hundred and twenty one (1) okay that’s one to try (.) any other counter-examples (.) Sara

S [ten]

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Who? Dialogue/event EpisodeTA ten (.) what did you say (.) tenS yeahTA okay (.) anything else (.) BenS minus [various voices]TA okay we’re going to hear from Alex and Tommy Brain at the

start of next lesson please (.) hand your books in

163

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INDEX

Awareness, 3, 4, 12–15, 22, 24, 27, 30, 46, 62, 64, 71, 76, 82, 84, 93, 98, 106, 110, 111

Awarenesses, 4, 93, 99, 120Bateson, Gregory, 4, 5, 7–10, 13, 15,

18–21, 48, 71, 72, 74, 97, 98being alongside, 101BERA (British Educational Research

Association), 26Brown, Laurinda, 4, 6, 7, 13, 22, 46BSRLM (British Society for Research

into Learning Mathematics), 13Conjecture, 22, 23, 44, 45, 60, 62, 65,

67, 79, 80, 89–93, 95, 96, 98, 102, 104, 119, 125

Critical theory, 13Constructivism, 13Dialectical contradictions, 8Dichotomies, 4, 5, 7–9, 18, 85, 86, 94Difference, 7, 18–20, 51Dualisms, 8, 87Enactivist methodology, 18, 64, 65Systemic circuit, 19Equifinality, 21– 23, 31, 89Cybernetic, 15, 21Equilibrium, 22Enactivism, 11–24, 33, 48, 51, 117Epistemology, 11, 13–16, 19, 32, 33,

86–88Ethics, 25, 26, 28, 29, 34Facilitator, 17, 24, 28, 37, 38, 40, 41,

43–45, 50, 54, 57, 60–65, 68, 69, 72, 76, 97, 101, 110, 115, 116, 119, 120

Gattegno, Caleb, 4, 10, 29, 32, 39, 57, 98

Gödel, 5Hypothesis, 14, 15, 41Indecipherable speech, 58

Interrupted speech, 58Interview, 29–31, 34, 79, 90–93, 147Jaworski, Barbara, 38–42, 45, 49, 50,

54, 61, 64Laughter, 65, 68–76LEA (Local Education Authority), 27learning, 11, 13, 17, 18, 21, 23, 28, 29,

34, 37–43, 45, 46, 48, 50, 53–64, 68, 71, 72, 76, 79–84, 86–88, 94, 98, 99, 107, 110, 115, 117–120

Listening, 26, 67, 101–111, 116–119Man of achievement, 5mathematical thinking, 11, 17, 24, 28,

30, 43, 45, 81, 106, 110, 115, 122mathematician, 6, 22, 61, 79, 85, 91–95,

97–99, 104, 116, 118, 119, 121, 122Maturana, Humberto, 13–16, 19, 21,

34, 99 metacognition, 79–89, 92–94, 97–99,

101, 104, 107, 110, 116metacommenting, 61–64, 94, 95, 115Metacommunication, 9, 10, 48, 71, 74,

81, 95, 96, 116Meta Experiential Authority, 14Methodology, 11, 13, 17–18, 29, 64, 65Negative capability, 5NSPCC (National Society for the

Prevention of Cruelty to Children), 26Paradigm, 13, 42Participatory, 13Pattern, 4, 9, 10, 15, 16, 21–23, 25, 27,

29–34, 50, 53–64, 79–81, 89–93, 98, 107, 115, 118, 121, 126

PME (International Group for the Psychology of Mathematics Education), 13

Positivism, 13Post-positivism, 13

169

INDEX

170

purposes, 46–48, 50, 53, 56, 62, 64, 68, 75, 106, 108, 116, 118, 119, 121, 129

researching, 28self-regulated learning, 81, 86story, 3–4, 8–12, 27, 47–48, 79–81,

89–99, 101, 107–111, 120Tahta, Dick, 39teacher discussion, 25, 29, 32, 34, 53–76,

94, 104, 106–108, 110, 119, 129, 130

teaching, 4, 5, 7, 11, 21, 23, 28–30, 37, 38, 41, 42, 47, 48, 54–58, 67, 68, 73–75, 82, 85, 86, 98, 102, 106–108, 116, 129

Varela, Fransisco, 8, 13, 14, 20, 25, 26, 28, 42, 46, 49, 117, 118

video, 11, 23, 25, 29, 30, 33, 34, 37–51, 53–55, 57–61, 63, 64, 70–73, 81, 95, 115, 119, 129