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DG11, ICME-10 Paul Andrews _____________________________________________________________________________________

International comparisons of mathematics teaching: searching for consensus in describing opportunities for learning Paul Andrews University of Cambridge, England and colleagues from the European Union funded mathematics education traditions of Europe (METE) project: JoséCarrillo, University of Huelva, Spain Nuria Climent, University of Huelva, Spain Erik De Corte, University of Leuven, Belgium Fien Depaepe, University of Leuven, Belgium Kati Fried, Eotvos Lorand Tudomanyegyetem, Budapest, Hungary Gillian Hatch, the Manchester Metropolitan University, England George Malaty, University of Joensuu, Finland Peter Op ‘tEynde, University of Leuven, Belgium Sari Palfalvi, Eotvos Lorand Tudomanyegyetem, Budapest, Hungary Judy Sayers, University College, Northampton, England Tuomas Sorvali, University of Joensuu, Finland Eva Szeredi, Eotvos Lorand Tudomanyegyetem, Budapest, Hungary Judit Török, Eotvos Lorand Tudomanyegyetem, Budapest, Hungary Lieven Verschaffel, University of Leuven, Belgium Contact: Paul Andrews University of Cambridge Faculty of Education, 17 Trumptington Street, Cambridge, CB2 1QA, UK Email: pra23@cam.ac.uk Tel

+44 (0) 1223 336290

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Abstract This paper outlines the processes by which a multinational team of researchers developed a descriptive schedule for use with video-recordings of mathematics lessons in five European countries: Flemish Belgium, England, Finland, Hungary and Spain. Located within a theoretical framework concerning learning and the negotiation of meaning, we discuss initial problems concerning assumptions about shared understanding of educational terminology and how these were resolved. We discuss, also, drawing on grounded theory, an iterative process of classroom observation and schedule development and refinement which was undertaken in all countries by teams of researchers from each country. This process enabled the production of a schedule which all participants can use with confidence, understanding, reliability and which satisfied the project’s desire to describe lessons not only simply but also in ways that highlight both similarities and differences between teachers and countries.

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Introduction Small-scale comparative studies, which are generally classroom- focused (Postlethwaite, 1988; Jenkins, 2000), frequently yield insights unlikely with large-scale studies (Theisen and Adams 1990 Kaiser 1999, Postlethwaite 1988). They seek to account for the influence of context (Schmidt and McKnight 1995) although they may be prone to problems of unacknowledged cultural influences (Theisen and Adams 1990) and, at their worst, may lead to "unwarranted correlation" and "unfounded speculation" (Jenkins 2000, p 137). However, despite such concerns, they have the advantage over large-scale studies which are often "frustratingly deficient in detail" (Theisen and Adams 1990, p 278) and little more than "attempts to quantify the unquantifiable" (Jenkins, 2000, p 137). This paper is an attempt to describe one element of one such small- scale study's attempts to make meaning from the data it collected and the insights brought to it by a multinational team of researchers. The work described here derives from the European Union funded mathematics education traditions of Europe (METE) project; a comparative study of the teaching of mathematics in five countries: Flemish Belgium, England, Finland, Hungary and Spain. These countries represent well the social and cultural diversity of the continent and, as is emerging from our work, its mathematics education traditions. A particular aim of the project was to examine how learning was structured over the duration of a topic by focusing on sequences of lessons taught on topics thought representative of all curricula. The main data set was to be derived from video recordings of such sequences made by each home team. However, to ensure the elimination of cultural bias from subsequent analyses, data collection was preceded by an extensive programme of live observations to facilitate the development of a descriptive framework for the analysis of videotaped lessons. This paper is concerned with the processes by which this framework was developed and the insights generated into the problems and rewards of working in an international team. As the paper will show, this was far from a simple process as even the simplest assumptions about the vocabulary of mathematics classroom activity proved to be far from unproblematic. Indeed, even the assumption about topics being taught in sequences of lessons proved less certain than previously thought. It is generally accepted that the mathematics classroom is a complex social setting in which all actors, present and absent, have a part to play in the search for common understanding or the construction of meaning (Lerman, 2001). It is also clear that the research conducted in classrooms, involving not only the ordinary classroom actors but additional adults in the form of observers, videographers, sound recorders and interviewers, creates an even more complex social milieu. Inevitably, the data derived from any classroom will have been interpreted not only by the actors involved in the lesson, including researcher influence on actors' actions, but also by the various observers and, ultimately, those involved in the interpretation and analysis of that data. In such circumstances it would seem important, in order to avoid further layers of interpretation and subjectivity, to ensure that the constructs used to describe classroom activity and their 3


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associated vocabulary were understood and shared by all participants. However, as other studies have indicated, such agreement is far from straightforward. The survey of mathematics and science opportunities team (Schmidt et al, 1996) found that many assumptions about shared meaning were challenged by colleagues' unfamiliarity with terms thought common to all classrooms. Mindful of this problem, and acknowledging both our belief that learning is a social construction and the fact that we were a team for the majority of which, English was not the mother tongue, we set out to develop an agreed and understood framework for describing mathematics classroom activity.

Theoretical framework In some ways our theoretical framework owes much to David Clarke’s “learners’ perspective”project which has shown well how (relatively) small- scale studies, while not necessarily affording the replicability, reliability and validity of larger projects, can offer important insights into classroom activity and its impact on cognition and affect. Firstly, our work would not have been possible without its being “predicated on a conception of the researcher as learner”(Clarke 2001a, 13). Secondly, in respect of our being both researchers and learners, our practice "enacts a theory of learning… that structures the data that is collected and affords certain interpretations of that data while constraining other possible interpretations”(Clarke 2001a, 14). However, we diverge from Clarke's perspectives in our goal of consensus. Writing in his autobiography on world football's desire for uniformity of referees' practice for international matches, Pierluigi Collina (2003), one of the world’s leading referees, suggests that consensus in respect of interpretation and application is the only means by which both the regulation and integrity of football can be maintained. In respect of an international study of mathematics classrooms, conducted by a multinational team of researchers, our experience has indicated that similar consensus is essential for the integrity of our work. In accord with most current thinking, not only do we acknowledge that learning is a social construction but also that understanding of learners' constructions requires an acknowledgement of the socio-cultural setting in which learning takes place. This can be viewed as causative of learning in that "social interactions, along with physical and textual interactions can cause disequilibrium in the individual, leading to conceptual reorganisation" (Lerman, 2001, 55). Lerman goes on to discuss his notion of learning activity, which is "certainly the task set by the teacher but is also a function of the style of classroom interaction, the texts, the ethos of the school, the possibilities arising out of the particular mix of the actors that day… It does not exist, for the class, for the teacher, or for the individual students, before the interaction." (57). This sense of context and learning activity, as will be discussed later, resonate strongly with the ways in which our project developed. All research is, in one way or another, conceptual and the development of our descriptive framework for use with videotaped lessons was rooted in the concepts that colleagues brought to the negotiating table. Lerman (1996) writes that as concepts derive their meaning from being used, their acquisition can be interpreted as the result of an 4


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individual coming to share meaning through negotiation. However, we must be careful not to assume that terms such as negotiation are unproblematic. According to Cobb and Bauersfeld (1995) negotiation is the "interactive accomplishment of inter-subjectivity" (295) or shared understanding. However, while one product of negotiation may be enhanced intersubjectivity, the nature of the negotiative process requires a prerequisite intersubjectivity that enables and sustains that negotiation (Clarke 2001b). Moreover, negotiation is dependent on language which is itself intersubjective (Clarke 2001b). As will be shown below, this prerequisite sense of intersubjectivity, particularly in respect of language, was initially missing in our work to the extent that it had to be developed over several months before significant progress could be made.

The research process This paper describes the collaborative development of a framework for describing and analysing mathematics classroom activity, as recorded on videotape, by an international team of educational researchers. Unlike many international projects which began with pre-determined categories of classroom activity, often constructed by researchers from the project's lead country, our intention was to generate a descriptive framework drawing on the collective experience of classroom observers from the five METE countries. Each team includes colleagues experienced in teacher education, research and extensive classroom observation. The team acknowledged work undertaken elsewhere but, in that acknowledgement, came to realise that educational concepts, and associated vocabulary, are more culturally located than previously it had realised. As found by the survey of mathematics and science opportunities team (Schmidt et al, 1996) educational terminology frequently thought common to educational systems appeared to have less shared meaning than was generally accepted. The aims of the project were not only to identify nationally characteristic patterns of classroom activity but also to determine effective approaches to the teaching of key topics in the age range 10-14. Initial results of the project’s work will be made available once preliminary analyses of project data have been completed. A decision made early in the project was to videotape sequences of lessons on key topics taught in all curricula. This was, in part, a pragmatic decision concerning the need for, in some sense at least, objectively obtained data that could be shared among participating teams. Such a decision, however, necessitated the development, or appropriation, of instruments that would facilitate the analysis of video recorded mathematics lessons in ways that were accessible and meaningful to an international audience for whom English was not its first language. Indeed, the linguistic imperialism of the mainstream educational research literature has, it became increasingly apparent to us, colluded in the construction of the myth of terminological conformity. Consequently, we aimed to develop an instrument that all colleagues could use with confidence, secure in the knowledge that they had participated in constructing the meaning of the descriptive categories that were to emerge. Research instruments are frequently likened to the tools used by craftspeople and described as too blunt or too sharp for their intended purpose. We adopt a different 5


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metaphor in accordance with our desire to illuminate. If the classroom is considered a fabric then the light we need must not be too sharply focused or the beam will highlight only individual threads and show nothing of the fabric's pattern. If the beam is too wide then all it does is alert the observer to the existence of a fabric indistinguishable from other, similar, fabrics. Our intention was to find a beam that offered sufficient illumination to distinguish between the details, patterns and colours, of one fabric and another. This did not mean, for example, that existing frameworks were ignored. Our intention was to acknowledge that the mathematics classroom is a complex set of interactions between teacher, students and mathematics (Kilpatrick et al, 2001). We were also conscious that the subject knowledge (Ma, 1999) and pedagogic content knowledge (Shulman, 1986) of the individual teacher are significant distinguishing factors between those who teach effectively and those who do not, and that observers bring to their work culturally embedded and frequently unconscious assumptions about the nature of effective teaching (Schmidt et al, 1996). It was decided at the outset that the project's instruments could only be developed through the use of systematically conducted live observations involving colleagues from each country. The processes employed reflected those of grounded theory (Glaser and Strauss, 1967). Put simply, lessons were observed and categories of activity that emerged were then tested against further observations in the same and other countries. Each observed lesson highlighted the tension between acknowledging new forms of activity while continuing to provide acceptable accounts of those observed earlier. In respect of this study, lessons were observed on five separate occasions in five countries over the course of eight months. Each phase of observation lasted a week, with one or two lessons observed each day. Each lesson, which was videotaped for later use, was observed by a team of at least five observers, one from each participating country. Afterwards observers returned to the host institution to discuss what had been seen and to relate the lessons’ activities to the developing descriptive framework. This was accompanied by the video recording made the same morning. The first playing of the videotape was explicitly focused on clarifying issues of fact. This allowed colleagues to refine and amend any notes made during the observations and typically lasted upwards of two hours. Once colleagues were satisfied that they understood the dialogue the process of analysis began. The tape was replayed and colleagues invited to code each episode –a discussion on the development of a definition of which follows below - against the current schedule, the tape was played episode by episode according to our emerging sense as to the nature of an episode. Each episode was discussed, particularly discrepancies between individuals’ coding, in order to compare that which was observed with the categories currently thought adequate. This process led to new categories being identified, old ones being amended or discarded and a greater sense of intersubjectivity in respect of the meaning of the descriptive framework. Clarke (2001b) stratifies classroom discourse into six levels - the lesson, the activity, the episode, the negotiative event, the turn and the utterance. For his work with the learners' perspective project, the explicit focus was on the negotiative event, which he describes as comprising those utterances forming the totality of an interaction. Our view was that this equated too closely to the tightly focused illumination described above. The episode, the

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definition of which emerged gradually over several months, seemed to provide a more useful unit of analysis in its providing sufficient illumination for us to distinguish between fabrics. For us, an episode is a sequence of observed interactions, negotiations and activities with a single didactic or managerial purpose. Thus, for example, an episode could be short and managerial, as in a teacher's calling a register, or long and didactic as in a period of seatwork during which learners complete exercises posed from a text book. Also, a period of exposition could comprise several episodes according to teachers' intentions a nd, of course, observers’interpretations. It became our view that in the totality of a lesson's episodes can be found the characteristics of teachers' conceptions of their professional identity. It is interesting that during the development of our schedule substantial variation in colleagues’ understanding of educational vocabulary was exposed and mediated. For example, it emerged during initial discussions that the words pedagogic and didactic held substantially different meanings for different members of the project team. For most colleagues, teachers’ decisions about the teaching of their subject to students were understood to be didactic, while for a few they were pedagogic. Consequently, as in many instances during our work, the minority acceded to the majority and adopted the common practice – pedagogy was accepted as the totality of the curriculum and the didactic strategies employed in its delivery, which was at some variance with the use of the word among the English participants. It is also interesting to note that during the first fact-establishing viewing of videotapes, home colleagues with responsibility for providing the translation frequently drifted into evaluative rather than factual reporting. In some respects this may have been inevitable, as there seemed, initially at least, a desire on the part of such colleagues to present their compatriots' efforts in a positive light. However, as our goal required that each lesson be viewed dispassionately, it became apparent that team members had to trust the professional and personal integrity of the other team members. As with much of the work we did, the necessary trust was achieved, but it was not something that should have been assumed or left to chance.

Observation week 1: Cambridge, England The first observations took place in England in order to give colleagues reasonable access to the language of classroom discourse. Others took place in Finland, Hungary, Belgium and Spain respectively. In each country a range of lessons was observed with the intention of providing a representative sample of the teaching found in different schools and across grades 5 through 8 respectively. A different school was visited each day. In respect of the processes of our research, the experience of the English week proved an effective model for the others. Observations were undertaken in the mornings and discussed in the afternoons. An open approach was adopted with colleagues contributing whatever they felt was appropriate in respect of what was observed. At times it was difficult to avoid inference, but we reminded each other constantly of the need for observable evidence. As the observation weeks passed, colleagues’nationally located 7


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perspectives became less apparent as implicit assumptions became more explicit and less influential in respect of what observers were able to see in any given episode. Thus, for example, an English observer was able to see beyond an apparent lack of exercises set or a Finnish colleague could suspend concerns about the lack of manipulative use. That said, it was clear during the first week that colleagues tended to assume that what they were seeing, informed by their respective national lenses, was also seen by others. Consequently, the first schedule constructed reflected this, as yet, unacknowledged diversity of perspectives. The first schedule comprised almost four pages of activities or interactions which were loosely structured under broad headings. The first of these related to lesson phases or what were called structural episodes. Among these were introduction, exposition, problem posing, problem solving/sharing, seatwork, plenary and what we called transitional activities. Within each of these were a number of subcategories. For example, the exposition category contained occasional input from students, frequent input from students, explicit conceptual development, activating prior knowledge and skills, invoking visualisation, use of mathematical terminology, use of manipulatives, use of teaching aids, use of teacher’s body, explicit structural development, demonstration, modelling, process oriented and performance oriented. The instrument, at this stage, looked as it should, the result of open-ended discussion about observable behaviour which might both typify and distinguish mathematics classrooms. At this early stage the problem posing category –concerning the ways in which teachers set up the tasks on which their learners would work - included the presentation of realistic problems, quasi-realistic problems, mathematics proble ms, open contexts/situations, multiple response problems, single response problems, process orientation and performance orientation. These were intended to reflect the diversity of the tasks presented and expose the underlying conceptions of mathematics and its teaching held by teachers. However, the vocabulary caused problems in respect of our ability to apply the schedule. Consequently, by the end of the week this section had been dropped from the schedule. The schedule experienced other changes as further lessons were observed. In particular we noticed that the instrument, and the behaviours it acknowledged, was located in classrooms in which teachers operated within the framework of current government advice concerning good practice. Such a realisation confirmed the decision to work in a grounded manner –to refine the instrument as new categories emerged and old ones were challenged by new evidence. By the end of the week the original list had been transformed into a two dimensional array. The columns of this array represented our perspectives as to the phases (structural episodes) of a lesson and the rows the behaviours or activities that might be observed. At this stage, this included only classroom interaction as all reference to the tasks set had been removed. During the Cambridge week other issues emerged. Firstly, the notion of coursework, a task on which learners work independently of their teacher for several lessons and for 8


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which a report is submitted, appeared unique to English classrooms – both as a form of mathematical activity and means of assessment. Secondly, the practice of inserting specific lessons on the development of, say, thinking skills rather than integrating them into everyday lessons, was unusual. Thirdly, the idea of an introductory warm up activity, independent of a lesson’s content, was unfamiliar to most colleagues from outside England. The view emerged quite early in our discussions that any descriptive framework would need to acknowledge such distinctive activity without being so particular as to be useful in only the single context.

Team meeting, Leuven, Belgium Before the second round of observations a team meeting was held in Leuven, Belgium. These meetings, which generally last between two and three days and typically involve two or three colleagues from each country, are held every few months to discuss the project’s progress and facilitate its management and strategic direction. At this particular meeting, the second of the project, the schedule developed during the first week of observations was exposed to the scrutiny of those not present during its development. This was facilitated by each team’s providing video recordings of typical lessons against which the schedule would be evaluated. The major issue to emerge was that the number of categories of interaction and activity were too many to be constructively helpful. Consequently much time was spent discussing how almost one hundred forms of interaction might be simplified and other significant issues, concerning, inter alia, the nature of the tasks set, might be acknowledged. Eventually, the rows of the original schedule were replaced by four broad categories, each with a small number of subcategories, related to the mathematical focus of a task, the mathematical context of that task, the nature of the learning induced by the task and the didactical strategies employed by the teacher. Such terminology, as in all cases, was discussed at length in order that all colleagues not only understood it but were able to work with the conceptual issues it represented. The use of video in this regard was invaluable. Additionally, concepts related to the mathematical context of a task were reintroduced, having been discarded earlier, and discussed at some length. During this time it became apparent that realistic mathematics was not a universally held conception. For some, realistic implied a sense of application or the basing of mathematics teaching in real world contexts while for others it was perceived as in the Dutch tradition of realistic mathematics education (Van den Heuvel-Panhuizen 2003). This, as did a lengthy discussion as to the nature of scaffolding, confirmed the disparity of understanding that existed within the group. In respect of scaffolding, some colleagues argued that any act of any teacher was some form of scaffolding and that it was a redundant category in its embracing everything else. Others adopted a more particular perspective derived from the didactic literature. Eventually a consensus was reached although this was not a straightforward process. It was interesting that the literature was not always cited as definitive as it b ecame clear that particular associations with particular words were not necessarily appropriate for our purposed. Similar processes were

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employed in respect of coaching and other didactical strategies where initial divergences of understanding were made convergent through discussion. During the meeting considerable time was spent on establishing our categorical definitions. This was to ensure, as far as such things can ever be ensured, a reference point for the meaning of each term to reduce the possibility of idiosyncratic interpretation later in the project. This led to the production, in addition to the coding sheet being developed, an explicated version showing the definitions currently agreed. As is shown below, these sheets were subject to substantial revision as the project developed.

Observation week 2: Joensuu, Finland The second series of observations took place in Joensuu, Finland. During the course of the week several new issues emerged. The first concerned the use of manipulatives which we had seen during the first observations in Cambridge. It became clear that colleagues had different perspectives not only as to what might constitute a manipulative but also their didactic significance. On the one hand there were colleagues for whom a manipulative was any piece of concrete apparatus introduced into a lesson for any purpose. Such perspectives included the use of, say, compass and straightedge for work on constructions. On the other hand there were colleagues for whom manipulatives had a specific function in modelling, often as metaphors for, the mathematical structures under consideration with a particular didactic purpose in respect of facilitating learners’ understanding of such structures. This was an ongoing tension which was resolved only after further weeks of observation and another team meeting. The course of each day followed the format established in Cambridge. A lesson, simultaneously video recorded, was observed by the whole team before being discussed at the University and the schedule revised. One significant difference, which set the tone for subsequent observation weeks, was that after each discussion colleagues attempted to use the schedule to code each of the lesson episodes and then compare the outcomes. It became apparent that inter-coder reliability was developing well although there remained some ambivalence of understanding. One issue to emerge was how to manage those activities which could have been inferred but for which no explicit evidence was observed. After lengthy discussion it was decided that one approach would be to code only those activities for which explicitly observable evidence was available or for which no evidence was available. Those that might have been inferred were not to be coded. The rationale for this was that as much could be learnt about lessons from that which was explicitly missing as that which was explicitly present. Also, inference was agreed to be an inappropriate means of data collection as it relied too much on interpretation with a possible compromise to the integrity of project data. During the week additional categories were added to the four broad themes and others amended or deleted. Decisions were always based on the quality of the evidence they were likely to yield, focusing on two criteria: that they should not focus on characteristics not found only in atypical lessons but should be capable of distinguishing between lessons.

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Another major issue discussed during this week was the nature of an episode. Up to this point our definition had derived from our experiences in England and related to the phases of a lesson observed in that country. It was agreed, despite such phases resonating with colleagues’experiences, that it was inappropriate to classify episodes according to one system’s evolving traditions in respect of the ways lessons are structured. A lengthy discussion led to an understanding that episodes tend to involve either the whole class (plenary) or take place when individuals are working independently of the rest of their colleagues (seatwork). Thus episodes were redefined in a less prescriptive manner than before and the recording sheets altered accordingly. Each lesson was now construed as a series of episodes, each of which had a didactic purpose different from or independent of both preceding of succeeding episodes. Each episode would be coded as either plenary or seatwork with the distinct possibility that a period of, say, teacher-whole class activity might involve several successive plenary episodes. To facilitate this, the record sheet became a series of blank columns, representing the episodes of a lesson, which would be scored p or s before being coded against the developing schedule of rows. With the re-defining of episodes came the means by which different episodes occurring within the same phase of a lesson could be coded in different ways to show their different didactic purposes and approaches.

Observation week 3: Budapest, Hungary The project conducts much of its business through electronic mail and this has proved an invaluable means not only of enabling colleagues to keep abreast of developments but, importantly, to contribute to those developments. Prior to this week’s observations colleagues had been set the task of addressing concerns raised by team members subsequent to the Finnish revisions. Challenges had been mounted to some of the categories and new ones proposed. Thus, in addition to the ongoing process of observation, schedule testing and revision, colleagues were invited to consider additional suggestions and requests that had emerged during the previous month. This was achieved and the schedule amended accordingly. For example, it was decided that too few lessons had been seen in which practical work or manipulative use had been observed for it to merit a category within the didactics section. However, in order not to lose the distinctive and, possibly, distinguishing function of the use of concrete materials or apparatus, it was decided to include a section on the recording sheet to code whatever equipment or other materials, including textbooks and worksheets, that had been used during the lesson. During the week it became apparent that some ambiguity remained in respect of the coding of an episode. For example, it was not infrequent for teachers, during a period of exposition, to invite students to work for a few seconds on a problem before using the outcomes of that work as the basis for the continuing exposition. Uncertainty lay in whether such activity constituted an episode of seatwork independent of the plenary exposition. Our conclusion was that if seatwork was an integral element of the exposition and that it did not interrupt its flow then it would not be coded separately. If, on the other hand, it was not an integral element of the exposition, despite its possibly contributing 11


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obliquely to the discussion, it would be coded separately. Such problems of interpretation were a part of every discussion and represented a substantial element of our public work. Towards the end of the week the use of teacher questions was raised as a didactic strategy. Little time was available for a full discussion but the category was added in order to remind colleagues of the need for further discussion during the next round of observations in Belgium.

Observation week 4: Leuven, Belgium This week of observations preceded a team meeting. The week followed the well-established pattern of observation, discussion, testing and refinement. The category of questioning proposed in Budapest was explored both within the context of the lessons observed in Leuven and those elsewhere earlier. It was agreed that questioning as a didactic strategy related to systematic attempts to support children’s development of mathematical ideas and thinking. This was not an attempt to classify questions, as that had been rejected at an earlier meeting, but was seen as a didactic tool in which the form of question was largely immaterial. It emerged during the week that differentiation as a didactic strategy had been neglected in the development of the descriptive framework. This may have been due to the little evidence having been observed in previous weeks although it certainly assumed the importance of a significant omission once it became apparent that a Flemish colleague had been observed adopting it as a didactic strategy. Importantly, it also became apparent that several ideas discussed earlier were still not universally understood or internalised. For example, it was found necessary to refine our distinction between derivational and procedural mathematical foci, to readdress the issue of realistic mathematics and to clarify meaning in respect of several of the didactic categories. Such discussion emphasised well the importance of time spent on joint observations as colleagues invoked a range of shared experiences to support their arguments. In this regard the use of examples or illustrations drawn from these lessons was invaluable.

Team meeting: Leuven, Belgium In respect of the development of our schedule several issues were dominant. The first was the definition of differentiation. Some colleagues argued that differentiation included those occasions when learners worked through an exercise at their own pace with no expectation of all completing all the tasks. Others argued that differentiation occurred only when teachers offered different learners different tasks or tasks from which different outcomes were clearly expected to emerge. This was a lengthy discussion and its outcomes hinged on whether there was evidence of differentiation forming an explicit didactic strategy. That is, it was accepted that an exercise may provide a de facto form of differentiation but that it would not necessarily constitute a didactic strategy. Eventually it was agreed that the former form of differentiation would apply to almost every teacher in almost every lesson with the consequence that to code differentiation in that manner 12


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would fail to distinguish one teacher from another. Thus the definition would focus on explicit forms of differentiation. Another involved a lengthy discussion concerning the notion of mathematical context. It was decided, in order to emphasise differences between lessons, to split the category of realistic mathematics, currently incorporating a range of ideas found in the literature pertaining to the Dutch tradition realistic mathematics education, into several. This was thought to allow for greater differentiation between lessons and prevent any inference that the practices of a particular teacher were intentionally derived from the traditions of RME.

Observation week 5: Huelva, Spain This was the last week of observations and marked the end of the inductive approach to the development of our schedule. It had been decided at the preceding team meeting that no further opportunities could be given after the completion of this week’s work as the video recordings had almost been completed – four cycles of lessons in each country – and would soon be ready for analysis. The week followed the same processes as in previous weeks although the programme of discussion was tighter than the norm with colleagues being reminded that the schedule would have to be completed by the end of the week. This proved a helpful strategy and some minor changes were made to account for the particularities of the Spanish classroom. By the end of the week it was clear that colleagues had not only reached a consensus as to the content of the schedule but were sufficiently confident with it that measures of their functional agreement was high.

Concluding thoughts As a process, this was a hard year’s work. The concentration required during observations themselves and their discussion and analysis against the developing framework was immense, particularly in respect of interpreting colleagues’attempts to express difficult ideas in a second language. I remain in awe of the ability of those, who have become my friends, to present their ideas in such concise and sophisticated ways. As the person chairing the meetings –due to my being a native speaker of English –I was aware both of the privilege of working with such a team but also the responsibility of keeping colleagues focused on the task we had set ourselves. It was often very tempting to allow lengthy conversations about the qualities of a lesson we had observed, and at times this was inevitable, but we managed to keep ourselves on task for most of the time and discussed well the facts, as we saw them, of a lesson rather than subjective evaluations or inferences. The scheme we developed we believe is unique in the educational literature in its being a consequence of genuine collaboration and acknowledgement of the researcher as learner. Referring back to the theoretical framework discussed above, it seems clear to us that the development of our descriptive framework was a well-defined learning activity in the sense intended by Lerman. Each of us, through the observation of lessons in countries other than our own, discussion of what was seen and the laying bare of pre-conceived 13


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notions of effective practice, learnt much about the teaching of mathematics and the conceptual structures that underpin teachers' activity. Lerman (1996) writes that "intersubjectivity is a function of the time and place and the goals of the activity and the actors" (137) and goes on to say that the role of the teacher is to assist the learner "in appropriating the culture of the community of mathematicians as a further social practice� (146). It seems to us that this describes well the processes by which our descriptive framework was developed. Admittedly, our work is distinguishable from the conventional learning situation in that throughout the period of our learning activity there was usually no teacher with a clear sense of the solution to our problem. At different times we were all either learners or teachers (sometimes both) as we attempted to produce, collectively, a solution to the problem we had set ourselves. Indeed, the problem itself was prone to revision as new solutions created new possibilities or undermined old ones. That said, our efforts were clearly focused on establishing a community as social practice through, at different times for different colleagues, legitimate peripheral participation (Lave and Wenger, 1991). The nature of that community, although we have not dwelt on this here, was also part of the negotiative process. Among the project team were those whose main professional responsibilities lie in educational research, one who had spent the whole of his career as an academic mathematician and, as indicated above, the majority who derive from teacher education. Thus, our sense of community was informed by various professional traditions and imperatives. This augmented well our conversations and led us to define ourselves as a mathematics education research community of practice. As has been indicated throughout this paper, language has been both a barrier to and facilitator of learning. Indeed, ambiguity has dominated much of our work and reflects Clarke's (2001b) perception that humans interact as though they hold meanings in common but which are frequently misconstrued. One reason for this may have been the fact that the mediating impact of language, which facilitates learners' interactions with both other learners and themselves (Moll, 1990), may have been undermined by unacknowledged culturally- informed perspectives which may not have been different from those of colleagues from other countries. It is also not improbable that colleagues' linguistic tools, many of which are metaphors (Bills, 2001), may not have had the degree of inter- linguistic transferability necessary for the successful interchange and assimilation of ideas. However, the manner of our work, focusing on the common experiences of participants and drawing on video recordings to validate perceptions of events, acknowledged the futility of discussing "the meaning of a word or term in isolation from the discourse community of which the speaker claims membership" (Clarke 2001b, 36). Lerman (1994) writes that meaning, which is a socio-cultural phenomenon, is a product of discourse based on resolving conflicts and disparate positions; individuals are acculturated into meanings and thus the intersubjective becomes the intrasubjective. Our work was located securely in such a model of learning. However, in order to obtain the intersubjectivity that was our descriptive framework, and due to initial difficulties in respect of commonality of understanding, we first had to negotiate the intersubjectivity necessary for an initial negotiation of classroom vocabulary. That is, the prerequisite

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sense of intersubjectivity that Clarke (2001b) suggests is necessary for negotiation of meaning was missing in our early conversations and was itself the result of protracted negotiation based around a substantial number of lessons viewed in five countries. Additionally Lerman (2001) writes that a learning activity can set up a shared zone of proximal development (ZPD) which informs and is informed by individual ZPDs and which draws on intersubjectivity in order to create intrasubjectivity. Our work was, we argue, an example of a collective ZPD being exploited to the benefit of not only individual participants but the collective also. This accords with Jones and Mercer’s (1993) contention that successful learning occurs whe n two or more people share their knowledge or understanding so that a new insight is created that is greater than the individual contributions. In describing the development of our observational schedule we have attempted to show how our work reflected closely the process of learning as represented in the literature. It shows also, how our collaborative approach led to a schedule that was understood, satisfied our criteria in respect of illuminating lessons meaningfully, and was straightforward in its implementation for a multinational team of researchers drawn from diverse mathematics education traditions.

References Bills, C. (2001) Metaphors and other linguistic pointers to children's mental representations, In C. Morgan and K. Jones (Eds) Research in mathematics education volume 3, papers of the British Society for Research into Learning Mathematics, London, BSRLM. Clarke, D. (2001a) Complementary accounts methodology, in D. Clarke (Ed) Perspectives on practice and meaning in mathematics and science classrooms, Dordrecht, Kluwer. (33-52) Clarke, D. (2001b) Untangling uncertainty, negotiation and intersubjectivity, in D. Clarke (Ed) Perspectives on practice and meaning in mathematics and science classrooms, Dordrecht, Kluwer. (33-52) Cobb, P. and Bauersfe ld, H. (1995) (Eds) The emergence of mathematical reasoning: interaction in classroom cultures, Hillsdale, NJ, Lawrence Erlbaum. Collina, P. (2003) The rules of the game, London, Glaser, B. and Strauss, A. (1967) The discovery of grounded theory: strategies for qualitative research, New York, Aldine. Jenkins, E., (2000). Making use of international comparisons of student achievement in science and mathematics. In D. Shorrocks-Taylor & E. Jenkins (Eds.), Learning from others, Dordrecht: Kluwer. Jones, A. and Mercer, N. (1993) Theories of learning and information technology In P. Scrimshaw (Ed) Language, classrooms and computers, London, Routledge (11-26) 15


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Kaiser, G. (1999). International comparisons in mathematics education under the perspective of comparative education. In G. Kaiser, E. Luna & I. Huntley (Eds.), International comparisons in mathematics education (pp.140-150). London, Falmer. Kilpatrick, J., Swafford, J. and Findell, B. (2001) (Eds) Adding it up: helping children learn mathematics, Washington, National Research Council. Lave, J. and Wenger, E. (1991) Situated learning: legitimate peripheral participation, Cambridge, Cambridge University Press. Lerman, S. (1994) Metaphors for mind and metaphors for teaching and learning mathematics, In J. Ponte and J, Matos (Eds), Proceedings of the 18th conference of the international group for the psychology of mathematics education (PME 18, Vol 3, 144-151) Lisbon, Portugal Lerman, S. (1996) Intersubjectivity in mathematics learning: a challenge to the radical constructivist paradigm, Journal for Research in Mathematics Education, 27 (2), 133-150. Lerman, S. (2001) Accounting for accounts of learning mathematics: reading the ZPD in videos and transcripts, in D. Clarke (Ed) Perspectives on practice and meaning in mathematics and science classrooms, Dordrecht, Kluwer. (53-74). Ma, L. (1999) Knowing and teaching elementary mathematics, Mahwah, NJ, Lawrence Erlbaum. Moll, L. (1990) Introduction, in L. Moll (Ed) Vygotsky and education: instructional implications and applications of sociohistorical psychology, Cambridge, Cambridge University Press (1-27). Postlethwaite, T.N. (1988). Preface. In T.N. Postlethwaite (Ed.), The encyclopaedia of comparative education and national systems of education. Oxford, Pergamon. Schmidt, W.H., Jorde, D., Cogan, L.S., Barrier, E., Gonzalo, I., Moser, U., Shimizu, K., Sawada, T., Valverde, G.A., McKnight, C., Prawat, R.S., Wiley, D.E., Raizen, S.A., Britton, E.D. & Wolfe, R. (1996). Characterizing pedagogical flow. Dordrecht, Kluwer. Shulman, L.S. (1986) Those who understand: knowledge growth in teaching, Educational Researcher, 15 (2), 4-14. Theisen, G. & Adams, D. (1990). Comparative education research. In R.M. Thomas (Ed), International comparative education: practices, issues and prospects. Oxford, Butterworth-Heinemann. Van Den Heuvel-Panhuizen, M. (2003) The didactical use of models in Realistic Mathematics Education: An example from a longitudinal trajectory on percentage, Educational Studies in Mathematics 54(1), 9-35.

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Estudio comparativo de 5 paises europeos