A METHOD FOR CONDUCTING LONGITUDINAL

PAUL COBB and JOY W. WHITENACK

A METHOD FOR CONDUCTING LONGITUDINAL ANALYSES OF CLASSROOM VIDEORECORDINGS AND TRANSCRIPTS*

ABSTRACT. In this paper, we describe a methodological approach that can be used to ana lyze large sets of qualitative data such as classroom videorecordings and transcripts. The method emerged while conducting longitudinal case studies of four pairs of students' small group activity. In the first phase of the analysis, the data are dealt with on an episode- by episode-basis in chronological order. Sample episodes are given to illustrate how inferences made while analyzing one episode are viewed as initial conjectures that can be revised when analyzing subsequent episodes. In later phases of the analysis, these conjectures become data that are (meta-)analyzed to create chronologies that are structured by general assertions and yet are grounded in the particulars of students' mathematical activity. In the course of the discussion, we clarify the role of domain-specific psychological models and the influence of assumptions about the relation between psychological and social processes. We conclude by considering both the generality and the trustworthiness of the approach .

... The purpose of this paper is to describe a methodological approach that emerged while conducting case studies of four pairs of students' small group activity over a ten-week period (Cobb,

1995).

We first orient the

reader by clarifying the issues addressed when developing the analytical approach. We then outline the sample case studies and describe the various phases of the analysis. Against this background, we conclude by relating the approach to Glaser and Strauss'

( 1967) constant comparative method.

Additional issues considered in the course of the discussion include the trustworthiness and generality of the approach, the underlying assump tions made about the relationship between classroom social processes and individual psychological processes, and the role of domain-specific psy chological models.

1. ORIENTATION The primary methodological issue that we address is that of developing ways to systematically organize and structure analyses of large, longitudi nal sets of qualitative data. This issue is significant given that mathematics * The research reported in this paper was supported by the Spencer Foundation and by the National Science Foundation under grant No. RED-9353587. The opinions expressed do not necessarily reflect the views of the Foundations.

Educational Studies in Mathematics 30: 213-228, 1996.

©

1996 Kluwer Academic Publishers. Printed in the Netherlands.

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PAUL COBB AND JOY W. WHITENACK

education researchers are increasingly using methodologies such as teach ing experiments that generate large collections of video recordings and transcripts. In the case of the small group case studies, for example, the development of analytical procedures proved to be extremely problematic. We had previously completed detailed analyses of single small group ses sions of 15 or 20 minutes duration (e. g. , Cobb, Wood and Yackel, 1 992), and could have continued to conduct further analyses of isolated episodes. The challenge was to develop an approach that maintained a focus on the particulars of the students' mathematical activity while simultaneous ly making it possible to generate and test more encompassing assertions. General claims resulting from the longitudinal analysis could then be sub stantiated in an empirically-grounded way. It was as we struggled with this problem that methodological issues first became an object of reflection for us. These reflections contributed to the refinement of the methodological approach and ultimately, to the emergence of a viable means for analyzing the small group data corpus. It will be apparent when we present sag episodes that the analysis attempts to account for students' mathe� al learning as it occurs in social context. In this particular case, the context is that of the students' small group relationships as they were located within the classroom micro culture. The analysis therefore involves assumptions about the relationship between individual psychological processes and classroom social process es. This might seem self evident in cases where the data are collected in the course of a classroom teaching experiment. However, we would argue that any analysis that focuses on the process of students' mathematical devel opment necessarily reflects such assumptions regardless of whether they are articulated by the researcher. This claim, it should be noted, applies to a range of investigations conducted outside the classroom including teach ing experiments in which a researcher interacts one-on-one with students. During the discussion of the sample episodes, we will illustrate the influ ence that assumptions of this type have on the structuring and organization of the analysis. As a further point, it is readily apparent that empirical analyses of stu dents' mathematical development necessarily deal with their construction of particular mathematical conceptions. Domain-specific psychological models of students' activity therefore play a central role in the analysis. They both indicate which aspects of students' activity might be particularly significant and constitute a resource that can be drawn on when account ing for individual students' mathematical activity in social context. The sample episodes serve to illustrate these contributions.

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2. BACKGROUND TO THE SMALL GROUP CASE STUDIES

The initial motivation for conducting the four case studies was primarily theoretical. Neo-Piagetian and Vygotskian perspectives did not appear to be entirely appropriate to the goal of investigating the relationship between mathematical learning and small group interactions. From the neo-Piagetian perspective, social interaction is typically treated as a catalyst for otherwise autonomous mathematical development. Thus, although social interaction is considered to stimulate individual development, it is not viewed as integral to either this constructive process or to its products, increasingly sophisticated mathematical conceptions. Vygotskian perspec tives, on the other hand, tend to elevate interpersonal or social relations above psychological processes. In the case of adult-child interactions, for example, it is argued that the child learns by internalizing mental functions that are initially social and exist between people. In recent years, several attempts have been made to extend these arguments to small group inter actions between peers (Forman and Cazden, 1985; Forman and McPhail, A 1993). It was against the background of these two perspectives that an . I"' attempt was made to develop an alternative approach in which psycho logical and social processes were given equal emphasis. This approach involved identifying regularities in the children's small group relationships that remained stable over time, and relating them to analyses of the individual children's mathematical development. In addition to addressing this theoretical issue, the case studies touch on a pragmatic point of some significance. This concerns the extent to which small group collaborative activity facilitates children's mathemati cal learning. The viewpoint taken when conducting the case studies was that students' mathematical constructions have an intrinsically social aspect in that they are both constrained by the group's taken-as-shared basis for com munication and contribute to its further development. As a consequence, the analysis focuses on the learning opportunities that arise for children as they mutually adapt to each other's activity and attempt to establish a consensual domain for mathematical activity. The findings of several pre vious investigations indicate that small group interactions can give rise to learning opportunities that do not typically arise in traditional classroom interactions (Barnes and Todd, 1977; Davidson, 1985; Good, Mulryan and McCaslin, 1992; Noddings, 1985; Shimizu, 1993; Smith and Con frey, 1991; Webb, 1982; Yackel, Cobb and Wood, 1991). The case studies extend these analyses by relating the occurrence of learning opportunities to the different types of interactions in which the children participated. As a consequence, they indicate the extent to which the various types of inter-

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actions are productive for mathematical learning. A detailed presentation of the findings is given by Cobb ( 1995).

3. DATA CORPUS The data analyzed when developing the case studies consist of video recordings made during a year-long teaching experiment conducted in one American second grade classroom with nine-year-old students. In this classroom, the instructional approach developed in collaboration with the classroom teacher was generally compatible with recent American reform recommendations (e.g. National Council of Teachers of Mathemat ics, 1989). In each mathematics lesson, the students typically worked in pairs for 15 or 20 minutes and then participated in a whole class discussion of their mathematical interpretations and solutions. Two cameras were used to record these lessons and, during small group work, each camera focused on two group . a consequence, it wa possible to record approximately half of each o. ur groups' activity throughout the year. The recordings analyzed when conducting U1e case studies were those of 27 lessons conducted between January 23 and March 13 of the school year. These lessons were selected because, by mid-January, the classroom social norms for both whole class discussions and small group activity were relatively stable. As a consequence, most children were aware of and attempted to fulfill the teacher's expectations for small group activity. In addition, the corpus of 27 lessons proved to be extensive enough to address both the theoretical and the practical goals of the investigation. Additional data consisted of video-recorded individual interviews conducted with all the children in January just prior to the first small group session that was analyzed. These interviews further clarified the children's interpretive possibilities and provided a means of triangulating inferences made about their mathematical activity in the classroom. As a final point, we should stress that our motivation for introducing the small group analysis is not to report the case studies or to justify the findings. Instead, our intent is to use this analysis as a vehicle to clarify the methodological approach. We therefore omit a discussion of a number of issues including the teacher's role, the whole class and small group social norms, and the instructional activities. We would only note that the chil dren's small group activity was itself situated within the broader context of the classroom microculture (Bauersfeld, 1992; Schroeder, Gooya and Lin, 1993). For example, the children's small group activity was influenced by their realization that they would be expected to explain their interpretations and solutions in the subsequent whole class discussions. Conversely, their

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small group activity served as the basis for these whole class discussions. Thus, the classroom microculture both constrained and was sustained by the small group relationships the children developed (cf. Kmmmheuer, 1995; Wood, 1995).

4. THE SMALL GROUP SESSIONS 4.1. Initial Analyses The initial attempts to analyze the four pairs of children's small group activ ity were modeled after the work of Edwards and Mercer (1987) and Voigt ( 1985). In these approaches, the analyst attempts to identify regularities or patterns in the teacher's and students' interactions. However, concerted attempts to identify such patterns in children's small group activity proved unsuccessful because the regularities identified within a particular small group session frequently did not hold up in subsequent sessions. An exploratory analysis of one pair of children's activity across five consecutive sessions suggested an alternative approach. It became apparent that the expectations these two children had for each other's mathematical activity and the obligations they each attempted to fulfill were consistent across situations provided the relative sophistication of their individual mathematical interpretations was considered. In other words, although there were sometimes dramatic differences in the nature of the children's interactions both within sessions and from one session to the next, their expectations and obligations appeared to be stable once their individual construals of tasks and of each other's mathematical actions were taken into account. Further, these infened obligations and expectations indicated a way to characterize the social relationship the children established in an empirically-grounded way. The viability of this approach was confirmed when conducting the four case studies in that it was possible to identify cognitively-situated expecta tions and obligations for each of the four pairs of children that held across the ten-week period covered by the video-recordings. As this approach involves both psychological analyses of the children's individual math ematical activity and social analyses of their interactions, it necessarily addresses the central theoretical concern of the investigation. Below, we describe the various phases of the analysis. 4.2. Episode-by-Episode Analyses For each case study, the children's January interviews and their small group sessions were analyzed in chronological order. Within each small group

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session, the children's attempts to solve individual tasks were treated as distinct but related episodes. The analysis of each episode was structured by four themes. These were:

1. the children's expectations for and obligations 1 to the small group partner; 2. the mathematical meanings the children gave to their own activity, the partner's activity, and the task at hand; 3. the learning opportunities that arose for each child; and 4. the conceptual reorganizations that each child made (i.e. , their mathe matical learning). It is readily apparent that these themes reflect assumptions about the relationship between psychological and social processes. In particular, the notion of a learning opportunity indicates that mathematical learning is viewed as a process of conceptual self-organization as well as of encultur ation. Individual students' constructive activities are therefore considered to be socially situated in that they occur as they participate in classroom social processes. However, the proposed linkage between psychological and social processes is indirect in that, in the last analysis, it is the students who interpret others' actions and reorganize their mathematical activity. This view can be contrasted with alternative sociocultural perspectives in which is argued that internal psychological plane of mathematical thought is formed by internalizing social processes. In such accounts, the link between social and psychological processes is a direct one. As an illustration of the way in which the four themes structured the analysis, consider an episode that occurred in the first of over 100 small group sessions that were analyzed. In this episode, two children, Ryan and Katy, solved a sequence of horizontal addition number sentences that were sequenced so they might relate the current task to a prior result and thus move beyond counting by ones. The video-recording begins midway through the session as they are explaining their solutions to 48 + 18 =_to the classroom teacher. They have just solved 47 + 19 = 66. 1. Teacher: 48 plus 18. 2. Katy: That's just the same. 3. Teacher: What's just the same? 1

The notion of obligation is used here is

a

sociological construct that indicates a

regularity in social interaction (Voigt, 1985). This usage should be distinguished from more colloquial psychological discourse in which we talk of people feeling obliged to act

in certain ways. This latter usage focuses on an individual's interpretation of a situation, whereas the sociological use of the term is concerned with interpersonal relations.


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4. Katy: See, if you take one from 19, put it with the 47, it makes this just the same (points to 48 + 18 =_and 47 + 19 = 66). 5. Teacher: Do you see that Ryan? She's saying that 47 plus 19 equals 66. [She then repeats Katy's explanation in more detail]. She said it's just the same number. 6. Katy: Yes, cause you just take one from the 19 and add it to the 47, that makes .. . 7. Teacher: 48. 8. Katy: 48. 9. Teacher: 48, right. 10. Ryan: I know what she's trying to say, she's trying to say take one from here (points to 19) and add it to here (points to 47). 11. Teacher: Right. 12. Ryan: That would be the same answer. The preliminary analysis of this episode was organized according to the four themes listed above: 4.2.1. The Children's Social Relationship. Katy repeatedly attempted to explain how she had solved the task. How ever, the teacher's initial intervention, "What's just the same?" might have been crucial. Thus, the most that can be inferred is that Katy was obliged to explain her thinking in the teacher's presence. Ryan, for his part, seemed obliged to attempt to understand Katy's explanations, as indicated by his comment, "I know what she's trying to say." However, as was the case with Katy, the teacher's interventions might have been critical. Her com ment, "Do you see that Ryan?" might have been made with the intention of indicating the obligations she expected him to fulfill. 4.2.2. Mathematical Meanings. Katy's use of the compensation strategy (i. e., the use of a relationship such as 6 + 6 = 12 to derive 6 + 7 = 13, cf. Steffe & Cobb, 1988) did not seem novel and was consistent with her performance on thinking-strategy tasks administered in the January interview. As Ryan used the compensation strategy to relate 8 + 8 = 16 and 9 + 7 = _ in his interview, it seems rea sonable to infer that he did construct a compensation relationship when he interpreted Katy's explanation. 4.2.3. Learning Opportunities. Ryan had not solved the task at the beginning of the exchange and the possibility of relating successive tasks did not seem to occur to him. The manner in which he clarified what Katy was trying to say suggests that

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he did eventually comprehend how she had related the tasks. This in tum indicates that a learning opportunity might have arisen for Ryan when he interpreted her explanations.

4.2.4. Mathematical Learning. Ryan's final interpretation of the task appeared to be more sophisticated than that which he would have made if left to his own devices. However, the extent to which he reorganized his mathematical activity is unclear. His advance might have been specific to this particular exchange or, alternative ly, he might have made a major conceptual reorganization. The viability of these alternative conjectures can only be determined by taking into account his solutions to subsequent tasks. The overall goal of the entire analysis was to develop a coherent account of Ryan's and Katy's small group activity across the ten-week period cov ered by the videorecordings. The inferences made while analyzing indi vidual episodes were therefore viewed as initial conjectures that could be revised in light of the children's activity in subsequent episodes. In the next episode, Katy again used the compensation strategy, relating 49 + 17 to 48 + 18 66. = _

=

1. Katy: Easy, it's the same thing.

2. 3. 4. 5. 6.

7. 8. 9. 10.

Ryan: No it isn't. Katy: Oh yes it is. Ryan: It's one higher, it's one higher. Teacher: Let's take a look. Katy: This one's [48] one less than that number [49], so if you take one from here and add it to here that makes 49. There's 17 left so it has to be the same number. Ryan: It can't be because. . . Katy: I t i s Ryan. Teacher: Why can't it be? Ryan: You're adding two.

The same four themes were used to guide the preliminary analysis of this episode:

4.2.5. The Nature of Interactions. As was the case in the first episode, Katy again seemed obliged to explain her reasoning to Ryan, at least in the teacher's presence. However, on this occasion, Ryan did not appear to try to make sense of her explanation. Instead, he challenged her answer and attempted to explain how he had related the two tasks. In the first episode, there was no indication that he had developed a solution when the teacher asked Katy to explain her thinking.


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Here, in contrast, he had solved the task in a way that he believed he could justify. It therefore seems reasonable to conjecture that in situations where the children's independent solutions are in conflict, Katy was obliged to explain her reasoning to Ryan with the expectation that he would attempt to make sense of what she was saying. For his part, Ryan was obliged to both challenge Katy's explanation and explain his solution. The extent to which either child attempted to understand the other's explanation is open to question.

4.2.6. The Children s Social Relationship. Katy's interpretation of the task appears to be consistent with the infer ences made when analyzing the first episode. Fmther, Ryan's contention that "You're adding two," clarifies the conceptual advance he made when interpreting Katy's explanation in the first episode. Perhaps he had under stood how Katy had mentally transformed 47 + 1 9 into 48 + 1 8 in the first episode without reflecting on why, for Katy, it followed that the two sums would be the same. To reconcile this conjecture with his performance in the January interview, we might further speculate that he could only create compensating relationships of this type with smaller numbers, perhaps ten or less (cf. Neuman, 1 987). 4.2. 7. Learning Opportunities. As Ryan did not appear to try to make sense of Katy's explanation, there is no indication that a learning opportunity arose for him in this episode. Further, although Katy was obliged to explain her thinking in response to Ryan's challenges, she did not seem to modify her interpretation of the task when doing so. Consequently, it seems unlikely that a learning opportunity arose for her. 4.2.8. Mathematical Learning. The conceptual reorganization that Ryan made in the first episode has been clarified by considering the interpretations he made in the second episode. It would seem that his advance was highly situation-specific. It can also be noted that, in the second episode, Ryan did attempt to relate successive tasks, whereas, in the first episode, he seemed unable to do so independently. Thus, he had come to the realization that the tasks could be solved in this way, and, as a consequence, the development of thinking strategy solutions was now taken-as-shared by the two children. The two episodes illustrate that the analysis involved a continual move ment between particular episodes and potentially general conjectures. Spe cific instances of the children's mathematical activity that conflicted with current conjectures became refutations in that they led to an examination

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of suppositions and assumptions that underpinned the conjectures. The sample episodes also clarify the role of domain specific psychological models. In particular, the attention given to the two children's possible construction of compensating relationships reflects the influence of prior analyses of arithmetical development. These prior analyses indicate that the construction of compensating relationships involves a qualitative shift in children's additive conceptions when compared with both counting solu tions and more elementary thinking strategy solutions (Steffe and Cobb, 1 988). Thus, the construction of these relationships was seen to be signifi cant against the background of the psychological model that informed the analysis. It should be noted that, for ease of explication, the discussion of the sam ple episodes focuses on observable solution strategies. We have therefore not attempted to infer the quality of the children's mathematical expe riences as they developed thinking strategy solutions. Inferences of this type are, however, central to the case studies and readers interested in the psychological constructs that emerged in the course of the analysis are referred to Cobb (1 995).

4.3. Analyses of Analyses The episode-by-episode analyses of the video recordings and transcripts were, in fact, only the first of three phases of the analysis. In the sec ond phase, the chains of inferences and conjectures made while viewing the recordings of each pair of children's small group activity themselves became data that were (meta-)analyzed to develop chronologies of:

1 . two children's obligations and expectations (i.e., their social relation ship) and 2. each individual child's mathematical activity and leaming. These chronologies gave an overview of developments over the ten week period covered by the case studies. However, in the process of constructing the three chronologies for each pair of children, some of the identified relationships between psychological and social processes faded into the background. As a consequence, the chronologies were not fully integrated. In the third phase of the analysis, the three separate chronologies were synthesized to create a single, unified chronology that would serve as basis for the written case studies. It should be stressed that the delineation of the three phases of the analysis was made retrospectively. As we have noted, the methodology emerged as attempts were made to interpret the video-recordings. In the remaining pages of this article, we briefly outline the results of the case


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studies, discuss the fit between the methodology and Glaser and Strauss' constant comparative method, and consider its trustworthiness.

5. OUTLINE OF THE RESULTS In introducing the case studies, we noted that they were conducted for both pragmatic and theoretical reasons. For the sake of completeness, we briefly summarize the results. Pragmatically, our purpose was to investigate the extent to which various types of small group interactions are productive for mathematical learning. In the course of the analysis, several distinct types of interactions were identified. Further, it became apparent that the four groups of children had established markedly different social relationships. In those that were most conducive to mathematical learning, the children created a genuine basis for mathematical communication and neither child was established as the mathematical authority of the group. The theoretical goal of the case studies was to develop an interpre tive perspective on small group activity that brings both psychological and sociological processes to the fore. The view that emerged was that of a reflexive relationship between the children's mathematical activity and the social relationships they established. On the one hand, the children's situated cognitive capabilities, as inferred from both the interviews and their activity in the classroom, appeared to constrain the possible forms that their small group relationships could take. On the other hand, the relationships that the pairs actually established constrained the types of learning opportunities that arose and thus profoundly influenced the chil dren's construction of increasingly sophisticated mathematical ways of knowing. These developing mathematical capabilities in turn constrained the ways in which their small group relationships could evolve. Thus, in this account, children's developing mathematical capabilities constrain the nature of their small group relationship, and this in tum constrains their learning opportunities and thus their mathematical development. It is in this sense that psychological and sociological processes are seen to be reflexively related with neither dominating the other.

6. THE FIT WITH GLASER AND STRAUSS' METHOD With hindsight, it is apparent that the general approach taken when ana lyzing the small group data is consistent with Glaser and Strauss' (1967) constant comparative method. Glaser and Strauss argue that the devel opment of theoretical constructs should occur simultaneously with data

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collection and analysis. They call such constructs grounded to indicate that they are rooted in the data analysis process. In their method, data are initially treated as isolated incidents that form preliminary categories. As new incidents occur, these are compared or contrasted with previous inci dents to determine if a fit is possible. For example, in the case of Katy's and Ryan's small group activity, initial categories might involve inferences about each child's role in solving mathematical tasks. We saw, for ex�m ple, that while Katy seemed obliged to explain her solution methods to Ryan, Ryan seemed obliged to solve tasks using his own method when he had a way to proceed. In line with Glaser and Strauss' approach, these initial inferences were treated as conjectures about the children's social relationship. It is in this process of reconciling provisional categories with subsequent data that newly formulated categories become stable and evolve into explanatory constructs. Examples of such constructs include the vari ous types of small group interactions identified in the course of the analysis and the specific psychological constructs used to interpret the individual children's mathematical activity. In our prior discussion, it was noted that the process of analyzing the small group interactions could be described as a zigzag between con jectures and refutations. This movement between conjectures and refuta tions is compatible with Glaser and Strauss' account of the construction of grounded theory. For instance, initial conjectures about the children's small group interactions focused on their expectations for their partner's activity and the obligations implicit in their own activity. The process of interpreting the children's subsequent actions within a small group session involved refining, redefining, or refuting initial conjectures about their obligations and expectations. Assertions made about the individual chil dren's roles during small group activity emerged as a result of this process. Similarly, inferences about the children's mathematical meanings, possi ble learning opportunities, and conceptual reorganizations were developed by zigzagging between conjectures and refutations. As a consequence, the theoretical constructs that emerged in the course of this process were empirically grounded in the data. This approach therefore instantiates what Glaser and Strauss refer to as theory as a process (p. 9).

7. TRUSTWORTHINESS An important issue that remains to be addressed concerns the trustwor thiness of analyses developed by using the general approach we have outlined. The convention when reporting qualitative analyses of this type is to illustrate general claims and assertions by providing sample episodes


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(Atkinson, Delamont and Hammersley, 1 988; Taylor and Boydan, 1 984). However, a difficulty arises in that the interpretations of these sample episodes frequently do not seem justified when they are considered in iso lation from the rest of the data. In the case of the small group analysis, for example, alternative explanations were often ruled out only by referring to general aspects of a small group's social relationship. However, the latter were corroborated by the very regularities across episodes that the sample episodes are meant to illustrate. We saw, for example, that initial con jectures concerning Ryan's understanding of Katy's mathematical activity (i. e. , her use of 48 + 1 8 66 to solve 47 + 1 9 ) had to be reconciled with inferences made about Ryan's use of the compensation strategy during the preliminary interview. His activity in the first of the two episodes we dis cussed suggested that he might have reorganized his interpretation of the task as he interacted with Katy. The analysis of the second episode clarified the extent to which Ryan had reorganized his thinking and led to further conjectures about his mathematical activity. These conjectures were then investigated when subsequent episodes were analyzed. As this recapitula tion illustrates, the interpretation of specific episodes and the delineation of general regularities are interdependent in that each is formulated only in relation to the other. This interdependence of the general and the particular can be viewed as a strength of the analytical approach in that it acknowl edges the reflexive relationship between students' mathematical activity and the contexts in which they act- their activity is both context-specific and context-renewing. This approach is, however, at odds with alternative methods of analysis based on the assumption of a one-to-one mapping between observed behavior and cognition. It should also be noted that analyses developed by using the method we have outlined do not purport to reveal the objective essence of teachers' and students' activity. The trustworthiness of the findings depends on the extent to which they are reasonable and justifiable given the researcher's interests and concerns (cf. Erickson, 1 986). In the case of the small group analyses, for example, it is important to acknowledge that other plausible interpretations of the children's mathematical activity could be made for alternative purposes. Several considerations contribute to the reasonable ness of and justifiability of analyses developed by following the general approach we have described. The most important of these stems from the use of an ethnographic approach in which the data set is analyzed system atically by continually testing provisional conjectures. As a consequence, the resulting claims and assertions can be justified by backtracking through the various phases of the analysis. In the case of the small group analysis, for example, the results can be justified by referring to the chronologies, the =

= _

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initial interpretations and conjectures, and, if necessary, the video record ings and transcripts. In general, a record of the process of developing an analysis provides a means of demonstrating its empirical grounding (Gale and Newfield, 1 992). A second consideration that enhances the credibility of an analysis is the prolonged engagement of the researcher with the par ticipants of the study (Lincoln and Guba, 1985. ) The first-hand experience of observing and interacting with teachers' and students' over an exterid ed period of time constitutes a crucial source of insight when attempting to account for their activity. A third consideration concerns the extent to which the analysis has been critiqued by other researchers (i.e., peer debriefers). It is particularly helpful if some of the debriefers are familiar with the participants of the study as this enables them to judge whether the analysis "rings true" at a global level. For their part, debriefers who are not familiar with the setting might be able to offer alternative interpretations that bring to the fore apparently self-evident suppositions and assumptions.

8. CONCLUSION

Our discussion of the analytical approach that emerged while conducting the case studies is consistent with Glaser and Strauss' desire "to stimu late other theorists to codify and publish their own methods for generat ing theory" (p. 8). We have attempted to illustrate that the approach we have described has considerable generality for analyzing relatively large longitudinal data sets in mathematics education. In particular, analyses conducted by working through the three phases have the potential to tease out theoretically-significant patterns and regularities in an empirically grounded way. It should, however, be clear that the actual enactment of this approach involves a number of additional considerations. For example, we noted that the themes used to structure the episode-by-episode analy sis reflected specific assumptions about the relationship between psycho logical and social processes. Similarly, we observed that the analysis of individual students' activity was guided by domain-specific psychological models. The general approach might therefore be thought of as providing an overall general orientation, the details of which have to be worked out in line with the researcher's theoretical commitments and interests. We hope that others will follow Glaser and Strauss' lead and enter this discussion of research methodologies that fit the problems and issues of mathematics education.

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Dept. of Teaching and Learning, Vanderbilt University, Peabody College, Box 330, Nashville, 400, USA.