Journal of Mathematics Teacher Education (2005) 8:363–392 DOI 10.1007/s10857-005-3848-3
Springer 2005
RICARDO NEMIROVSKY, CARA DIMATTIA, BRANCA RIBEIRO and TERESA LARA-MELOY
TALKING ABOUT TEACHING EPISODES
ABSTRACT. This paper examines two types of discourse in which teachers engage when discussing case studies based on classroom episodes, and the ways in which the availability of video data of these episodes may motivate a shift in the mode of discourse used. We interviewed two pairs of secondary school mathematics teachers after they had read a case study based on a 16-minute mathematics classroom episode taped in a secondary school in the United States. During each interview, a multimedia version of the case study, including video of the original episode, was available to the participants. We identify two modes of discourse engaged in by the teachers during the interviews: Grounded Narrative and Evaluative Discourse. We examine and identify the characteristics of the two discourse forms, drawn from both video and textual analysis. These characteristics are self-reflective talk, perspective, ethics, and linguistic patterns. The identification of two modes of discourse is relevant for researchers and teacher educators using case studies or video recordings. In addition, the findings provide insight into how teachers are ‘‘seeing’’ classroom events in a video case study.
TALKING ABOUT TEACHING EPISODES Twenty years ago a distinctive strand of literature on the roles and uses of cases in teacher education emerged (Merseth, 1996; Shulman, 1986). Inspired in part by the long-term use of cases in management and medical education (Barnes, Christensen, & Hansen, 1994), this literature claimed that using cases of teaching could be pivotal for teacher education. A rationale given was the need for teachers to learn by reflecting in ways that are situated in the nature of teaching. I envision case methods as a strategy for overcoming many of the most serious deficiencies in the education of teachers. Because they are contextual, local, and situated—as are all narratives—cases integrate what otherwise remains separated (Shulman, 1992, p. 28).
Departing from the view of teaching as a process of applying theoretical principles and portraying it as a practical endeavor, several educators elaborate on the epistemology and learning associated with ‘‘practical knowledge’’ (Fenstermacher, 1994; Shulman, 1986; Sykes & Bird, 1992), whose growth has traditionally been associated with rich examples and engagement with specific situations of teaching.
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Recent research on teacher thinking has broadened the conceptualization of the teacher from the one who operates with a narrow set of prescribed theories of propositions to one who defines his or her knowledge as situation-specific, context dependent, and ever emerging.... Teacher action derives from induction from multiple experiences, not deduction from theoretical principles (Merseth, 1996, p. 724).
Different taxonomies have been proposed for the identification of different types of cases and ways of discussing them. For example, Sykes and Bird (1992) propose that cases can be created and treated as (1) instances of theories, (2) problems for deliberate and reflective action, (3) material for the development of narratives, and (4) material for the development of casuistry, that is, the internal and tacit logic developed through the consideration of multiple cases. Shulman (1992) distinguishes between cases ‘‘as occasions for offering theories to explain why certain actions are appropriate’’ (p. 3) and cases as ‘‘vehicles for inquiry and debate regarding proper ethical and moral behavior’’ (p. 7). In all instances, what counts is not only the content and structure of the case itself but also the ways in which it is discussed and talked about: ‘‘It matters both what is discussed and how it is discussed’’ (Merseth, 1996, p. 727). Many case studies are framed as research papers; therefore, teachers talk about case studies often relates to their views about research literature. A few studies have been conducted into how educational research literature is read and talked about by teachers, and how teachers tend to feel alienated from the academic styles that they see represented in these papers. Kennedy (1997) for example, gave samples of research papers adapted for teachers to read and then interviewed them about their reactions. She discussed the interplay between an authors conclusions in the paper and the prior beliefs and experiences that teachers often used to the evaluate them. However, this work did not address methods of discourse engaged in by the discussants, or the impact of the use of multimedia on styles of discourse. This paper will attempt to address these issues.
RESEARCH QUESTIONS The research questions addressed in this study are: • What types of discourse do teachers engage in when discussing classroom-based cases? • In what ways does the availability of classroom video motivate a shift or further development in their modes of discourse?
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• How do these discourse modalities relate to the mathematical content of the classroom episode? These research questions are of great relevance for researchers and teacher educators, so that they are aware of what discourses are being enacted, how to stimulate shifts across them, and how the use of video can play in role in reflective discussion. In this study, we identify and elaborate on two different ways in which talk about cases of teaching occurs. We call one of these modes of talk or discourse Grounded Narrative, whose aim is to articulate descriptions of classroom events accounting for the available evidence. We call the second type of discourse Evaluative Discourse. This type of talk centers on the values, virtues and commitments at play in the case. We find in the literature references to narrative construction (e.g., Richert, 1991; Shulman, 1991), and to teacher evaluative talk (Harrington & Garrison, 1992; Levin, 1993; Seago, 2000). However, a detailed analysis of these two types of discourse, and of how teachers project attitudes and meanings through these modes of talk, is missing in the literature. We also investigate the question of whether the availability of the classroom video referenced in the written case study motivates the development of the modes of discourse used. We conjecture that the actual use of video is likely to have a deep impact on how a case is ‘‘read’’ or ‘‘viewed’’. Thus, a case study that includes the classroom video allows the reader/viewer to ascertain the extent to which the conclusions of the case study seem well grounded and to generate alternative interpretations (Sherin, 2004).
METHODOLOGY The Study We conducted and filmed interviews with three pairs of teachers. All interviewees teach mathematics in schools located in the Boston, Massachusetts area of the U.S. Prior to their interviews, the teachers were asked to read a text-only research paper to prepare for discussion. During the interviews they had access to the multimedia version of the research paper, as well as their printouts of the original paper. In this way, we explored how the availability of the classroom video might make a difference in teachers modes of talk and reflection. The interviews were designed to be open conversations rather than scripted exchanges. The participants were not requested to analyze or
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discuss the video in any particular way. The interviewer did not follow a list of questions, but tried to generate interventions that would make clearer and more explicit the issues that the participants cared about. This interview approach requires that the interviewer is not a neutral observer, but a participant whose role is to motivate the conversation and to make it as explicit and clear as possible. In the following discussion, we will use the terms conversation and interview interchangeably since often the talk flowed freely without prompting from the interviewer. Apart from Jesse Solomon, all names used are pseudonyms.
Video Analysis Our approach to the analysis of the videotaped interviews shares a number of commonalities with Interaction Analysis as described by Jordan and Henderson (1995) and the interpretive approach described by Packer and Mergendoller (1989). Rather than approaching the filmed interviews with a predetermined coding scheme, we allowed the analysis to ‘‘emerge from our deepening understanding’’ of the events unfolding on the videotaped record (Jordan & Henderson, 1995, p. 43). We analyzed the interviews with a sociolinguistic approach (Goffman, 1981). Accordingly, we developed detailed transcriptions that annotated tones of voice, gestures, utterance overlaps, gaze direction, changing speed of utterances, and silences. Transcript annotations are described in the Appendix. With the assistance of a sociolinguist experienced in the multilayered study of conversations (Branca Ribeiro), we produced a full transcription for these three interviews. Through close readings of these transcripts, we worked to identify ways in which the participants position themselves in the discussion, as well as how the classroom events resonate in their life experiences. The goal of this study was to examine a few cases in depth with the idea that any conversation among teachers about classroom events reflects certain types of discourse. While a given conversation can be more or less clear in this respect, forms of discourse are always present. We were not interested in describing all possible forms of discourse; rather, we strove to compare just two. Our characterization of evaluative and grounded narrative forms of discourse does not rely on an isolated list of specific traits, but on the articulation of examples; that is, of excerpted transactions in the conversation which convey the full sense of these discursive traits. The reliability of the analysis is not given by the coincidence of interpretations among researchers, but by making available to readers rich examples
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transcribed in detail, so that it is possible for them to recognize (or not) discursive forms similar to the ones proposed by the authors.
The Research Paper: The Content In order to conduct the study we chose a research paper entitled ‘‘Mathematical Conversations’’, from which the authors had also developed a multimedia version of the text (Solomon & Nemirovsky, in press). The paper examined a 16-minute classroom conversation, captured on video in a high school math class. The two main themes of the ‘‘Mathematical Conversations’’ paper are the nature of open-ended problems and the sources for the ‘‘sense of direction’’ emerging in a classroom conversation. The authors argue that what makes a mathematics problem open-ended is not so much its textual definition but the classroom culture within which it is discussed and figured out. They also contend that the sense of direction of a mathematical conversation does not follow pre-planned paths; instead it is co-developed by the teacher and students. We include here a description of the classroom episode encompassing only those aspects that the reader would need in order to understand the transcript of the interview study. The episode followed a question posed by one student (Maria) about a homework problem that she could not solve. At the start of the class, the teacher, Mr. Solomon, asked the students whether they had had difficulties with the homework, intending to devote a few minutes to review the assignment. Maria said that she did have difficulties with Problem #18, which gave a sequence of four numbers: 1, 8, 27, 64,..., asked students to graph at least six points, and to decide whether the sequence appeared to have a limit. Discussion of this problem was not part of the lesson plan for that day. Maria started by saying: ‘‘I couldnt figure out what the next two would be. I figured- I did the differences, I did multiplication, I tried times 2 plus 4, times 2 plus 3. I tried that. I couldnt get it. My mom couldnt get it, and she told me just clean my room.’’
Students started to brainstorm possible rules including power and linear relationships (e.g., 8Æ8=64; (1+8)Æ3=27; etc.). Mr. Solomon recorded their ideas on the overhead projector. This interaction had complex emotional and interpersonal nuances. While all ideas were accepted for consideration, some were refuted with irony and others engendered laughs by the providers of the suggestions themselves. At one point Mr. Solomon asked the group whether they wanted him to
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reveal the 5th term in the sequence. The group was divided. Some students rejected that possibility because they felt that they should be able to get it by themselves. Jamal argued that he had the solution but would not tell it in order to ‘‘let others think.’’ After being pressured by the students and Mr. Solomon, Jamal agreed to tell his solution. Jamal thought that the solution was based on successive differences. However, Jamal soon realized that he had made a mistake (he had miscalculated one of the first differences as 21), therefore his idea would ‘‘not work.’’ However, having the first differences written on the transparency, other students started to play with patterns such as the second digit being 7, 9, 7,... and the first digit being the sequence of odd numbers, or the previous one plus 1, plus 2, plus 3, and so on. Several students gave their opinions along the way on the value of the ideas offered and of the conversation itself (e.g., whether Mr. Solomon should tell the 5th number, requests to ‘‘move on’’ when certain approaches seemed unfruitful, or assertions that four numbers are too few to work with). Students proposed different 5th numbers for the sequence: 121, 133, 113, and 123. Naomi made the case for 123 by postulating that the 4th first difference would be 59 because the second digit alternates 7 and 9 and the first digit, 5, is next in the sequence of odd numbers (so that 59+64 =123). From the researchers perspective, this defense of 123 was a turning point of the conversation because hitherto the class held a tacit assumption that there was a unique ‘‘right’’ number known by Mr. Solomon. This assumption was also shared by Mr. Solomon in the sense that he did have in mind a target value of 125. Mr. Solomon responded to Naomis argument for 123 by saying, ‘‘Okay, that could work.’’ His assertion marked an important shift for the class by acknowledging the multiplicity of possible sequences. Next, Nadia proposed the answer of 121. She started by asserting, ‘‘I didnt stop at finding the difference between the numbers given. I found the difference between the differences.’’ Apparently, Nadia had generated 121 by assuming that the third difference of 6 remained constant. After correcting an arithmetic mistake pointed out by Mr. Solomon, she changed the 121 into a 125. The bewildered reaction from some students was immediate, ‘‘This is ridiculous! Differences from differences!’’ ‘‘This is crazy.’’ Mr. Solomon recognized the connection between polynomial powers and the constancy of successive differences. This is how he described it in his journal: Although I knew something about the use of differences in determining the kind of polynomial equation, these ideas were not fresh in my mind and I was not
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completely confident that I knew how or why a constant third difference indicates a cubic function. It may have taken until the end of this episode for me to see clearly.
Mr. Solomon explained to the class that even though he or the textbook had an answer in mind, there are infinitely many possible rules to continue the sequence; it just happens that the set 1, 8, 27, 64 is a familiar one, suggesting to many people the rule 13=1, 23=8, 33=27, … Several students reacted with a sense of ‘‘of course’’ (‘‘Oh, yeah, I knew it!’’). Shortly afterwards, Mr. Solomon pointed out the mathematical significance he had recognized in this technique of successive differences: ‘‘So, what actually you just discovered is this method that people use to find out what kind of function describes their sequence. So, just by taking successive differences, if you kept doing this, and said, Oh! Its always 6, that would automatically tell you that youve found a 3rd degree function.’’
The Research Paper in Multimedia Format: VideoPaper Videopapers1 are multimedia documents that include a text frame, a video frame, and an image frame, the contents of which are interrelated in multiple ways. (see Figure 1). Videopapers can be seen with a web browser, and thus are readily accessible by a wide audience. All the components are linked and synchronized. The ‘‘Mathematical Conversations’’ videopaper contains the full 16 minute classroom video on which the text was originally based; the video can be played, scrolled through, stopped, rewound, or otherwise viewed according to the users preference. The video is captioned with the transcript of the conversation; the full transcript is also included within the text area of the videopaper. The full text of the original paper is also included, with the addition of buttons inserted in the text which play intervals of video associated with particular passages. A slide show of images is displayed alongside the video, showing an alternate view of the classroom and additional information about the mathematics being performed.
DATA ANALYSIS Interview 1 Grounded Narrative Discourse The Interview with June and Ron June and Ron are first-year secondary school mathematics teachers. They were classmates in graduate school and seem comfortable in
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Figure 1. Example of a VideoPaper with arrangement of video, image and text elements identified.
conversation together. They had both read the ‘‘Mathematical Conversations’’ case study before arriving at their interview, which lasted for an hour and a half. Throughout the interview, June and Ron need little intervention from Teresa, the interviewer. Typically, they begin to answer her specific questions and then turn to other issues and interests. They continue each others utterances and continually refer to incidents in their own experience. At the beginning of the interview, Teresa gives some background on how to navigate through the ‘‘Mathematical Conversations’’ videopaper. They experiment with some of the videopaper elements for a few minutes. June asks for background on what the class had been studying when this homework problem came about, and why they worked through the problem using differences of differences and not factoring. They then watch the video in its entirety. The next hour is spent discussing the videopaper. Teresa then brings up the idea of open-endedness, and Ron expresses his interest in this question, relating it to his experience as a teacher. There is some discussion about lesson planning and the definition of open-ended: whether Mr. Solomon had meant this to be an open-ended discussion, or had a particular answer in mind. Ron and June then had a lengthy conversation, a segment of which is presented below, in which they track the series of events leading to
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the idea of differences of differences and develop a narrative to explain the students thinking and participation. They continue tracking this idea of differences of differences until nearly the end of the interview. Annotated Transcript June came to the interview interested in figuring out why Solomons class had solved a problem of number sequences using successive differences. For June, this solution was an unusual one, and she wondered whether it would have ever taken place in her own class. This concern led Ron and June to work on tracing the origins of this idea within the filmed classroom conversation. Their data analysis was grounded in their non-linear examination of the video, with references back to the text paper, and took a form that we call ‘‘Grounded Narrative.’’ For this section, we chose a 4-minute segment during which June and Ron discussed the origins of the ‘‘differences of the differences’’ idea and developed a growing sense of what kind of persons the students were and of their subjective experiences. Our commentaries are inserted throughout the transcription, in an attempt to articulate how a grounded narrative unfolds. For clarity, the transcript is annotated using conventions defined in the Appendix. Segment 1. The following excerpt begins right after Teresa asked ‘‘Do you have any sense that there is one of those students that is a better student than the others?’’ Ron said no, but June said yes and that ‘‘Jamal and Nadia probably are used to struggling with stuff, working it through, playing with the numbers.’’ Teresa asked ‘‘Can you show me?’’ June pointed to instances of Jamals style of work, and then of Nadia: Junes observations in [1] and [3] about Nadia combine her memories from both her reading of the paper and her watching of the 1. June:
[looking at paper] and then Nadia just keeps plugging away, and then shes- she realizes she did (.) something ahm wrong with her subtraction, [turns page fwd.] 2. Teresa: mmm mmm. 3. June: and (.) she said ‘‘okay, wait (.) I might have a different number.’’ I forget where that is [Turns page back.] And shes- (.) like working away there, while yknow other people are discussing stuff.= 4. Teresa: =mmm mmm.=
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video. The image of Nadia ‘‘plugging away’’ and ‘‘working away’’ while everyone else is talking comes from the video (nothing is written to that effect in the paper); however, June looked for a specific, remembered quote by Nadia in the text [3]. June refers to the following piece of transcript: ‘‘Nadia: wait, my numbers might change then’’. Note Junes use of ‘‘where’’ in [3]: it alluded to a particular page in the paper, as well as to a particular moment in the classroom episode during which Nadia had uttered her request to ‘‘buy time’’ to correct her mistake and change her prediction. This overlapping page/moment was part of a broader classroom circumstance during which Nadia was intensely working out her ideas while the class discussed other issues, and of Junes attempt to validate her impression of Nadia being used to ‘‘struggle with stuff.’’ This is an example of what we call ’’zooming in’’, that is, examining the particulars of a specific event (e.g., Nadia fixing her mistake) while deepening the broader circumstances it is part of (e.g., Nadias style of work, her arithmetical mistake opposed to her reasoning, etc.). The conversation goes on: 5. June:
6. Teresa:
=okay, yeah. [Finds reference] Nadia says [quoting, see Figure 2] ‘‘okay. well I- I didnt stop at finding the differences.’’ Shes the one who went to the differences between the differences.= I think. (...) =mmm mmm
In [5] June found the text she was looking for (see Figure 2), but instead of reading the line she had been quoting in [3] she reads one that comes before (‘‘I didnt stop at finding the differences’’); her choice shifted the conversation from describing Nadias style of work to addressing the question Who had come up with the ‘‘difference of differences’’ idea? with the answer of Nadia. However, she immediately added hesitation to her answer (‘‘I think’’), which had the effect of leaving the question somewhat open. Ron, in his first utterance in this
Figure 2. Text of the case study that June quotes from and references in [5].
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segment (see [12]) will propose an alternative: ‘‘Maria said she did the differences’’. The point we want to highlight is that a grounded narrative discourse attempts to address many questions at once. While June was looking for a transcript line that would show Nadias work style, she found another line, next to the one she was looking for, that seemed to answer another question she had in mind: how did the ‘‘differences of differences’’ idea come about? In other words, a grounded narrative discourse can work on many layers in the text or video that strives to narrate, referring to students styles or the growth of ideas, and that it often shifts across them. Segment 2. 7. June:
8. Teresa:
[looking at transcription, see Figure 2] And then she got 12, (.) and 16, and then (.) Jesse says ‘‘maybe its 18.’’ (laughs) mmm mmm.
In [7] June describes the first two utterances reproduced in Figure 2. Note how her description adopts the form of an ‘‘and then’’ narrative: ‘‘And then she got 12, (.) and 16, and then (.) Jesse [Mr. Solomon] says maybe its 18. ’’ This is a ‘‘micro-narrative’’ because it narrates two utterances that took only 18 seconds. As such, it is another case of ‘‘zooming in’’. Another important quality of this micro-narrative to point out is the way in which the narrative reflects selective choices. For example, June did not include Nadias doubt about her 16 (‘‘wait, did I get 16?’’).
9. June:
10. Teresa: 11. June:
And then Nadia says ‘‘okay wait,’’ and then she she goes back. (..) [turns to next page] And shes shes plugging away at the numbers while everyone else is talking, okay, then she comes back. mmm mmm. [long pause, in which June seems to be reading the paper] And then she says ‘‘125.’’ (.) So, she obviously (.) made the correction and went back up, and did (.) the reverse of taking the differences by adding on 6 to get the next one, and then 24 to get the next one and then 125. Cause thats not recorded in the conversation, nor in the video.
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June continues extending her narrative to encompass the subsequent events. She says that Nadia ‘‘goes back’’ means that Nadia returns to her notes to work out the new prediction by herself. Junes assertion in [9] that while ‘‘everyone else is talking’’ is likely to reflect both Nadias absence in the ensuing papers transcription as well as Junes memory of having seen Nadia in the video working by herself. Note that June excludes all the talk among students between ‘‘she goes back’’ and ‘‘then she comes back’’: because she is tracking Nadia, what everyone else is saying becomes a background to Nadias being away. June marks Nadias ‘‘coming back’’ by making her publicly announce ‘‘125’’. In the episode Nadia says several times that she got 125, interposed with other dialogues going on in between. June puts aside this multiplicity and the intervening dialogues by compacting all of that into ‘‘and then she says 125’’. This is another aspect of a grounded narrative: compressing multiple transactions into one event or utterance. We call this ‘‘zooming out’’ in contrast to ‘‘zooming in’’ which, as we have seen, entails pulling apart all the aspects that take place simultaneously or immediately together in an event or utterance. June asserts that Nadias process of getting 125 is not documented in the paper or in the video. She feels that she is inferring how ‘‘obviously’’ Nadia had got it. However, there is an exchange in the video that documents it (see Figure 3). It could be that June knows how Nadia got the 125 because she thought of this result by herself, without a conscious recollection of the exchange. The point here is that a grounded narrative combines events experienced as ‘‘having been seen’’ with others experienced as ‘‘being inferred’’, and that in many cases the narrator makes this distinction explicit, as June did in [11].
Figure 3. Transcript of video segment included in the VideoPaper, the text of which is not included in the original paper.
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Segment 3. Grounded Narrative also changes the scope or boundaries of events. Ron has been listening to June and flipping through the paper. He suddenly jumps in:
12. 13. 14. 15. 16.
Ron: Teresa: Ron: June: Ron:
17. June:
Maria said she did the differences. (..) where are you? she said when she went home,= =yeah= =she did the differences. She did multiplication.(..) Alright, so thats where the word differences, I think, comes up first right?= =ah
So far Ron had been sitting quietly, apparently removed from the conversation, but what June had said in [5] (‘‘Shes [Nadia] the one who went to the differences between the differences.’’) sparks his search in the paper. In [12] he announces a different possibility. Teresa responds by asking for bearings in [13], ‘‘Where are you?’’ It is understood that one is wherever one is paying attention or referring to. ‘‘When she went home’’ was enough of a reference point to help June and Teresa locate Marias intervention at the beginning of the transcript included in the ‘‘Math Conversations’’ paper (see Figure 4). The point of this commentary is to illustrate the shift in topics (i.e., from Nadias style of work to the origin of the ‘‘differences of differences’’ idea), and the accompanying shift in the range of events encompassed by the narrative (i.e., Marias first utterance became a new beginning for a narrative that would end with the same return of Nadia with her ‘‘125’’). This segment also illustrates the use of multiple data sources for the examination and discussion of ideas.
18. Ron:
19. 20. 21. 22. 23. 24.
June: Ron: June: Ron: June: Ron:
okay. (.) Then she says- wait- no then Margaret says theres no sequence, [flipping pages] the only comment (...) but then for some reason, where does the differences come out? Jamal again. (...) like who says 1 8 27 64? (...) I mean, 7 right here. 27 (..) Jamal, here.= =yeah. the differences between, the differences between=
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Figure 4.
Text of the case study that Ron alludes to in [12].
In [18], Ron closes the first instance of ‘‘differences’’ and inserts a ‘‘then’’ asserting a bridge to the ensuing instance which he does not find as he flips pages. June finds it in [19] (‘‘Jamal again’’) and points at its location on the paper [21, 23] (see Figure 5). Ron and June have embarked on the development of a new narrative to account the successive appearances of the term ‘‘differences’’. Turns taken in the discourse [18–24] reflect another case of zooming out through which all the events that took place between Marias utterance and Jamals are shifted to the background.
25. June: 26. Ron:
27. June: 28. Ron:
29. June:
30. Teresa: 31. June:
32. Teresa: 33. June:
34. Teresa:
=yeah. (..) /he says-/ he comes up with 7 (.) then he ahm- Molly (.) ah- quickly says the other two numbers. Molly saidshe had she had [pointing to the paper and quoting] ‘‘she had probably already takenthe first differences.’’ alright yeah. She reacted thats why. but Jamal was the first one that brought (that up) the differences (.) in the discussion (there).= =yeah and I think he sensed that, yknow, there were other people (.) since Molly saw it, yknow, he said ‘‘oh yeah yeah yeah.’’= =mmm mmm.= =he wanted to see them [the numbers] up there.(..) and I think that sometimes kids do do that. Theyll- theyll say ‘‘I know theres something in there.’’ (.) They dont have to own it, necessarily though, yknow.= =own it? own, like- (..) y know, what is the pattern in the differences, but theres something there, yknow= =mmm mmm.=
In [25] June notes that ‘‘Molly quickly says the other two numbers’’, an observation that prompts Ron to read aloud a line in the papers commentary [26]. They aligned themselves with the interpretation of
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Figure 5.
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Text of the case study that June alludes to in [19].
Mollys utterance articulated in the paper (‘‘alright yeah’’). Right after asserting that Jamal had been the first to introduce this idea into the discussion in [28], Ron moved toward searching for the next instance of ‘‘difference of differences’’ idea, but June began to further elaborate on Jamals contribution. June thinks that Mollys parallel uttering of the sequence of differences added to Jamals ‘‘sense’’ that there was something significant and moved him to request that the numbers to be written ‘‘up there’’ on the overhead projector. Jamals request triggered a feeling of familiarity in June [31], ‘‘I think that sometimes kids do do that’’. June attempts to make explicit what that is by voicing what, she thinks, kids sometimes think: ‘‘I know there is something in there’’. Elaborating a narrative account is not merely a matter of highlighting a sequence of events; each event has its own implications and colors what the whole is about. This particular allusion to differences led June to reflect on the dynamics between individual contributions in a group conversation. It suggested the centrality of the common overhead projector to shape group interaction and the inherent negotiation of what deserves to be projected in front of everyone. Note how a narrative account expresses at once grounded evidence (e.g. ‘‘here, he says’’), interpretations (‘‘he sensed that there were other people...’’), and the background of life experience (‘‘kids sometimes do do that’’). The segment goes on as follows: 35. 36. 37. 38. 39. 40.
Ron: June: Ron: June: Teresa: Ron:
=he just messed up the difference between 8 and 27.= =yeah.= =he said 21.= =right.= =mmm mmm. okay (..) so he knew that there was some kind ofwell he knew (.) to go to the next step to take the difference.= 41. June: =mmm. [We skip a few lines in which Ron comments on technical issues] 42. June: where did you see Molly say (..) 43. Teresa: right here [pointing to paper, see Figure 6] 44. June: ‘‘theres no pattern?’’ Oh. (.) oh, okay. so shed given up on that series. 45. Teresa: mmm mmm.
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From [35] to [40] Ron examines Jamals intervention from a different angle. According to him, Jamal felt he knew that the sequence of differences would solve the problem. Seeing that Jamal gave up on the sequence of differences because of his arithmetical mistake, June tries to understand why Molly had also dismissed it. But Teresa only finds the line in Figure 6. There are limits to how far one can zoom in, given the availability of evidence. June could elaborate nothing further on Mollys decision to abandon the path of the differences. 46. June:
47. Teresa:
48. June:
49. 50. 51. 52.
Teresa: June: Teresa: June:
yeah, thats funny, because:: yknow, who would think to go to the next difference, but (.) Nadia decided to. but how: (..) after how long? (...) I mean the secondthe next thing that they do is they go after the ahm- pattern of the numbers, going to the 7 the 9 andyeah, right, and talking about primes and theyre all chatting away there, but you dont see Nadia talking. I think shes like (.) scribbling away and seeing something. (...) can we see that in the video? mmm. No, every time she does speak the camera pans to her.= =mmm. (.) shes like out of the picture.
Teresa and June zoom out all that happens between Jamal and Mollys dismissal of the sequence of differences and Nadias talk of the differences of differences. Nadia was, as June had noted before, working quietly: ‘‘plugging away’’. The two narratives, the one that attempted to account for what type of student Nadia is and the one tracing the origins of the differences of differences idea, end up overlapping. Since Nadia is ‘‘out of the picture’’ they cannot zoom into how she decided to use constant second differences, an unusual decision to June (‘‘because: yknow, who would think to go to the next difference’’). The fact that June herself would not have attempted this solution was a crucial aspect of Junes perception, prompting her to wonder why Nadia had tried it.
Figure 6.
Transcript from paper which Teresa points to in [43].
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DISCUSSION Through our commentaries, we tried to characterize the development of a Grounded Narrative discourse. We call it ‘‘narrative’’ because it highlights a sequence of events over time with a common plot or issue that unfolds throughout, and ‘‘grounded’’ because the narrator invokes the available evidence to make the case that the narrative is ‘‘real’’ rather than fictional. The narrators themselves motivate the issues that the narrative attempts to account for. For instance, the unfamiliarity of the ‘‘difference of differences’’ idea was a crucial motivation for June to understand where it came from and whether this idea could have emerged in her own classroom. We have noted several traits of how a Grounded Narrative unfolds. The development of a grounded narrative encompasses characteristic shifts that we have called ‘‘zooming in’’ and ‘‘zooming out’’. Zooming in entails dwelling in a particular event while keeping in mind the broader occasion, pulling apart the different aspects involved, and sometimes developing a micro-narrative that takes place ‘‘within’’ the event. Zooming out establishes direct continuity across events separated in time while shifting utterances occurring in-between to the background. There are limits to how deep the narrator can zoom in or out imposed by the available evidence, a fact expressed by June in [52]. A Grounded Narrative attempts to make several points at once and at times shifts its focus from one to another. A Grounded Narrative also changes in scope or in the range of events that constitute it. An example is [12] where Ron redirects the narrative from what kind of student Nadia is to the origins of the ‘‘differences of differences’’ idea. Because of the ‘‘grounded-ness’’ of the narrative, the narrator is careful to distinguish between events that have been seen from the ones that have been inferred (e.g., [11]), to point out the evidence (e.g., [12–14]), and to signal when evidence comes from life experience (e.g., [31]). Grounded Narrative is a discourse prevalent among certain research traditions dedicated to the production of ethnographies and case studies. The fact that June and Ron engaged spontaneously in such a discourse suggests that it can be an activity of common interest for teachers and researchers, perhaps, as it was the case here, when the evidence can be examined in flexible, non-linear ways. Interview 2: Evaluative Discourse The Interview with Carol and Cher Cher and Carol are secondary school mathematics teachers with 30 years of experience. They teach at the same school and seem to be
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on friendly terms. They both read the text version of the paper prior to the two-hour interview. Carol comments that she had been ‘‘anxious to see in the video’’ how an open-ended question played out. Cher then brings up the issue of ‘‘telling versus not telling’’. Both refer to Mr. Solomon in positive ways. Carol, quoting text from the paper to ground her opinion, says that he had created a level of comfort needed to work with open-ended questions. They then comment on techniques to encourage participation and student discovery. Annotated Transcript Toward the end of the episode in the ‘‘Math Conversations’’ videopaper, Mr. Solomon stood up as he said, ‘‘let me tell you something’’ and started an explanation about successive differences as a technique to identify the degree of a polynomial. Up to that point he had been seated next to the overhead projector, annotating students suggestions and asking for ideas. Cher and Carol commented three times during the interview on this shift in stance. We include in this section these three excerpts that occurred at the beginning (right after watching the video), in the middle, and toward the end of the interview. Segment 1. 1. Cher:
But then I thought- that it- after letting them do all that, then hes- he explains it to them.
2. Teresa: 7. Cher:
8. Teresa: 9. Cher: 10. Teresa: 11. Cher: 12. Teresa: 13. Cher:
He should let them (..) continue to (.) to get the rest of that out, mmm. .... ..... ..... I think he should have let them (..) do that. If he was going to let them do all that hard part, he should have let them finish it. (.) You know? /You think so?/= =Yeah! Wha- Why? DoesI mean, I think that would have been appropriate in the context of the way that lesson was going. /mmm mmm/ If I had something going on like that, I would have tried to (.) get them to (.) to tell me that.
Cher made the first allusion to the shift from ‘‘doing all of that’’, namely, letting the students experiment with their ideas, to explaining them. This is a change from receptive to authoritative
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teaching. She uses the subjunctive mode in her talk: ‘‘He should let them continue...’’ Her ‘‘should’’ implied a course of action that she deemed preferable. Cher qualified this assessment with a conditional: ‘‘If he was going to let them do all that hard part...’’ suggesting that, during the previous phase of the episode, momentum was building for the students to figure out the idea by themselves, but Mr. Solomon cut them short through explicit telling. In other words, Cher pointed out an inconsistency between Mr. Solomons intent throughout the episode and the way he concluded it. In [13] Cher introduced another element that we take to be characteristic of an evaluative discourse: ‘‘If I had something going like that ...’’, which is a contrast between what the other did and what one would have done.
18. Carol:
19. Carol: 20. Cher: 21. Carol:
22. Cher:
26. Carol:
27. Cher: 28. Carol:
[acc] [acc acc] But he would have had to give them more examples. He would have had to give them an example where the difference was one. You know what I mean? At that point. [dec] And then, he might be, the math teachers that you and I are ‘‘Oops, this is still homework. Ive got my plan.’’ [laughter] Im sure he realized this was a valuable lesson. yeah. But I think at some point, yknow, we all get to the points where, ahm, ‘‘this has been good |but weve got to move the ball.’’ |Yeah ( ) We dont know where he was in the lesson. But it just seems after (.)dragging that all out, .... .... ..... /yeah/ I honestI honestly think that wasnt- that if if he had gone in planning to teach finite differences, this is a great way to bring up finite differences. yeah. If he had gone in planning to teach that, Im sure (.) he would have had appropriate examples. But this just kind of came out of the blue, Im sure, (.)for him.
Carol initiated in [18] a line of response to Chers disappointment through which she justified Mr. Solomons shift into a ‘‘frontal’’ teacher: he was not prepared to continue the discussion and in the end, he told them the answer. A second element in Carols justification
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was the likelihood that time constraints had forced Mr. Solomon ‘‘to move the ball’’. Note that Carols arguments are all framed as a matter of identification between Mr. Solomon and themselves [18]. At times, the discussion reached an impasse in the face of the unknown: ‘‘We dont know where he was in the lesson’’. These unknowns prompted Carol to build hypothetical scenarios to be contrasted with the filmed events, ‘‘If he had gone in planning to teach that, (...) But this just kind of came out of the blue.’’ Carol agreed with Cher that it would have been good, ideally, to let students come up with the final explanation but that the circumstances prevented Mr. Solomon from doing so. She makes the case that she, Cher, and Mr. Solomon share similar values as to what good teaching is, but that they all face constraints forcing them to deviate from the ideal course of action.
30. Teresa:
31. Carol: 32. Teresa: 33. Cher:
34. Carol: 35. Cher: (.)
36. Teresa: 37. Carol: 38. Cher:
39. Carol:
40. Cher: 41. Carol:
well (.) also I think maybe if you gave them (.) a square- a series of squares, theyd figure it out. yeah. But theyd have to see it. ( ) maybe, who knows. Because you might find ( ) And the next thing would be how to get that- how to get some terms of the ahm, yknow, the coefficient? Right, where its not- where its not the(the greater) In other words, its ahm (.) like the 6. Youre dividing like by the 6 ( ) 3 factorial, so thats gets you a 1 for a coefficient. So thats like the next (.) the next level up on the formula, how to get the coefficients. mmm mmm. [to herself] Like 2, 16, 54... But they couldve discovered- they couldve discovered that too. But I dont know, of course we dont know how much time ( laughs) yeah! I mean, but they have the MCAS theyre going to have to take too. I mean, I can see where you you do (.) get on your podium and start giving them the answers.= =yeah.= =And thats the whole (.) I gathered from reading this, this was a nicean example where (.) the teacher doesnt feed the answers, the students kind of get them.
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42. Cher: 43. Teresa: 44. Carol: 45. Cher: 46. Teresa:
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AhmWell, thats what it says, [background voices] until until the end there. /mmm mmm./ But it might have been the bell was going to ring. [talking over each other:] So I dont know if that was him running out of time. ahm, right.
In [33] Cher described a hypothetical instructional sequence: after ‘‘getting’’ that the successive differences tell the power degree, the next stage is to figure out how the constant in the successive difference gives us the value of the coefficient.2 She deems in [38] that the students could have discovered this next idea too. Carol restated possible constraints that Mr. Solomon might have faced—the same ones that they encounter in their own teaching—leading him to ‘‘feed’’ students with the answer. At the same time she gathered from the paper that this is a case of the teacher not feeding answers. This last remark offered Cher [42] an opportunity to point out the inconsistency in the ending of the classroom episode. Like Carols previous assertion that Mr. Solomon would have had examples to guide the students to reach the conclusion [18], Chers definition of ‘‘the next thing’’ [33] implies the notion of instructional sequences formed by examples and steps that lead students to discovery. The authors of the paper had not been trying to exemplify situations in which the teacher avoids feeding answers, but rather situations in which the teacher works with the students to develop an emerging sense of direction, as opposed to pre-designed instructional sequences. Carol and Cher both admit that things can come ‘‘out of the blue’’ making the teacher unable to enact a good instructional sequence because of lack of preparation or time; their disagreement was on whether unknown circumstances had possibly justified Mr. Solomons departure. The crux of the matter was the interplay between values and actual states of affairs. An Evaluative Discourse is, on the one hand, about values and commitments (e.g., it is good to let students discover ideas by themselves, it is good to prepare sequences of examples to guide students discoveries, etc.) and, on the other hand, about the practical circumstances that allow for or block the fulfillment of these values; in between, there is the more or less
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skillful performance of the instructional agents. Evaluative Discourse deals with matters of ethics. 47. Teresa:
48. Carol: 49. Teresa:
Do you- do you think – the students earned- ( ) it was okay to tell them because they earned (laugh) the right to know at this point. Gee, I never thought of that! [overlapping with background voice from Cher] (The idea of) y know, once youve worked hard enough on it, may be its okay for you to find out the answer: oh- or the explanation of why this works.
In [47] Teresa introduced a new dimension to the analysis: it might be the case that Mr. Solomon stood up and explained because the students deserved to be told. This remark brings into play a wide range of issues (e.g. How does one know when students have won the right to be told? Is there such a ‘‘right’’?), which make ‘‘the shift’’ in his teaching appear as a genuine and complex ethical dilemma. Segment 2. 50. Carol:I really like the way the teachers ah- a little bit braver than I am (.) [laughter] [dec] in letting them go all these wrong ways, /and letting them / (..) [acc] [dec] I mean wrong from getting the correct answer, and: (.) the false starts, all theseall the false starts that, ahm, he allowed- (.) .... .... .... I dont know if Id- (.) be that brave (laughs) for that amount of time, to be letting them flounder. But it was amazing how (.) how they- (.) were so far off in some cases, /but they/ came right back without (.) intervention on his part. [acc] /I mean thats what amazed me in this whole thing./ He really- didnt intervene that much, he just kept writing down ( )
We skip several utterances in which they talk about two students; Carol returns to Mr. Solomon:
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51. 52. 53. 54. 55. 56. 57.
Carol: Cher: Carol: Cher: Carol: Cher: Carol:
58. Cher:
59. Teresa: 60. Cher: 61. Teresa:
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It looked like he was sitting right with them. Like he was one of them. Well, he was sitting down at the (.) right. yeah! And he didnt get up. (.) Until right at the very end. ahm: When he became a physical, right, authoritative (.) he knew it! He () the answer. You could see the difference, right there. [overlapping conversation with Cher and Teresa in the background] [gesturing towards computer screen] (and thats- see thats- that was- right-) He was one of them, and then he was- the teacher. So ahm (.) ahm (.) I would have liked to see him continue, like I said, either (..) get more out of them, let them finish it off, or (.) if there were a time constraint, have them (.) have him (.) set that as the next challenge mmm mmm. to them and then (.) part of the homework. mmm mmm.
Carol and Cher described the shift in diverse ways, from recorder of students suggestions to being authoritative, or from being one ‘‘of them’’ to being the teacher. They bring their own appreciations of the before/after based on what they themselves would do. Carol tells of an admiring perception of what he did before, and Cher of a critical view of what he did afterwards. To articulate this issue, Carol talked about being ‘‘brave’’. That a teacher can be more or less brave implies that there are risks that a teacher might decide to embrace or to avoid. In this case, the risk stemmed from allowing students to ‘‘flounder’’ for so long that they could lose themselves with flawed ideas without getting ‘‘anywhere’’. The danger was that it might end up with a failure situation with no sense of teaching and learning accomplishment. Carols talk shows how an evaluative discourse brings into play all ethical virtues, including courage, tolerance, and confidence. Segment 3. 62. 63. 64. 65. 66.
Cher: Carol: Cher: Carol: Cher:
Yeah, I didnt like him changing roles at the end. [laughs] I thought the bell was going to ring. Im not as harsh as you are. She [Carol] was giving him the benefit of the doubt. Because you dont know. I was thinking he should have just stayed there and just let it continue, but youre probably right, probably it was the end of the class (time).
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‘‘Being harsh’’, ‘‘Giving the benefit of the doubt’’: as with any judgment, what counts is not only the deed that is being judged but also the interpretive framework of the judge. One can be more or less strict or lenient; moreover, this distinction is itself an expression of values subject to debate. An evaluative discourse brings to the surface the overall orientation of the speakers towards the events under discussion—in this case, teaching, classroom dynamics, authority, and so on. This is the subject of the ensuing transactions.
67. Carol:
71. Carol: 72. Teresa: 73. Carol:
74. Cher: 75. Teresa: 76. Carol:
=Look at it this way. How many classes, how many times do you not stand up in front of your (class)? .... .... .... and doing something. To be- to have- to have a new principal, mmm. an assistant principal walking ah- in the building now, they walk the corridors a littlea little more frequently than the previous ones did. And every time they walk by my room Im in study hall. So Im sitting there. They might not know its a study hall.3 I feel very self-conscious when theyre walking by my room and Im seated. [dec] As a teacher, I still take a negative connotation to being seated during a class. You get that mentality. mm mmm. [acc] I mean its ingrained in me, you have to be out there performing. So that was my first observation of the film [looking at the computer monitor.] was, gee! /He was seated/ and it was a good thing. I mean I er- as I took it as an imp- a supportive, [dec] hes part of the dis- not part of the discussion, hes letting them do it, [acc] and hes (.) just writing down their observations. And he changed roles. But its so hard. Weve been doing that other role (.) for so long. Give him the benefit of the doubt, [laughs] the bell was going to ring, [laughs]
Carol strove to deflect Chers criticism of Mr. Solomon by referring back to their practices. She described a situation that seems familiar to Carol and Cher: the principal walking by and seeing that she ‘‘is
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sitting there’’. Carol and Cher suggested that over the years a certain form of and outlook for teaching develops [74, 76] and that it is very difficult to step outside of it. Because ‘‘you have to be out there performing’’, any sign of not commanding the stage could put them at risk for negative perceptions. Also implied in these reflections is the role of those to whom teachers are accountable. These thoughts reflect a large theme of ethics: where virtuous performance results from. One might decide that being seated is the right thing to do, but a background of life experiences could prevent one from doing so (‘‘But its so hard. Weve been doing that other role for so long’’).
77. Carol:
78. Teresa: 79. Carol: 80. Cher: 81. 82. 83. 84. 85.
Carol: Cher: Carol: Teresa: Carol:
[dec] I mean I think he (.) gave them a lot of leeway. [acc] I would have been tempted to say things myself in this dialogue. mmm. There were places where- his name isnt there thatI would have been saying something.= =but it didnt seem like there was a lot of- a lot of- dead airspace in there though. Oh, there was (none). You probably would have that. It went a lot faster than I thought. I thought there was more silence,= =mmm mmm. y know as I was reading it.
In [76] Carol had commented that her first reaction after seeing the film was ‘‘gee, he was seated’’. In reading the paper Cher and Carol had imagined many aspects of the interaction on the basis of what was ‘‘normal’’ and to be expected from their ways of being in class. In [80] Cher introduced another one of these aspects: the overall rhythm of the interaction. When reading the paper they had imagined a slower pace of exchanges. Watching the video led them to question their tacit assumptions (‘‘I thought there was more silence’’). This type of talk reflects perhaps one of the most interesting contributions of good video documents: to make the viewers aware of their own implicit assumptions. A written transcription, no matter how detailed it is, leaves out countless elements of the situation (facial expressions, tones of voice, pace) that the reader tends to ‘‘fill’’ with whatever seems natural.4 Discussion When speakers engage in an Evaluative Discourse, issues of values, judgments, and commitments arise. The participants, as in this
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example, examine teaching episodes in light of their concepts of what good teaching is and strive to assess the present circumstances. The following are the main characteristics that the annotated transcript allowed us to point at: (1) The use of the hypothetical past construction (conditional, subjunctive) is widespread. Speakers repeatedly elaborate a contrast between how things were and how things could/should have been. Sometimes this contrast is a comparison between the teaching episode and a hypothetical instructional sequence, which is deemed closer to an idealized course of events. At other times it is a distinction between teachers: what s/he did and what I would have done. (2) All the central issues of ethics are in the foreground of an Evaluative Discourse: the trade-off between ideal actions and practical circumstances (‘‘Weve got to move the ball’’ [21]), the playing out between life experience and force of will (‘‘But its so hard. Weve been doing that other role for so long’’ [76]), and the conflicting impulses to be consistent with and to depart from past traditions (‘‘If he was going to let them do all that hard part...’’[7]). Skillful performance, risk-taking, and coherence are some of the ‘‘factors’’ that speakers try to notice and account for. (3) Interpretations aiming to relate what is and what ought to be pivot on the ethical outlook of the speakers. They express themselves as being more or less strict, lenient, sensitive, principled, and so forth. Nevertheless interpreters can be flexible because they understand that reality can force departure from optimal courses of actions (e.g., Carols many arguments for how the circumstances were likely to motivate Mr. Solomon becoming authoritative), and questioning tacit assumptions may prompt radical interpretive shifts, such as Teresas argument in [47] that the students might have ‘‘deserved’’ to be told. Evaluative Discourse is in our experience, by far, the most prevalent mode used in conversations about videotaped teaching episodes. Teachers, administrators, and researchers all tend to engage in Evaluative Discourse. While the focus on hypothetical situations created by this form of discourse offers an easy ‘‘way out’’ of the need to attend carefully to the filmed events (because the speakers can effortlessly skip to their own personal experiences or self-contained statements of values), we tried to show in this annotated transcript that this is not necessarily the case, and that conversants engaged in Evaluative Discourse can work hard to take the perspective of the filmed participants.
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CONCLUSION Two Types of Discourse In the literature, we find references to teachers being ‘‘evaluative’’ and ‘‘judgmental’’ in their conversations about classroom episodes, but we did not find a detailed characterization of how these attitudes are expressed in their discourse, nor of what alternative discourse might emerge when they are not being evaluative. In this paper, we have distinguished two modes of discourse that emerge in conversations about teaching episodes. The contribution of these findings is not the mere stipulation of the two types of discourse but their sociolinguistic characterization. We believe that, with the refinement of our ability to recognize forms of discourse and to motivate or respond to them, both types of talk may be used to contribute to the professional development of participants with any level of experience. The Grounded Narrative Discourse flows in a conversation whose aim is to formulate narrative accounts of classroom events. These narratives, which strive to distinguish fiction from actuality by pondering the available evidence, are at the same time acts of interpretation that participants debate. The talk takes the form of ‘‘this...and then...and then...’’ which correspond to highlighting successive ‘‘snapshots’’—like a discrete sequence of photos portraying a continuous trip—whose continuity is imagined by speakers and listeners. By choosing particular moments and voicing them through a temporal sequence, the narrator conveys a sense of the whole: a sense of the ‘‘point’’ that she tries to elucidate and better understand. The grounding of these choices and temporal arrangements takes place through instances of ‘‘zooming in’’ and ‘‘zooming out’’; the former allows the narrator to separate aspects of a complex interaction and the latter to put in touch events that occur at different times. The Evaluative Discourse emerges among participants striving to ascertain ethical questions around the classroom episode. We use the term ‘‘ethics’’ in the customary sense of dealing with matters of values, virtues, and commitments. Ethics is of course pervasively present in everyday life and talk, but it takes a particular significance in the talk on teaching episodes because these episodes are often taken as exemplary of good or bad practices. The speakers in an Evaluative Discourse have a propensity to use conditional and subjunctive speech; they tend to posit a contrast between the events described in the episode and other hypothetical courses of action, as well as between what teachers and students actually did in the episode and what they or
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their students would have done. The participants in an Evaluative Discourse bring to the subject not only their judgments but also the criteria on the basis of which, they think, one should judge.
Use of Videopapers as Case Studies Our analysis suggests that multimodality is relevant in both discourses. The video allows for a depth of zooming in and out that goes beyond what is possible using only textual descriptions. Videopapers also allow the speakers to richly grasp the overall ‘‘climate’’ of an interaction, and notice when things diverge from what participants tended to project on the basis of their own previous experiences. The textual analysis, on the other hand, offers a point of view: ideas and questions with which to watch the episode. The text also conveys the authors take on how the classroom episode relates to broader issues that matter to the speakers. Both discursive types, Grounded Narrative and Evaluative Discourse, emerge through talk about teaching episodes that include an ever-present desire to ‘‘know more’’. No matter how detailed the video portrayal of the classroom is, or the extents of the available background information, participants always feel that countless questions remain unanswered.
APPENDIX Transcription Conventions (.) (..) (...) . ? , : underline /words/ //words// ( )
noticeable pause or break in rhythm, less than 0.5 second half second pause or longer a second or longer pause sentence-final falling intonation sentence-final rising intonation phrase-final intonation (indicating more talk to come) glottal stop or abrupt cutting off of sound lengthened sound (extra colons means more lengthening) emphatic stress spoken softly spoken very softly transcription impossible
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(words) = —|—[acc] [dec]
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uncertain transcription two utterances linked by = indicate no break in flow of talk(latching); overlapping speech: two people talking at the same time spoken quickly (appears over the line) spoken slowly (appears over the line)
ACKNOWLEDGMENTS This research has been supported by the ‘‘Bridging Research and Practice’’ project funded by the National Science Foundation, Grant REC-9805289. All opinions and analysis expressed herein are those of the authors and do not necessarily represent the position or policies of the funding agencies. The authors would like to thank Jesse Solomon for his generosity in providing us with access to his classroom and his self-reflections.
NOTES 1
In order to produce a videopaper we have developed a software tool called ‘‘VideoPaper Builder’’ that allows authors to interconnect and synchronize the different components without having to be a programmer or even technically confident. The VideoPaper Builder can be downloaded from http://vpb.concord.org/. 2 Let us call n the power of a polynomial; Cher is expressing the result that the constant successive differences will be the coefficient of the nth term multiplied by n! 3 The phrase ‘‘Study Hall’’ is used to describe a class period which is used by students to work individually; although a teacher is present, they do not ‘‘teach’’ during that time. 4 A video also keeps innumerable aspects out of view (whatever is outside of the camera angle, out of focus, blocked by people standing in between, etc.); there is a great potential disparity between the range of features that can be documented in text or in video media.
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Teresa Lara-Meloy
