An investigation of communicative competence in an upper-secondary ...

PATRICIA A. FORSTER1 and PETER C. TAYLOR2

AN INVESTIGATION OF COMMUNICATIVE COMPETENCE IN AN UPPER-SECONDARY CLASS WHERE USING GRAPHICS CALCULATORS WAS ROUTINE

ABSTRACT. This paper presents an exploration of conversation in a single episode of whole-class work in a Year 12, all girls’ Calculus class. The analysis is cast in terms of Habermasian communicative competence, which requires students to evidence critical thinking and the ability to express their views. Favourable learning outcomes in the class are explained by the teacher’s mode of questioning, cooperative endeavour by students, the instructional sequence and use of technologies including the graphics calculator. The circumstances of problematic outcomes are also identified. KEY WORDS: communicative competence, graphics calculator, mathematical inference

I NTRODUCTION The topic of instruction was vector calculus. The setting was a Year 12, all girls’ class of 13 students. The conditions included student ownership of Hewlett Packard HP38G calculators with limited symbolic capabilities. We observed 21 lessons in the class and were impressed by the lively conversation that occurred. Sometimes it had the pulse of shared inquiry and other times the rhythm of teacher questions. Sometimes it was harmonious, with a concern for relationship, and at other times it sounded discordant. It seemed that the high majority of students joined in. Therefore, in view of the current research interest in classroom discourse, our purposes in writing this paper were to explore critically our impressions, describe the conversation in theoretical terms and explain the overt student involvement. Our approach was to select a classroom episode in which insightful inferences were made and conduct a micro-analysis of it. Our principal theoretical referent for the analysis was Habermasian communicative competence, as described by Young (1990, 1992). In brief, communicative competence is the capacity to express and defend knowledge claims and validity judgements in social situations of unequal power relations (Young, 1992, pp. 54–60). Communicative competence typifies the empowered learner and emerges through social engagement (p. 57). We were informed Educational Studies in Mathematics 52: 57–77, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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PATRICIA A. FORSTER AND PETER C. TAYLOR

also by other social perspectives on learning (e.g., Cobb, 1998; Mehan, 1979). The significance of the paper for mathematics education is that we identify what communicative competence can comprise in an uppersecondary mathematics classroom, and identify conditions that we judge were conducive to the exhibition of communicative competence and mathematical inference. These include the instructional sequence of which the episode was part, the teacher’s questions and cooperation amongst students. In addition, we problematise practices that from our point of view were antithetical to students’ communicating their mathematics through conversation. The paper follows others on conversation related to graphics calculator use in the same class (e.g., Forster et al., 2002a, 2002b).

R ESEARCH METHOD Selection of the class The selection of the teacher and his class was purposive (Cohen and Manion, 1994, p. 89). We sought his involvement because we knew from his professional development activities for other teachers that he was active in using graphics calculators in his teaching; and his students achieved outstanding results in public examinations. These attributes suited our main purpose for the study which was to investigate learning with graphics calculators. Fieldwork The first author attended 21 50-minute lessons in the class, over five weeks, and took the role of observer-participant, observing whole-class work and acting as an assistant teacher during group-work and individual work. Data generation strategies were introduced gradually during the first three lessons and set in place for the following 18 lessons. The one-to-one conversations of all 13 students were audio-recorded. A video-camera captured graphics calculator screen-displays that were projected onto a whiteboard during instruction. Fieldnotes were made of (a) the teacher’s board-work that was not in the field of view of the video-camera, (b) students’ use of their calculators and what they wrote in their workbooks, (c) informal follow up with students and the teacher about issues that arose in class and (d) formal interviews with the teacher about our analyses of the action. A research journal was kept and shows records of early impressions and maturing perceptions of the classroom practices.

COMMUNICATIVE COMPETENCE

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The analysis and literature that informed it Our analysis of the classroom conversation was based, in the first instance, on understandings that we gained in our previous investigations of the development of social and subject-related competencies, in a variety of educational settings (e.g., Forster, 2000; Forster and Taylor, 2000; Taylor, 1996, 1998). Then, we chose Habermasian communicative competence as our main theoretical referent for it is a highly desirable outcome of education. Hallmarks include students’ being open to and respecting others’ ideas, while being critical, so that inconsistencies and information unsupported by evidence are recognised (Young, 1992, pp. 54–60). As well, communicative competence involves knowing how to enter a conversation, seek information, question and challenge other’s ideas, and defend personal views (ibid). Cobb (1998, pp. 33–34) describes similar classroom ideals whereby students attend to the explanations made by others, indicate understanding or non-understanding, ask clarifying questions, articulate alternatives when differences in interpretation become apparent, and explain and justify their conclusions. These types of student response are to be valued for they signify preparation for full participation in democratic society and, according to Cobb et al.’s (1997, p. 264) constructivist view, they can advance mathematics learning. When individuals reflect on each other’s responses, individual and collective learning can occur. So, we scrutinised the transcripts of the audio-recordings of classroom conversation for instances of communicative competence. The recordings revealed students’ utterances to the class as well as conversations between individuals. The video-tapes offered corroborating data. As well, we searched for mathematics inquiry between the teacher and students. Inquiry allows the exhibition of communicative competence and, according to Young (1992, pp. 57–58), can encourage its development. A feature is that students’ cognitive work is heard. The teacher asks questions to elicit student ideas, rather than only to elicit answers that match his own preconceived views. Students’ ideas are taken seriously and the distortion of control (of the conversation) being vested only in the teacher is avoided. Cobb et al. (1992, pp. 590–597) emphasise also that inquiry is a shared enterprise. Teachers aim at shared understanding in conversation between themselves and students, and between students. In summary, for both Young (1992) and Cobb et al. (1992), inquiry learning is based on the rationale that reasoning about the subject matter is the responsibility of the teacher and the students. We were also guided by Mehan’s (1979) foundational work on classroom interaction and competence in mathematics. Mehan identifies the predom-

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inant interaction pattern in instruction of teacher initiation, student reply and teacher evaluation (pp. 81–125); and summarises competent participation as “the production of academically correct and interactionally appropriate replies to teacher-initiated actions and the generation of effective initiation acts” (italics added, p. 169). Academic correctness, in the narrow sense used by Mehan (1979, p. 169), comprises factually correct answers. We assumed for our analysis that academic correctness in relation to inquiry learning encompassed responses that were reasoned and relevant to the mathematics being discussed, without factual correctness being necessary. Factual correctness could emerge through negotiation. Students’ responses are interactionally appropriate if their form, as enacted through gesture and conversation, is consistent with preceding utterances (Mehan, 1979, p. 169). For example, students need to recognise to raise their hand if answering “Who knows . . .?”, and to supply reasoning if asked directly “How do you know?”. Effective initiation acts involve students’ introducing topics of discussion themselves. To be successful, first, they need to get the floor (i.e., gain the right to speak), then, they must hold the floor. Getting the floor normally requires that students wait until the completion of any ongoing interaction (Mehan, 1979, p. 149). Holding the floor depends on the introduction of new material (p. 152). Otherwise, when students contribute ideas, it is likely they will receive no response from the class and perfunctory treatment, at best, by the teacher. We viewed academic correctness in a broad sense, interactional appropriateness and effective initiation as constituting facets of Habermasian communicative competence. Replies with academic content indicating discernment were highly relevant as were interactional forms comprising challenge, explanation and justification. Further, Watson and Mason (1998, p. 42) list exemplifying, completing, correcting, sorting, changing, generalising, verifying and refuting as processes that typify mathematical thinking and, when communicated, we took these as indicators of Habermasian communicative competence. The exemplification etc. is constituted by the form of an utterance (i.e., interactional structure) as much as by the content. In addition, we subscribe to the view of Mehan (1979, pp. 128–130), Young (1992, pp. 57–58), and Cobb et al. (1997, pp. 266–269) that competencies emerge in social interaction (see Forster, 2000). They are fundamentally social phenomena, but are determined also by individual thought and action.


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We were also are guided in our inquiry by Sfard’s (2000) analysis of ambiguity in mathematics conversation. She describes ambiguity in terms of discrepancies between what a speaker intends to say, what is said and how the text is interpreted (pp. 302–307), and describes how negotiation can lead to agreed to, more-precise, new mathematical understandings (pp. 308–318). However, productive negotiation always depends on mutual (i.e., shared) background knowledge (Gumperz, 1995, p. 106–114). Mutual knowledge of relevant content and of conversational form (e.g., the obligation to answer a question) underpins the resolution of discrepancy. Selection of the episode for reporting The criteria that we used to select the episode were that students displayed attributes of communicative competence and inference was evidenced. Episodes on displacement and paths of motion, speed, velocity, acceleration, and relationships between these variables; and on the meaning of the integral of velocity with respect to time, and the integral of the absolute value of velocity were available. We chose one on acceleration because it was relatively short and so was practical to report, yet it entailed consideration of several aspects of acceleration. In relation to how well the episode represents the classroom action, student responses and teaching actions similar to those described were evidenced on other occasions, usually in relation to new properties or procedures. However, the nature of interaction in the class varied and included traditional (non-inquiry) instruction, direction (e.g., about efficient calculator use) and rationalistic argument (e.g., about alternative methods of solution). Furthermore, actions consistent with communicative competence and inquiry learning were more diverse than those discussed. Research standards Regarding the quality of the research, the quoted classroom conversation is verbatim, to the extent that it could be ascertained from the audio and video recordings. Moreover, we held to openness and consultation with the teacher so that we discussed our early impressions and initial analyses with him, which influenced our thinking. Later, adhering to a relational ethic of care for research participants (Brickhouse, 1993), we asked the teacher (a) to scrutinise the paper in order to check that he did not object to publication and (b) to be guardian of the students’ interests. However, we did not specifically check that the teacher agreed with us on every point. Hence, we do not claim consensus with the teacher, nor do we presume to infer intention for the teacher or the students and, in these regards, our approach was informed by the methods of ethnomethodology as practised by Roth

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(1998, p. 10, pp. 36–38, 108). Thus, the text is our subjective interpretation of the action and, calling on the notion of intertextuality (Fairclough, 1992, pp. 104–106), the piece stands in relation to the literature that informed the interpretation. Therefore, we carefully documented the relevant literature.

C LASSROOM ACTION Lead up to the episode, August 22nd–August 30th, 2001 On August 22nd the teacher (Mr D) introduced vector calculus by asking students to graph r(t) = 2ti + (t − 1)j, r(t) = 2ti + (t 2 − t)j, and r(t) = sin ti+ costj one at a time in the parametric aplet of their calculators and observe and discuss the speed of objects with position vectors given by r(t). The rate at which a parametric graph is plotted on the calculator reflects speed. A student’s calculator was connected to an overhead projection panel and the display was sometimes the focus of discussion. The class had studied a unit on vectors in Year 11, for which motion was limited to straight-line motion with constant velocity. Other background is the class had completed units on rectilinear and simple harmonic motion that addressed displacement, velocity and acceleration. The graphing activity continued in the next lesson (August 28th). Mr D asked the class to plot circular and parabolic graphs, and the ‘bow-tie’ graph given by r(t) = 10sinti + 5sin2tj, on their calculators and on paper. They were to describe the properties of the graphs. Students mentioned intercepts, initial positions, directions of rotation and co-ordinates of turning points. For homework Mr D asked the class to draw the bow-tie graph on graph paper, and draw the position vectors for t = 0, π /8, π /4 . . . 2π using components. Whole-class work in the double (100 minute) lesson on August 29th was based on the bow-tie graph. Mr D projected the image of the graph on his calculator onto the whiteboard and drew r(t) vectors on it, using students’ component values. He led discussion on limits and continuity in relation to the graph and other vector functions. As well, he asked students to differentiate the r(t) components for the bow-tie relationship and asked what the result could mean. Homework was to calculate the components of velocity at t = 0, π /8, π /4 . . . 2π and to use the components to draw the velocity vectors v(t) on the graph-paper graph, at the positions given by r(0), r(π/8), etc. Figure 1 shows the graph that students were building up. Next lesson (August 30th), using student data, Mr D drew the r(t) and v(t) vectors on an overhead transparency that he had made of the graph. He


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Figure 1. The graph of r(t) = 10 sin t i + 5sin 2t j showing position vectors r(t) and velocity vectors v(t). Note r(0)=0i+0j and r(π/8)=3.8i+3.5j.

Figure 2. The graph of speed against time for r(t) = 10 sin t i + 5sin 2t j, [0, 2π].

asked the class to predict from the graph where a particle whose motion was given by r(t) would speed up and slow down. Students judged this visually from the length of the velocity vectors on the graph. They also calculated speed at selected points, and graphed speed as a function of time on their calculators (see Figure 2). As a class, students identified the points on the bow-tie graph that corresponded to the relative maxima and minima on the speed graph. Then, Mr D told the class that differentiating velocity v(t) gave acceleration a(t), and immediately set homework, which was to calculate the components of acceleration for the bow-tie relationship at t = 0, π /8, π /4 . . . 2π and use the components to draw the acceleration vectors on the graph. The episode that we present started at this point. In summary,

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students had available their graphs showing r(t) and v(t) vectors but no acceleration vectors. They had calculated some speed values, but no acceleration values. They had linked points on the speed graph to points on the bow-tie graph. Our interest is in the competencies that students demonstrated and the circumstances of the demonstration. The episode, August 30th Part 1 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Mr D Can anyone guess where the acceleration vectors might be? Several replies were heard: “On the graph”, “Off the graph”, . . . Mr D indicated the vectors were to be drawn “on the graph”, starting at the points corresponding to r(0), r(π/8), etc. Then, he restated his question: Mr D Does anyone want to take a wild stab in the dark as to where these acceleration vectors will be? [pause] Yes, Kim is going to have a go. Kim For the first bit, they will be going [pause] backwards, because it’s slowing down. Mr D It will be going backwards. What do you mean, backwards? Kim I don’t know. But it’s slowing down, so it’s going the opposite way to the velocity. Chris Velocity and acceleration vectors are opposite. Some students were listening, others were talking about the task and others were off task.

Our attention is on Kim. With “backwards” (line 9), she distinguished that “where the acceleration vectors might be” (line 1) could mean the direction of acceleration. She limited her answer to “the first bit” of the graph (line 9) and explained her rationale (“because it’s slowing down”). When asked to define “backwards”, she gave a more precise answer (“it’s going the opposite way to the velocity”, line 12). Hence, from our point of view, the form of Kim’s response was entirely appropriate (i.e., interactionally appropriate). It was academically correct in that it was reasoned and relevant to Mr D’s question. Further, Kim exhibited critical thinking and contributed to reasoning in the public domain, which are requirements of Habermasian communicative competence (Young, 1992, pp. 54–60). However, the criterion of respecting others’ ideas was in jeopardy for Kim did not respond when Chris offered clarification (line 14), and neither did the teacher.


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Next we turn to the circumstances of Kim’s performance. Mr D’s question addressed a new domain of mathematics activity in the class, namely, acceleration under curvilinear motion where the angle between velocity and acceleration changes. With rectilinear and simple harmonic motion, which the class had already studied, velocity and acceleration were in the same or opposite directions. With circular motion, acceleration is perpendicular to velocity, but this had not been discussed in the mathematics class, although some students, including Kim, would have met the property in physics. Therefore, predicting acceleration required extended abstraction, in the way that Hershkowitz, Schwarz and Dreyfus (2001, p. 202) define the term. Students needed to proceed back and forth between considering a moving object, the varying speed that had been discussed, the phenomenon ‘acceleration’ and the representation of it as a vector in graphical form. So, we suggest an important aspect of the action was that Kim had time to think while Mr D answered other students. Having the motivation to think is also important and the video-recordings show that the development of the properties using the bow-tie graph involved a lot of debate and widespread participation in debate across the class. Explanations for the high level of activity include that: the bowtie relationship was more intrinsically interesting than a simpler graph; the development was sequential and based on computation, rendering the vector properties accessible to students; and class discussion had relied on individual student endeavour at home and in class, which allowed shared ownership of the graph and its properties. Mr D’s actions were another factor of Kim’s performance. He accepted Kim’s response without confirming it or negating it, explicitly or by implication through his intonation, and maintained this non-evaluative stance which, according to Young (1992, p. 90), is crucial for provoking students’ critical thinking. By way of comparison, in replying to other students at the beginning (lines 2–4), he confirmed “on the graph” to be a valid answer by explaining the meaning of it, in his terms; and he ignored “off the graph” and other responses. In these cases, the students did not voice what they meant. Mr D’s partialness is explained by the traditional and practical action that only one student is chosen when several students respond (Mehan, 1979, p. 91). Further, if teachers perceive that students do not offer anything new (Mehan, 1979, p. 152), or break into a conversation when it is the teacher’s turn, typically they are ignored (p. 143), which could explain why Chris was not acknowledged (line 14). Another explanation is Mr D and Kim did not hear Chris because of chatter around the class.

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In summary, the excerpt (lines 1–16) illustrates interactional appropriateness (good form), academic correctness (reasoned content) and critical thinking by a student, that is illustrates the communicative competence that influenced and served learning in the Year 12 class. We have argued that time to think, motivation to think, the task and the development within which it was embedded, the opportunities for participation and requirements for explanation by the teacher were conditions of the performance. On the other hand, the teacher explaining students’ answers or ignoring them closed the possibility of students’ competence being exhibited in the public domain. Hence, a challenge in teaching is to manage the classroom action so that competence can emerge for all students. Part 2 17. 18. 19. 20. 21. 22. 23. 24. 25 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

Mr D

Let’s put this thing up here. [He placed a transparency on the overhead projector showing the bow-tie graph and r(t) and v(t) vectors]. Girls can you all listen please. Students who were talking lowered their voices or stopped talking. Kim [to Abi, who was sitting next to her] It will be going backwards as it goes towards the next one, won’t it? Abi I don’t know. Kim [still talking one-to-one] So, it goes like that [as she gestured over her own graph]? Abi What goes like that? Kim Acceleration. By now Mr D had said: Mr D Okay, listen to what Kim is saying. So, what did you say, “It will be going backwards”? Kim Not, not Tanya [interrupting] I think it will start off big and then decrease. Mr D was looking at Kim and she carried on: Kim [inaudible] will be going backwards as it goes to the next point round. Mr D Say, like that maybe? [as he drew a vector starting at the origin and pointing down, see Figure 3]. Kim Yes. [pause] I don’t know.

Again, we focus on Kim. When Mr D moved temporarily out of the conversation (line 17), she took the opportunity to test the validity of her ideas with Abi. In addition, she advanced her spoken reasoning by specifying that “backwards” was relative to “the next one” (line 22). Then,


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Figure 3. The graph of the acceleration vector that Mr D drew.

she gestured over her bow-tie graph, so, moved to establish meaning visually, which can be crucial to the advancement of ideas (Sfard, 2000, p. 324). The gesture was the first appearance of a vector for acceleration. Then, Abi recognised the ambiguity of “it” (line 26), the pronounced focus (Sfard, 2000) or (mathematical) thing that was being talked about, and Kim identified “it” to be acceleration. “It” had been used for ‘acceleration’ and ‘the moving object’ since line 11. Before line 11, acceleration vectors were referred to in the plural and the object was the singular “it”. So, throughout the one-to-one interaction Kim was discerning, which is a quality of communicative competence, but it is not sufficient to demonstrate the quality with a sympathetic peer. Students must be able to sustain performance in the presence of unequal power relations (Young, 1992, p. 58), such as exist in the whole-class domain between students and the teacher, and between students. Mr D facilitated Kim’s move back into the public forum by asking the class to listen to her (line 29). Moreover, he had organised the graph on the whiteboard (line 17). It was available for Kim to speak to, and to assist other students as they listened. Kim attempted to correct Mr D (line 31) but she wasn’t heard because Tanya interrupted. Then, Kim’s “as it goes to the next point round” (line 34) clarified the “next one” (line 22) was the next point. Mr D gave form to Kim’s idea by drawing an acceleration vector on the graph (line 36) and, at the same time, assumed control of the acceleration definition. However, he sought Kim’s authentication and, based on our viewing of the videorecording, he asked if he had interpreted her suggestion correctly, or asked if she agreed with his suggestion. Whichever was the case, he passed control and responsibility for the definition back to Kim. It was a moment of negotiation. Kim’s “yes” (line 38) suggested that she accepted joint

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ownership and responsibility for the diagram, then she expressed doubt (“I don’t know”). Doubt was warranted. Near the origin, the direction of acceleration is close to being opposite to the direction of the velocity (and the alignment of the curve), but not exactly opposite. Otherwise the object would not turn. At the origin, acceleration is undefined if t is restricted to values greater or equal to zero, or has zero magnitude if negative t is allowed. Taken overall, it seems to us that the interaction between Mr D and Kim had many of the qualities of inquiry mathematics described by Young (1992) and Cobb et al. (1992). Mr D played a key role as facilitator from the moment he projected the graph onto the whiteboard, to when he passed control back to Kim. Furthermore, with “Say like that, maybe?” (line 36) he invited ‘confirmation/disconfirmation’ by a student, and this teaching action, according to Young (1992, p. 118) is rare. Shared inference was salient, then, Kim doubted Mr D’s diagram. Her action was consistent with symmetry in the teacher-student power relation, which is an ideal for inquiry learning. Another circumstance of the performance was Kim’s definition to the class (line 36) evolved in conversation with Abi (line 21). The early wholeclass conversation (lines 1–16) led to their one-to-one conversation (lines 21–27), then the one-to-one conversation fed into the whole-class conversation (lines 29–38). This speech pattern was noticeable across the class and was a regular feature of the classroom practice (see other examples in Forster et al., 2000b). Mr D endorsed it by allowing private (one-toone and small group) conversation during class discussion. Indeed, some students started discussing the acceleration task when Mr D asked his first question and continued until he called for attention the second time (line 29). It is relevant that Mr D had taught the class for two nearly years, so there had been time for Mr D and the students to establish when private conversation was in order and when it wasn’t. Last we turn to Tanya’s interruption (line 32). She spoke out of turn. Her timing was inappropriate. We could say that she was more concerned with the mathematics than with considering others. She was ignored, which is a typical response to interruptions (Mehan, 1979, p. 118). In summary, the excerpt (lines 17–38) illustrates that interactional appropriateness extends to taking advantage of ‘moves away’ by the teacher and to testing validity with peers. It also illustrates that one-to-one conversation can support the necessarily public appearance of communicative competence. In addition, we saw ambiguity appear through mishap (“it”) and in the expression of a new idea (“backwards”). When Kim addressed the ambiguities, in conversation with others, she defined a new mathemat-


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ical object (an acceleration vector) for the class (with questionable validity). Progress was assisted by the availability of projected and hand-drawn graphs and by Mr D’s interventions, and responsibility for the definition passed between him and Kim. Part 3 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.

Tanya

I think they will start off big and then they will start getting smaller and smaller. Mr D They will start off big and get smaller and smaller. Why would they get smaller and smaller? Tanya Because the acceleration decreases in size. Kim No, because isn’t it slowing down the particle? Tanya That’s why acceleration is decreasing in size. Abi It has to slow it down less as it gets to the extremum point. Tanya Yes Abi It has to slow it down less as it gets to there. Mr D Yes, and it’s. Okay, what Kim is saying, because this thing is slowing down, Kim is hypothesising that this acceleration vector will be behind the velocity vector because along here the thing is slowing down. I wonder where it will be when we get to here [as he pointed to the relative maximum in the first quadrant]? A new round of discussion began.

First, we look at Kim. She disrupted the teacher-student-teacher turn taking, yet was acknowledged. However, according to tradition, if students reply incorrectly the floor is open to other students (Mehan, 1979, p. 116), and Kim’s “No” presumed an error. Further, when Kim challenged Tanya (line 44), she framed the challenge as a question. She invited negotiation more than defense. So, rather than distancing herself from Tanya, Kim’s response appeared, to us, to indicate a preference for maintaining relationship, which is common amongst female students (Belenky et al., 1986, p. 84). In addition, Kim provided reasoning to support her challenge. Hence, in summary, the form of Kim’s response in line 44 was interactionally appropriate, in the setting of the all-girls’ class. The content of her reasoning “because isn’t it slowing down the particle?” was ‘correct’ in that the particle was slowing down, but she did not recognise that Tanya’s analysis (line 43) partially described the situation: “acceleration decreases in size” was consistent with changes in speed as the particle approached the first turning point but not with changes in speed at the start.

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Then, Kim did not accept or reject Tanya’s next explanation (line 45), which included an error in the logic. Slowing down does not require a decrease in acceleration. Neither did Kim acknowledge that she accepted or rejected Abi’s claim (line 46). When Abi repeated her statement (line 48), Kim still did not acknowledge whether she agreed. So, Kim did not conclude her challenge satisfactorily and Mr D did not demand that she did. Now we turn to discuss the actions of other students. Tanya instituted her own re-entry appropriately (line 39), after Kim’s “I don’t know” signaled an opportunity for others to join in. However, Tanya did not explain why the size of the acceleration vectors varied and, when Mr D asked for explanation, she merely substituted ‘getting smaller’ with ‘decrease’ (line 43). In responding to Kim (line 45), she reiterated what she had already said and introduced an error. Last (line 47), she agreed with Abi, but it is not clear that her “Yes” was based on accurate reasoning for she didn’t add to Abi’s statement. In summary, Tanya introduced a new property for the class but overall, the form and content of her responses were less sophisticated than Kim’s. Abi joined the discussion (line 46) after Kim upset the turn taking order (line 44), and consecutive student turns are normal after a disruption (Mehan, 1979, p. 154). Her clarification (“slowing down less as it gets to the extremum point”, line 46) conflated Tanya’s and Kim’s suggestions: it coordinated the ideas of slowing down and decreased acceleration. Hence, overall, Abi demonstrated good form and discernment in this excerpt, as well as in the previous one. Throughout the action, Mr D continued his facilitating role. He sanctioned Tanya’s re-entry, took up her idea about size without judging it, and placed the onus on her to explain (line 41), as he had done for Kim. He also allowed the consecutive student turns when students talked to each other, rather than to him. Strings of student turns are characteristic of an inquiry environment (Young, 1992, p. 119). In line 49, Mr D launched into judgement (“Yes”), but stopped in midsentence. This was an instance when he didn’t press for explanation. Returning to Kim to secure a defense or to Abi and Tanya for further reasoning would have held the students responsible for the properties they were claiming. The excerpt finished with Mr D returning to Kim’s idea for direction. In doing so, he privileged Kim’s idea over the other students’ reasoning, but he used it to lead into his next question (“I wonder where it will be when we get to here”, line 49). “There” was the first turning point, so the new question moved the discussion forward by giving students cause


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Figure 4. The graph of r(t) = 10 sin t i + 5sin 2t j showing the velocity vectors v(0) and v(), tangential to the graph, and acceleration vector a(). The starting point was the origin.

to consider how the moving object would turn. In addition, Mr D’s explicit use of “hypothesising” addressed the issue of rigour. Students had the opportunity to check the properties that had been discussed when they graphed the acceleration vectors for homework. In summary, lines 39 to 55 illustrate that interactional appropriateness extends to students speaking consecutively instead of taking turns with the teacher, and illustrate progression in inference by students. Further, we suggested a role for the teacher was to follow up Kim’s challenge. Concern for relationship by students, insistence on explanation by the teacher, mathematical rigour, and the sophistication of student responses were other issues that were touched on. We note that persistently expecting explanation can result in it being given spontaneously, that is, can result in students like Tanya developing in interactional competence (e.g., Cobb et al., 1997, pp. 266–269). Mr D’s last question started a new round of discussion that brought the lesson to a close. The transcript is in Appendix 1 for readers who would like to see the directions the conversation took. An excerpt from the discussion on the homework the next day (August 31st ) is included below. It shows that other students joined the conversation and illustrates the advancement that is possible when individual work at home complements class discussion. The bow-tie graph was projected onto the whiteboard. Mr D asked for the acceleration vectors at t = 0, (0i + 0j), and at t = π/8, (−3.83i − 14.14j), and he drew the acceleration for t = π/8 on the graph (see Figure 4) which already had displacement and velocity vectors on it.

72 Mr D

Rebecca Mr D Abi Anna Mr D Tanya Mr D Doreen St Mr D Tanya Mr D Tanya Mr D Abi Mr D Chorus Mr D Tanya


Okay. How does this fit in with what Kim was saying yesterday as we were about to leave here? [pause] What is happening to the particle at this point? It’s slowing down. It is slowing down. Notice where the acceleration vector is [pointing to a(π/8)]. It’s pulling back and down. It’s pulling in the opposite direction. It’s pulling in the opposite direction, isn’t it? It’s not really, hardly it is. Well, it’s behind it. Cause it’s negative x. Not directly in the opposite direction. Not directly in the opposite direction? No, no, it’s pulling in that direction so it can make it turn. It’s pulling this way, isn’t it? So it can make it turn. It’s doing two things. Making it? Slow down. Making it slow down, and making it? Turn. And making it turn. Oh, yes [suggesting satisfaction].

Reflection on the instructional sequence From our constructivist perspective (Taylor, 1998), strengths of the instructional sequence of which the episode was part, and which gave rise to the episode, include the following: 1. The development relied on individual endeavour (at home and in class) and on one-to-one and whole-group social endeavour (in class). 2. The bow-tie context occupied several lessons, thereby giving students time to think about the vector-motion properties without the distraction of new contexts for each new property. 3. Students had concrete (or at least virtual) experience of the speed relationships that they were investigating through viewing the speed at which graphs were plotted in the parametric aplet of their graphics calculators.


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4. Graphical and numerical approaches were combined for the investigation of velocity and acceleration. Using component values that they calculated, students drew the bow-tie graph accurately on graph paper, together with position, velocity and acceleration vectors; and the magnitudes and directions of the vectors could be gauged visually and some speeds were calculated. The speed against time graph was plotted and connections made with the bow-tie graph. 5. The overhead projector was used to display calculation on the calculator, calculator graphs and the graph-paper graph. The displays supported discussion. With them, the spoken word was less ambiguous, for example, Mr D’s diagram of the acceleration vector gave precise form to Kim’s “backwards”. Sometimes the teacher’s calculator was linked to the overhead panel and sometimes a student’s calculator. 6. The procedural calculation and drawing activities were tempered with the challenge of prediction that involved extended abstraction. 7. Other activities on the bow-tie relationship and the corresponding graph of speed against time followed the discussion on acceleration. Then, velocity and acceleration properties for the simpler cases of circular and parabolic motion were deduced from the graphs of the motion and numerical data. This is the reverse of starting with properties of simple cases, which is more usual.

C ONCLUDING DISCUSSION During the fieldwork, the conversation in the Year 12 class caught our attention: the pulse of shared inquiry and the rhythm of the teacher’s questions; harmony and concern for relationship, and the sometimes-discordant sounds. Analysing the short 45-line episode led us to identify a range of qualities of the interaction and to identify implications for participation and inference by students. The rhythm of ‘teacher question, student reply, teacher response’ was heard at the beginning and heralded the shared inquiry. Mr D’s questions included three calls to guess (or infer) and three requests for explanation. Asking for inference and then requiring explanation, as Mr D did of Kim and Tanya, indicates that students’ ideas are valued. Their cognitive work is heard, which is conducive to critical thinking and communicative competence. If the teacher expects inference and then evaluates students’ responses, speaks for students, and ignores responses, which occurred at the beginning, students are slaves to and victims of the teacher’s agenda. They supply answers but the opportunity to (a) articulate further their

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ideas, (b) develop their ideas with an ‘expert’ scaffolding the progress, and (c) develop expertise in public debate is closed off. The pulse of shared inquiry was evident when the teacher drew a diagram in response to Kim’s suggestion of “backwards”, and in the consecutive student-turns in conversation. Thus, conditions for inquiry are the teacher works with students’ ideas, and students feel they can take the initiative. Issues raised for inquiry learning were satisfactory conclusion to challenge and the importance of validity checks. Harmony or accord was evident when Kim consulted with Abi. They knew how to work cooperatively, and their transitions from public to private conversation and vice-versa were smooth. Mr D facilitated Kim’s reentry to the public forum, so the private conversation informed the public discussion, which was a common practice in the class. We view the practice as highly empowering because it allowed more students to express their ideas verbally than is possible in ‘whole-class’ work, and ideas became available to the class that otherwise might not have been heard. Discord appeared when Tanya interrupted Kim and when Kim disrupted the teacher-student turn-taking with “No”. Timing and content determine whether discordance is an interruption and is ignored, or whether the utterance is heard and taken up. Furthermore, Kim’s “No” was followed by a question and we suggest the form showed concern for relationship with others. However, concern for relationship is problematic if it amounts to reluctance to resolve differences in opinion. Kim not moving to pursue debate with Tanya and Abi could be cast in these terms. Thus, the analysis led to identification of communicative competence and inquiry. It also revealed very different outcomes for the students across the class. If verbal communicative competence is viewed as an important outcome of education, a challenge in teaching is to make opportunities for all students to hold the floor. This is a challenge in view of the grip of traditional order and in view that there was a culture of dialogue in the Year 12 class, yet, scrutiny of the transcripts revealed 4 of the 13 students did not voluntarily contribute to class discussion. In conclusion, we commend the instructional sequence as creating a rich context for the development of vector-calculus relationships. We suggest that the nature of the tasks, the teacher’s questions, and the use of graphics calculator and projector technologies, were fundamental to students’ formulating and articulating their ideas. Consequently, new mathematics relationships were made available to the class. Moreover, Kim’s actions illustrated the complexity of competent performance. That is, performance that was interactionally appropriate (rising to a challenge, gaining and holding the floor, limiting an answer, offering explanation) and


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advanced the academic content (through discernment which resolved ambiguity), while she sustained relationships with peers. Finally, we acknowledge that the classroom practices we described in relation to Kim’s performance were not necessary or sufficient for the development of communicative competence, but suggest that regular appearance of them could assist the development.

A PPENDIX 1: T HE LAST ROUND OF CONVERSATION , AUGUST 30 TH Mr D Abi Mr D

Abi Mr D Abi Mr D Kim Mr D St Mr D

Anna

Sophie Mr D Anna

Mr D Anna Tanya

. . . . . . I wonder where it will be when we get to here ([the turning point in the first quadrant, see Figure 1? It won’t be. It won’t be. What do you mean it won’t be? It’s got to be somewhere. There will be some acceleration acting on the body. It will go forward. Like down. The opposite direction, increasing. Going this way? Increasing in the opposite direction. What about right at this instant [pointing to the turning point]? Down, maybe, I don’t know. Might be down? Down this way? Down that way? Straight down. Why straight down? [pause] Okay. Right, I am going to let you discover that tonight and tomorrow we will talk about what you find tonight. [to Sophie who is sitting next to her] I think the acceleration, if you think of the acceleration graph. [pause] When you differentiate cos [referring to cosine in v(t) = 10 cos t i + 5cos t j], the acceleration is going to be sine, so won’t the velocity and acceleration be at right angles to each other? [to Anna] I reckon it will have to be going back a bit. What are you saying, Anna? I was thinking if it’s at right angles and it’s something to do with the fact that if velocity is cos, then acceleration is going to be sine. So, you think it will always be at right angles. Yes. I don’t know, I could be guessing. What are you guessing?

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Anna

I am guessing it’s at a right angle because in acceleration, the cos’s are going to become sines. St But it’s got to stop it going up, otherwise it will fly off. Mr D Yes. You can’t have it flying off here. The lesson came to a close.

ACKNOWLEDGEMENTS We thank Mr Craig Davis of St Hilda’s College for Girls and his 2001 Year 12 Calculus students for welcoming us into their class, and thank the principal Mrs Joy Shepherd for welcoming us into the college. We acknowledge reviewers’ feedback that led us to deepen the analysis considerably. The study was supported by a Curtin University of Technology Small Grant funded by the Australian Research Council. R EFERENCES Belenky, M.F., Clinchy, B.M., Goldberger, N.R. and Tarule, J.M.: 1986, Women’s Ways of Knowing, Basic Books, New York, NY. Brickhouse, N.W.: 1993, Evolving Conceptions of Ethical Practice in Interpretative Research, keynote address at the 1993 International Conference on Interpretative Research in Science Education. National Taiwan University. Cobb, P.: 1998, ‘Analyzing the mathematical learning of the classroom community: The case of statistical data analysis’, in A. Oliver and K. Newstead (eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, Vol 1, South Africa, pp. 33–48. Cobb, P., Boufi, A., McClain, K. and Whitenack, J.: 1997, ‘Reflective discourse and collective reflection’, Journal of Research into Mathematics Education 28(3), 258–277. Cobb, P., Wood, T., Yackel, E. and McNeal, B.: 1992, ‘Characteristics of classroom mathematics traditions: An interactional analysis’, American Educational Research Journal 29(3), 573–604. Cohen, L. and Manion, L.: 1994, Research Methods in Education (4th ed.), Routlege, London. Fairclough, N.: 1992, Discourse and Social Change, Polity Press, Cambridge, UK. Forster, P.A.: 2000, ‘Katie thought she couldn’t do it but now she knows she can’, Educational Studies in Mathematics 43(3), 225–242. Forster, P. and Taylor, P.: 2000, ‘Emergence of mathematical (in)competence and identity’, in J. Bana and A. Chapman (eds.), Mathematics Education beyond 2000: Proceedings of the 23rd annual conference of the Mathematics Education Research Group of Australia, Perth, Australia, pp. 65–71. Forster, P., Taylor, P. and Davis, C.: 2002a, ‘ “One hundred and sixty geckos and one without a tail”: Wit and student empowerment when class discussion is centred on the display from a student’s graphics calculator’, Australian Senior Mathematics Journal 16(1), 56–64.


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Forster, P., Taylor, P. and Davis, C.: 2002b, ‘The sherpa-student role with a graphics calculator: Empowering or disempowering?’, in B. Barton, K.C. Irwin, M. Pfannkuch and M.O.J. Thomas (eds.), Mathematics Education in the South Pacific, Proceedings of the 25rd annual conference of the Mathematics Education Research Group of Australasia, Auckland, New Zealand, pp. 276–284. Gumperz, J.J.: 1995, ‘Mutual inferencing in conversation’, in I. Markova, C. Graumann, and K. Foppa (eds.), Mutualities in Dialogue, Cambridge University, Cambridge, pp. 101–123. Hershowitz, R., Schwarz, B.B. and Dreyfus, T.: 2001, ‘Abstraction in context: Epistemic actions’, Journal for Research in Mathematics Education 32(2), 195–222. Mehan, H.: 1979, Learning Lessons: Social Organization in the Classroom, Harvard University Press, Cambridge, MA. Roth, W.-M.: 1998, Designing Communities, Kluwer Academic, Dordrecht, The Netherlands. Sfard, A.: 2000, ‘Steering (dis)course between metaphors and rigor: Using focal analysis to investigate an emergence of mathematical objects’, Journal for Research in Mathematics Education 31(3), 296–327. Taylor, P.C.: 1996, ‘Mythmaking and mythbreaking in the mathematics classroom’, Educational Studies in Mathematics 31, 151–173. Taylor, P.C.: 1998, ‘Constructivism: Value added’, in B. Fraser and K. Tobin (eds.), The International Book of Science Education, Kluwer Academic, Dordrecht, pp. 1111–1123. Watson, A. and Mason, J.: 1998, Questions and Prompts for Mathematical Thinking, Association of Teacher of Mathematics, Derby, UK. Young, R.: 1990, A Critical Theory of Education: Habermas and our Children’s Future, Teachers College, New York, NY. Young, R.: 1992, Critical Theory and Classroom Talk, Multilingual Matters, Adelaide, South Australia. 1

Edith Cowan University, Joondalup, Western Australia 6027 E-mail: [email protected] 2

Curtin University of Technology, Bentley, Western Australia 6102 E-mail: [email protected]