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FOR041047 Listening in the Mathematics Working Relationship of Two High-School

Students

Patricia A Forster Edith Cowan University

p.forster@ecu.edu.au

Abstract This paper presents an inquiry into listening by two senior (Year 11) high-school students, as they worked together, one-to-one. It is based on classroom observations and audio-recordings of the students' conversations over ten lessons. Listening is inferred for repeated patterns of interaction. Paired listening acts where the students evidenced shared understanding, or disagreed, or where one student sought to understand the other are described. Inquiry indicated the latter type of listening was prevalent. Learning outcomes associated with the different modes of listening and implications for teaching practice are identified.

Introduction

This paper presents an investigation of the listening between two students in a Year 11

Geometry and Trigonometry class, over 10 lessons. The students, Katie and Jenny, regularly worked together and I propose that they engaged in evaluative, interpretative and hermeneutic listening, which Davis (1996) defined to characterise listening between teachers and students. The different types of listening mediated different types of mathematical endeavour by Katie and Jenny, including peer-tutoring and collaborative work.

The assumptions and methods of ethnomethodology (e.g., Holstein & Gubrium, 1994) guided the research writing. This interpretative methodology requires on-site observations so that the researcher has first hand experience of the life-world of the researched. The purpose in the method is to describe social order and the research account is known to be personal to the researcher (Roth, 1998). The inquiry presented in this paper was directed at the order in Katie's and Jenny's one-to-one interactions. Listening was inferred from conversation data. Order in the whole-class conversations in which Katie and Jenny participated is discussed in Forster (2000a).

The significance of the paper resides in the identification of the students' paired listening acts while working together, and in the identification of ways the students' supposed listening enabled and constrained their mathematics advancement. Implications for teaching are described.

Listening in the Literature

Davis (1996) proposes 'evaluative, interpretative and hermeneutic listening' in the teaching act. He defines evaluative listening to be the "taking in of information which is out there" (p. 53), followed by judgment. Teachers cognise, then, judge as right or wrong the information that students offer and, whatever the verdict, deviate little from their plans. The repeated cycles of 'teacher questioning, short student response, teacher evaluation' associated with traditional teaching (e.g., Mehan, 1979; Young, 1992) are consistent with evaluative listening.

Interpretative listening is "a sort of reaching out rather than a taking in . . . whereby the listening is deliberate and aware of the fallibility of the sense being made" (Davis, 1996, p. 53). With interpretative listening, the teacher's focus is on accessing rather than assessing students' ideas. Constructivist literature (e.g., Confrey, 1990) promotes this type of listening, and

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questioning by the teacher is likely to motivate the endeavour. The questions are directed at prompting students to think, and explain what they are thinking, and not at short responses.

With both evaluative and interpretative listening there is a split, a felt separation, between the teacher and the student (Davis, 1996). They act separately. In hermeneutic listening the split is transcended:

This manner of listening is more negotiatory, engaging, and messy involving the hearer and the heard in a shared project. . . . The focus is on the dynamic interdependence of agent and setting, thought and action, knowledge and knower, self and other, individual and collective--rather than on autonomous constitution or construction . . . hermeneutic listening is a participation in the unfolding of possibilities through collective action. (Davis, 1996, p. 53)

The notion of hermeneutic listening is consistent with learning being "fluid, co-emergent, and dynamic comprehension" (Pirie & Thom, 2001, p. 49), and "co-operative change in relationship with others" (Young, 1992, p. 4).

Listening can also be distinguished in terms of monologue and dialogue (Coulter, 1999). With monologue, one speaker assumes control and places others exclusively in the listening role. Meaning is the expression of one person's ordering of experience and not the product of the interchange of ideas. In dialogue, there is genuine communication between participants. This requires listening to one another, mutuality in the listening (i.e., interactive listening), and mutual respect. However, to understand dialogue is to understand that in every utterance there are unifying, convergent, centripetal forces that bring participants to common understandings, and stratifying, divergent, centrifugal forces that are attributable to speakers' different histories (Bakhtin, cited in Coulter). In not giving recognition to others, monologue presumes divergent forces only. Dialogue between teachers and students is problematic because of the asymmetrical teacher-student power relation that arises because teachers know the subject matter and because of their pedagogical responsibilities (Young, 1992). Power relationships also exist between students, which may mediate against dialogue where all freely offer their ideas (Watson & Chick, 1997).

Schweickart (1996) theorises the role of listening in caring relationships. She reminds us that for meaning to be produced between people the listener must be open to the speaker's utterances and actively engage with them, in an appreciative but not selfless way. She holds that generally there is over-evaluation of assertive argument, and under-evaluation of understanding and agreement that depend on listening. Along with other feminist scholars and following Belenky, Clinchy, Goldberger and Tarule (1986), she points to males finding rationalistic argument more relevant than females, while females favor empathetic relations, in which listening is paramount. Schweickart links the prioritization of argument to the male preference for it.

Schweickart (1996) also recasts listening and speaking, to reverse the action that is dominant. Drawing on Noddings' (1984) analysis of caring, she proposes that, in the act of listening, one is caring for the speaker; and the cared-for, the speaker, is dependent on and vulnerable to the interpretative agency of the listener. As well:

The one cared for must acknowledge and accept the care given, use its enabling power to advance her own projects, and freely share accounts of her own life and her activities with the one caring. (Schweickart, p. 320) McNiff (1996), similar to Schweickart (1996), holds that listening is a "marginalised

practice" (p. 3). She describes also how she undertakes listening holistically; and that for her, listening is a mixture of gesture and experience, is trying to catch the unspoken words, and is striving to be at one with another.

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Gordon Calvert (2001) speaks of listening between two friends (Stacey and Ken) when undertaking a mathematical investigation. She says, 'Stacey's act of listening and responding to Ken's suggestions entered them into an inviting conversational space. It is often the listener who creates the space for conversation' (paraphrased, p. 97). Gordon Calvert (2001) recommends that conversation is an opportunity to bring ideas into play, to change ideas and expand one's understanding. Conversation does not preclude disagreement, but after it effort may be needed to re-enter conversation with the other.

Gordon Calvert (2001) goes on to contrast mathematics conversation with mathematical argument. She says: "In the presence of my mathematical argument: I address the other because I want to convince him or her of its truth. . . . The response I expect and listen for in the other's gestures is either agreement or disagreement with what I have proposed" (p. 135). According to Gordon Calvert, argument is a platform for enforcing personal viewpoints.

Wood (2002) considers listening by students in whole-class conversation. She characterises listening in a traditional paradigm as "paying attention" (p. 64) and suggests an expectation in a reform culture is that students as listeners will feel responsibility to participate: in the presence of inquiry, they will ask questions and acknowledge that things make sense; in the presence of argument, they will disagree and make challenges.

In summary, a person in monologue does not listen to others and is closed to others' views (Coulter, 1999). Evaluative listening (Davis, 1996) and listening in mathematical argument (Gordon Calvert, 2001; Wood, 2002) entail evaluation in relation to personal views. With interpretative listening (Davis, 1996) one person directs her attention towards another's ideas but mutuality in the listening is absent. Hermeneutic and holistic listening (Davis, 1996; McNiff, 1996), and listening in dialogue and conversation (Coulter, 1999; Gordon Calvert, 2001) seem to me to intersect. They are synergistic and involve shared endeavour.

The above definitions of listening underpinned my interpretation of Katie's and Jenny's actions. I was guided also by definitions of argument in The New Shorter Oxford Dictionary (Brown et al., 1993), namely, "a connected series of statements or reasons intended to establish a position" and "a reason urged in support of a proposition" (p. 112). I assumed that positions and propositions could be shared in the production of a mathematical argument or could be put forward by one person only, and the whole range of listening stances could be evoked.

Bateson's (2002) theory of mind/experience/learning also informed my analysis. He proposes that what we hear, see or otherwise sense is created by our sense organs and that sensory perception, in interaction with others, consists of random processes so that we experience some, but not all stimuli to which we are exposed. Experience is also evolutionary so that what we perceive depends on our past experiences and development. Consequences for listening are that personal stances develop as do patterns of listening between individuals. I did not presume the evolution of any patterns in Katie's and Jenny's listening in the relatively short time of my inquiry (10 lessons), but rather that patterns were in place.

Research Method

The assumptions of ethnomethodology as articulated in mathematics and science education

literature (Jungwirth, 1991; Livingston, 1987; Mehan, 1979; Roth, 1998) and literature on research methods (e.g., Holstein & Gubrium, 1994) guided my inquiry. Ethnomethodology is suited to the investigation of social order in groups. Order is assumed to be organised internally and not the product of rules set by outsiders. Group members are seen to interactively constitute and reconstitute the objects of their experience, and the same applies to researchers.

I inferred order in Katie's and Jenny's listening, in their one-to-one interactions. The main source of data was audio-recordings of the students' talk, which I collected for 10 lessons, out of the 15 50-minute lessons that I attended in the Year 11 class. I attended the class as an observer-participant, observing whole-class work, and questioning and responding to students as though

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an assistant teacher during seatwork. Other data were photocopies of the students' written work; video-recordings of the 10 lessons from the front of the room, with three- quarters of the class (including Katie and Jenny) in the field of view; field notes in which I had noted my impressions of the classroom action, and journal entries of my reflections on the action.

In starting to write the paper, I searched the audio-transcripts for repeated speech patterns. Consistent with ethnomethodology (Holstein & Gubrium, 1994), I searched for ways the students themselves interactively produced 'recognizable, intelligible forms they treated as real' and not for patterns from the literature. Several patterns caught my attention and I noticed that they occurred when one student was helping the other, viz during peer-tutoring (Linn & Burbules, 1993), or when they were producing a solution to a problem together, viz during collaborative learning (ibid). So, I divided the extensive data into discrete interactions and classified each as peer-tutoring, collaborative endeavour, off-task interaction, initiation with no response or minimal response (e.g., "Mmmm"), and monitoring. Monitoring was on-task but any mathematical advancement was peripheral. I counted as a single interaction conversation that started after a period of whole-class work, discussion with the teacher, discussion with other students, or silence; and finished when conversation expired or was terminated by whole-class work, was interrupted by the teacher or was interrupted by other students.

The classification motivated wider recognition of speech patterns for which I inferred qualities of listening. Then, I made a purposeful selection (Cohen & Manion, 1994) of four ‘discrete interactions’ for reporting in this paper. I chose ones that portrayed different speech patterns and allowed definition of different modes of listening.

Last, because research is a subjective exercise, I need to justify why my account should be accepted as reasonable. To this end, I have defined the inquiry process so that the reader can audit it. In the analysis section, I quote classroom conversation verbatim, so that the reader can know on what my propositions were founded; and I cite the literature that influenced my judgments. As well I sought feedback on the analysis from critical friends involved in education. Our discussions alerted me to aspects of listening that I hadn't thought of, some of which I included in the paper. Others I rejected as not being consistent with the uniqueness (van Manen, 1990), as I experienced it, of Katie and Jenny.

The Classroom Context

The Year 11 class comprised 18 students. The school was a private, church-sponsored

girls’ school in Western Australia. The Year 11 Geometry and Trigonometry course that the class was studying was designed for students "desiring a strong mathematical preparation for tertiary studies" (Curriculum Council, 1998, p. 13). A topic on vectors in two-dimensional space was treated while I was with the class. All students had owned and used a graphics calculator for one month prior to my attending the class.

Instruction in the class was traditional in that reduced questions and evaluation by the teacher predominated. The teacher sanctioned joint endeavour on textbook exercises and on other tasks that he set by permitting students to work together with students of their own choice. The action was distinct from the joint effort described by Anderson, Holland and Palinscar (1997), where the teacher organises it and assigns students to specific roles.

The teacher and I assisted students on an individual or small-group basis during seatwork, in the Australian way described by Clarke (2003), which entailed us moving between desks to discuss students work with them--rather than the students coming forward to the teacher’s desk. The teacher did not ask students to present their individual or joint work to the class, in the formal way described, for example, by Anderson et al. (1997) and Brown and Renshaw (2000) but he facilitated class discussion on problematic solutions, when Katie and Jenny voiced their joint endeavours (e.g., see Forster, 2000a, 2000b; Forster & Taylor, 2000).

5

!

500km56km/h

350km/h

10°

40°A

B

Jenny and Katie always sat next to each other and were friends outside the mathematics classroom as well as in it. They regularly discussed their work, and generally other students weren't involved. In the assessment test that was relevant to conversations that are presented in this paper, Jenny was first with 80%, Katie was seventh with 71% and the class average was 62% (N= 18). Two months later, in the mid-year examinations, Jenny retained first place with 85% and Katie advanced to third place with 82% (class average 66%).

Listening in Katie's and Jenny's Working Relationship

Four ‘discrete interactions’ are discussed below. They varied in length from 11 to 54 turns in conversation, so, in the interests of space, I provide excerpts only from transcripts. I have included interpretative comments in the transcript, to assist the reader. The comments were based on my understanding of the conversations in retrospect, which took into account how the conversations unfolded in their entirety. Example 1: Evaluative and Interpretative Listening

Katie and Jenny were working separately on a textbook question (see below). It required them to analyse the flight path of a plane using displacement and velocity vectors. The students had recently worked a similar question together, which had been discussed in class. They each drew a diagram for the question and the parts of these that were relevant to the conversation below are replicated in Figures 1a and b (the complete diagrams are provided in Appendix 1). Angle measurements are not shown in Figure 1b because the original ones in Katie’s workbook were erased and were not discernable. They were erased as a result of the conversation.

In what direction must a plane head to fly 500 km in a direction 040° if it can maintain 350 km/h in still air and the wind is blowing at 56 km/h from 100°? How long would the journey take to the nearest minute? (Sadler, 1993, p. 46)

a b Figure 1. Replication of (a) Jenny's diagram and (b) Katie's diagram for the problem.

A few minutes after starting a solution to the question, Katie turned to Jenny and said: 1. Katie I can't get the direction. 2. Jenny Okay. Katie, it's going to fly 500 km in a direction of 40, it's going to fly that way at 350 [see

Figure 1a]. 3. Katie Yes. I got that. But see my diagram I did it here [Figure 1b]. 4. Jenny Okay, okay. The wind is blowing from 100 degrees so that's 90 [rotating clockwise from the

vertical at B on Katie's diagram], that's 100 [the angle from the vertical to the vector labeled with 56]. My diagram

5. Katie [interrupting] No, the thing is, isn't it from the horizontal? 6. Jenny Ehh!

A

B

500km

350km

56

56km/h

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7. Katie Aren't the angles from the horizontal? 8. Jenny Mmm. No, isn't it [pause]? If you want to fly in a direction of 40 degrees, the direction is

there [clockwise from the vertical]. Is that right Mrs Forster?

I propose that Jenny and Katie engaged in evaluative and interpretative listening respectively in this example, and that the conversation constituted peer-tutoring of Katie by Jenny. My rationale is as follows.

In listening to Katie’s first utterance (turn1), Jenny discerned that help was needed, then, drew on her own reasoning for the information that she gave (turn 2). After Katie agreed and presented her diagram (turn 3), Jenny (turn 4) provided more information and an explanation, according to how she (Jenny) viewed the set up for the question. In the presence of the disagreement (turns 5 and 7), Jenny (turn 8) evaluated Katie's suggestion in terms of definitions she (Jenny) knew and she sought confirmation of her own view from an authority. Hence, consistent with evaluative listening (Davis, 1996), Jenny always drew on her own thinking when responding to Katie, and did not seek Katie’s views.

Having sought assistance (turn 1), Katie's role was to interpret, and her confirmation (turn 2) and negation (turn 5) of Jenny’s responses evidenced interpretation. She was critical in the sense that she did not accept Jenny's responses at face value: she evaluated them in terms of her own understanding (turn 2 and 5). However, evaluative listening was not operative, for Katie did not prioritise her own ideas--she was not self-absorbed. Neither was hermeneutic listening operative, for it resides in synergistic interaction, mutuality and inter-dependence. Rather, Katie’s actions were consistent with interpretative listening, which is directed at understanding the other (Davis, 1996), especially because she persisted in accessing Jenny’s opinion about the diagram (turn 2) and the angles (turns 5 and 7).

As a result of the action, Katie's mistaken understanding of bearings was uncovered. Upon seeking outside help, Jenny had her understanding of bearings confirmed and Katie corrected her mistake. Peer-tutoring (Linn & Burbules, 1993) where one helps another gain expertise seems an appropriate descriptor. Example 2: Hermeneutic Listening Leading To and Repairing Peer Tutoring

In this second example, Katie was working a textbook question that Jenny had completed for homework. The question is provided below and Jenny’s solution is given in Appendix 1. The solution is replicated in Figure 2 below, with descriptive words added to simplify interpretation for the reader. The two students had agreed on the diagram in a previous lesson. They had set the horizontal component of the plane's velocity equal in size and opposite in direction to the horizontal component of the velocity of the wind (see Figure 2). Katie initiated conversation (see the transcript below) as she checked her solution from the answers at the back of the textbook.

A helicopter can fly at 75 m/s in still air. The pilot wishes to fly from airport A to a second airport B, 300 km due North of A. If i is a unit vector due East and j a unit vector due North find (in the form ai+bj) the velocity vector that the pilot should set and the time the journey will take if there is a wind of (21i + 8j) m/s blowing (Sadler, 1993, p. 72).

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r

75m/s

21

8

!

r = 75 cos ! = 21cos ! = 21/75! = 73·74j component of the plane's velocity = 75 sin 73.74

= 72 m/svelocity of the plane = -21i + 72jd = s x t300 000 = 80 x tt = 300 000 / 80= 3750 s= 62·5 minutes

300 km

Figure 2. Problem solution.

The opening sequence

1. Katie Ummm. Oh dear. Mine's wrong, I knew [as she checked the answer]. 2. Jenny Ahhh. I seriously cannot do G and T [Geometry and Trigonometry] at home. I sit here, yeah,

yeah, then I get home and can't do it any more. 3. Katie Well, I can't do it ever. So [pause]. Well Jenny, I can't do it, I don't know what I am doing. 4. Jenny Well, Katie [firmly]. 5. Katie I have got that and that [the velocity of the plane and wind]. Are they in metres per second,

are they? My reading of the text is that Katie called for peer-tutoring in turn 1. However, unlike in

Example 1, she did not state her problem, so Jenny could not respond with advice. Instead, Jenny claimed that she was unsuccessful like Katie (turn 2). Hence, Jenny acted to be in sympathy with Katie: she attuned herself to her, to be with her. Their non-success was mutual: there was mutuality in the interaction, and dialogue between the students. It is noted that, by definition (van Manen, 1991), sympathy is based on similar experience and, with empathy, attempts are made to understand another, in the absence of similar experience.

Katie (turn 3) continued the dialogue. She claimed her ineptitude was more acute (“I can’t do it ever”), and in doing so she indicated that she had attended carefully to Jenny's turn 2 response. On all these counts, I suggest that the utterances in turns 1 to 3 indicate synergistic, hermeneutic listening. Then, the style of interaction changed when Jenny moved to peer-tutor Katie (turn 4) and Katie moved quickly to secure the help (turn 5):

Peer-tutoring in process Jenny confirmed the velocities were in metres per second, then, compared Katie's solution

with her own. She identified that Katie's j component for the plane's velocity was incorrect. Katie reworked the calculation and obtained 72j (see Figure 2). Conversation continued as given below.

13. Jenny Now you just have to work out the time. 14. Katie But I've got to add it [the velocity of the plane] to that vector [the velocity of the wind]. 15. Jenny No, because the question said what vector should he set the plane at. 16. Katie Oh. 17. Jenny Imagine where the plane goes. So how many horizontal this way [to the left] and how

many vertical. 18. Katie Okay. Now I just have to work out the time. Does it ask for bearings? 19. Jenny Nope. Because that [the velocity vector of the plane] gives you the bearing. 20. Katie Okay. Time. Speed equals distance by time. 21. Jenny No. Distance equals speed times time.

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I suggest that, as for Example 1, the peer-tutoring was underpinned by evaluative listening by Jenny and interpretative listening by Katie. Jenny explained from her own viewpoint (turns 13, 15, 17, 19 and 21), made judgments (“No” or “Nope”, turns 2, 19 and 21) and did not question Katie on her reasoning. Katie stated her thinking (turns 14, 18 and 20) and thereby accessed Jenny’s opinion about the method of approach (turns 15, 19 and 21).

A problem though, is that Jenny’s advice in turn 13 was incorrect. The velocities of the plane and the wind had to be summed before calculating time. Jenny’s written solution shows she had realised this after an initial error (see Appendix 1), but it seems she had lost sight (Watson & Chick, 1997) of the method in explaining it to Katie.

Katie’s response in turn 14 compromised further the effectiveness of the peer-tutoring, in that her purpose for adding the vectors was not clear. Possibilities were: 'she had to add the vectors to find the velocity vector that the pilot should set' or 'she had to add the vectors before she could calculate time' (see the question above). Then, Jenny (turn 15) mentioned the vector that the pilot should set, so, appeared to assume the first purpose, and it is likely that she did not perceive the possibility of two purposes. Typically, people take unique meaning in conversation and assume others take the same meaning (Holstein & Gubrium, 1994): they experience the world objectively (Bateson, 2002). Furthermore, lack of clarity in what is spoken, and misinterpretation by listeners of the intended meaning is a common problem in mathematics (Sfard, 2000). .

Katie might have realised that something was awry, judging by her "Oh" (turn 16), but she did not hold to her suggestion or voice any other concern. In Watson and Chick’s (1997) terms, she failed to hold onto her ideas in the presence of persuasive argument by another—and holding on tenaciously to personal viewpoints can be important for success in working together.

Hence, it was a case of communication in the peer-tutoring being flawed because: Jenny seemed to lose sight of a calculation that she had already performed; Katie’s purpose for adding vectors was not clear; Jenny, assumed the non-viable alternative; and Katie did not hold to her ideas in conversation. A second discrepancy "Speed equals distance by time" (turn 21) was addressed, and gave rise to another three turns that indicated another shift in the listening of the two students:

22. Katie Isn't it: speed equals time times distance? 23. Jenny Draw a little triangle. 24. Katie This is your beautiful little magic triangle [see Figure 3]. 25. Jenny You like my magic triangle. I like my magic triangle. I haven't drawn my magic triangle this

year yet, though.

DS T

Figure 3. The magic triangle.

I propose that mention of the "little triangle" (turn 23) evoked a return to interactive,

hermeneutic listening. It evoked mutuality in the interaction, which was evidenced in echoed words that were based on the students’ shared history of working together.

Their reminiscing served the ongoing peer-tutoring in several ways. First, the three-turn sequence brought them together after Jenny had negated Katie's suggestion (turn 21): in other words, the sequence repaired (Gordon Calvert, 2001) the students' conversation. Second, Katie's response "This is your beautiful little magic triangle" (turn 24) was highly complementary and

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suggested respect for Jenny and her mathematics achievement. Hence, Katie acknowledged and affirmed Jenny which, according to Noddings (1984) and Schweickart (1996) are necessary actions for the continuation of care in a relationship. Third, Jenny was self-affirming (turn 25). Expressing her feelings could have made her vulnerable to taunts by others, but she was safe with Katie, and I suggest her openness signified interest and enjoyment in the working relationship.

The meaning of the triangle, as Jenny explained later to another student was "DST, meaning distance equals speed times time, or time equals distance over speed". So, the triangle's magic lay in the form it gave the distance-velocity-time relationship which is widely reported as causing students difficulty (e.g., Hale, 2000).

The peer-tutoring continued as follows: 29. Jenny Time is what we are trying to find. And speed is 75. You have got the distance. . . . . 34. Katie So, then its [inaudible]. Oh it's not [upon looking up the answers in the textbook?]. Ohh

[pause]. 35. Jenny Oh sorry. The speed [of the plane in the j direction] is 72. 36. Katie Why is it 72? 37. Jenny Because the plane is going to be traveling . . . . . . . 43. Jenny . . . But it [the 72] would also have to be added with the other vector [the wind]. 44. Katie So, that's 80 [the j component of the combined velocities, see Figure 2]. 45. Jenny No. 46. Katie But you have to add it to the wind. 47. Jenny Yes. [pause] Yes. Katie proceeded to calculate the correct time, 3750 seconds, see Figure 2.

In this excerpt, Katie persisted in seeking Jenny’s ideas (turns 36, 44) but as well she challenged Jenny (turn 46). She forced Jenny to listen when Jenny disagreed, and Katie's rejoinder (turn 46) seems consistent with a shift from interpretative listening to evaluative listening. In listening, she changed from focusing on Jenny's reasoning to focusing on her own.

Hence, I am suggesting that, in disagreement, Katie and Jenny both assumed an evaluative listening stance directed at receiving confirmation of their own views, in the way that Gordon Calvert (2001) describes. However, in their evaluative listening, they were open to the standpoint of the other. In particular, Jenny evidenced openness by considering and accepting Katie’s rationale (turn 47).

So, in summary, performance fell (Watson & Chick, 1997) in this example because of a convincing, yet erroneous, argument by Jenny, with which Katie went along (turns 13-34). Then, Katie evidenced tenacity in argument and Jenny evidenced openness to her viewpoint, and their actions were mathematically productive. Discussion

Evaluative/interpretative listening Katie initiated 12 peer-tutoring interactions over the 10 lessons of my inquiry, and Jenny

initiated one. In 8 of the 12 instances when Katie spoke first, she asked a question or specified a calculation or mathematics property that she did not understand, and Jenny gave information or confirmed or negated what Katie said, as in Example 1. In subsequent responses, Jenny persisted in drawing on her own thinking and did not seek Katie’s views. So, as argued in Example 1, the actions implied mainly evaluative listening by Jenny and interpretative listening by Katie.

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Another time when Katie spoke first, she asked Jenny how she had solved a question and she detected an error in Jenny’s written solution. Jenny moved to take up Katie’s reasoning. So, their roles in the peer-tutoring were reversed as were their modes of listening.

The three other peer tutoring interactions that Katie initiated included Example 2 and all started like Example 2. Katie expressed failure but did not identify her difficulty. Not knowing the source of difficulty, Jenny spoke to be in sympathy with Katie, and Katie claimed her failure was worse than Jenny’s. Jenny offered help, and the peer tutoring commenced. Subsequent conversation implied interpretative listening by Katie and evaluative listening, except that in one episode Jenny did not listen. She spoke aloud a method as she worked, as would a teacher demonstrating a solution. Jenny was effectively in monologue (Coulter, 1999).

The one time Jenny initiated peer-tutoring, she had finished a task that the teacher had set and was watching Katie work. Jenny lent over and said of Katie's method "That's wrong Katie" and explained what to do. Katie compared their methods and accepted Jenny’s.

Hence, the conversation in peer-tutoring seemed mainly motivated by evaluative listening by the tutor, and interpretative listening by the one-tutored. The listening acts were complementary (Bateson, 2002): the students were in a complementary relationship where one sought information or an opinion and the other was prepared to respond, and listening underpinned their actions. Further, in my experience as observer in the class and upon analysis of the data, the peer-tutoring seemed to mediate Katie's mathematics progress favorably in that she usually proceeded on her own after her impasses had been addressed. Jenny benefited from the occasional resolution of discrepancy and possibly she clarified her understanding through providing explanation.

An important aspect of success in the peer-tutoring was that Katie was active in thinking through the mathematics. Her questions arose from her attempted solutions and, in listening to Jenny’s responses, she clearly engaged with them. She sometimes voiced critique, which contributed to resolving for her and Jenny the issues being discussed (see Example 1).

Jenny, with her evaluative, judgmental stance, often dealt efficiently with Katie’s calls for help, although the approach suffered the limitation that Jenny sometimes made wrong assumptions. In the case of discrepancies not being noticed, or being noticed and not being voiced (Example 2), communication was compromised and performance fell. Hence, Examples 1 and 2 illustrate that critical listening by the one being tutored, and holding to and (re)voicing personal views can be crucial for efficient and productive peer-tutoring.

Other listening acts The action in the peer-tutoring also indicated hermeneutic listening between the students.

The listening was inferred twice in Example 2. The first instance was for the three-turn sequence of demise that secured peer tutoring for Katie. Two other episodes started with similar sequences. The demise was of both of them, but more-so of Katie. Hence, the interaction reinforced mathematical superiority for Jenny and inferiority for Katie and, from my viewpoint, the enacted power-play was undesirable.

The other instance of hermeneutic listening in Example 2 involved Jenny’s magic triangle. Possible benefits of the interaction were repair of conversation after disagreement, sustenance of the working relationship, enjoyment in the artefact, and the artefact gave form to the distance/velocity /time formulae. The affirmation by Katie of Jenny, and self-affirmation by Jenny reinforced mathematical superiority for Jenny. Counterbalancing this, a different episode of peer-tutoring involved affirmation in the opposite direction (of Katie by Jenny, and self affirmation by Katie).

Peer-tutoring also gave rise to evaluative listening by both students, when they disagreed (Example 2). Both listened for confirmation of their own views. Resolution of the disagreement occurred through being open to each other’s reasoning. Thus, relinquishing personal views can benefit peer-tutoring, as can holding on to them. Another aspect of the disagreement was it did

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not seem to suffer the limitation identified in the literature of being motivated by goals of winning, which result in negative affect for the loser (Gordon Calvert, 2001). Competition did not seem to be an issue between the two students, maybe because their prevailing belief was Jenny was more capable. Example 3: Hermeneutic Listening and Collaboration

This third example is taken from the first lesson on the vector topic in Year 11 class. Magnitude and direction of vectors had been discussed and the students were working through textbook questions that required the calculation of magnitude for vectors drawn on a two-dimensional grid. The teacher said to mix use of the Pythagorean relationship and the absolute value function ABS on the graphics calculator. The excerpt of conversation below is about using the ABS function to calculate the magnitude of the vector (1.76, 1.48) (see Figure 4).

Figure 4. Calculation of vector magnitude.

1. Jenny Katie, do you reckon you could put this [the ABS calculation] into our G and T things

[Geometry and Trigonometry equations stored in the calculator], if you went ABS brackets I J? [see Figure 5a]

2. Katie Yes. That could work. 3. Jenny I am going to try that because I think it is much quicker. 4. Katie Yes. But you would have to need it a lot. 5. Jenny If you had to do it real quick, I think you would do it, I think you would. [pause] Ahh. 6. Katie Did it work? 7. Jenny I made a mistake. 8. Katie You have to make an M. 9. Jenny Yes. That would be okay. M could work. Then I could go equals M [see Figure 5b]. 10. Katie You have to have all capitals don't you? 11. Jenny Okay. Then, I, J, M [pause]. Hey. It works [see Figure 5c].

a b c Figure 5. Formulation of the ABS equation.

I propose that Jenny and Katie engaged in hermeneutic listening in this example, and that

the conversation constituted collaborative endeavour (Linn & Burbules, 1993). My rationale is as follows.

In starting conversation (turn 1), Jenny invited Katie's opinion. They discussed the merit of having a stored equation (turns 3-5) and how to get the calculator to compute magnitude (turns 8-11). Therefore, Jenny motivated hermeneutic listening in Katie, through way she addressed her, and the listening style was perpetuated by both of them, in the ways they accepted and responded to each other's suggestions. I would argue that attentiveness and synergy in the interaction were accomplished through the listening (in turns 1-5, 7-10) and, as well, characterised the listening.

I classified the endeavour as collaborative because the move to generalise from specific examples signifies advancement in mathematical understanding (Sfard, 1991), and both students contributed to the ABS(I, J)=M formulation. The action entailed an idiosyncratic focus within a

12

set task and in this respect resembled the successful collaboration between students that Williams (2001) describes.

The students saved the equation as an aplet in their calculators and it became known to them as the "little aplet". The diminutive signifies affection (Brown et al., 1993), and perhaps for Jenny and Katie, "little" also signified magic in discovery (as for the triangle). Ownership of the aplet ended up being assigned completely to Jenny, in interaction with the teacher. The assignment is another story and was remarkable (see Forster, 2000a).

Example 4: Monologue and Collaboration

This last example illustrates ‘joint’ or ‘collaborative’ effort on a task for which the teacher had suggested to the class that students’ work together. The task involved (a) measuring distances from Cartesian axes to an ‘invisible’ object with a detector attached to a graphics calculator, (b) recording the distances on a diagram, and (c) calculating co-ordinates of the object with respect to the axes (see the worksheet in Appendix 2). Katie and Jenny made the measurements (see Figure 6 and Appendix 1) with two other students then attempted the calculation together. The transcript below relates to the calculation.

91cm(71)

77cm(64)

91

77

64

71

a b

Figure 6. Replication of Jenny's diagrams showing the measurements. Starting calculation

1. Jenny Okay. How do we calculate it? 2. Katie . . . the co-ordinates. We have to find the co-ordinates [of the object--the dot in Figure 6a]. 3. Jenny Okay. Okay. Here are our y and x axes [as she started to draw the graph in Figure 7a]. 4. Katie Mm. Mm. Go on. 5. Jenny They [the axes] will be at right angles. Right? 6. Katie Yes. 7. Jenny Then 77 points up. 8. Katie My diagram isn't very to scale. 9 Jenny And 91. 10. Katie Okay, so then 11. Jenny [interrupting] This [the right side of the quadrilateral] goes this way a little bit [to the left, see

Figure 7a]. . . so, this is 71 and this [the top of the quadrilateral] is 64. 12. Katie Then, if we got 13. Jenny [interrupting] That is our resultant [from the origin to the object?]. If we made some little

triangles [see Figure 7b]. 14. Katie Look Jenny, won't this be 91 cm from here to here [the top of the rectangle, Figure 7b]? 15. Jenny But that doesn't help us. We have to know what this and this is [the horizontal distance to the

object, and the height of the object, Figure 7b]. 16. Katie Is there any way of finding the angles? 17. Jenny That's just what I am trying to think. . . . I am going to redraw the triangles. [See Figure 7c.

After drawing them, Jenny proceeded to refer to the co-ordinates of the object as (x, y) and identified x, y, 64, 71 and 91-x on the triangles].

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9 1

7 7

7 1

6 4 x

y

6 4

7 1

- 9 1 - x

b c

[object]

9 1

7 7

7 1

6 4

a

Figure 7. Replication of the diagrams Jenny drew for the solution.

Jenny took the lead in formulating the solution (turn 3) and maintained it until turn 26 (see

below). Initially, she was inclusive of Katie and sought her confirmation (turn 5). Then, her stance seems to fit Coulter's (1999) definition of monologue. She interrupted Katie and didn't listen to her (turns 11-13); peremptorily rejected use of the 91 cm measurement (turn 15); and kept control of the solution ("We have to know…", turn 15; "I am going to draw…", turn 17).

Katie, for her part, agreed to Jenny having the lead and positioned herself as listener (turn 4). Interpretative listening to access Jenny's method was salient rather than evaluative listening to confirm her own thinking. When Katie moved to contribute, she was unsuccessful (turns 10 and 12), then, she secured Jenny's attention through personal appeal ("Look Jenny . . . ", turn 14). However, Katie did not challenge Jenny's rejection of her 91 cm suggestion. So, Katie's actions (the positioning and personal appeal) and failure to act (not pursuing her 91 cm suggestion) influenced the ways Jenny attended and listened to her. In fact, working with the 91 cm could have taken the solution forward, and Jenny's later identification of 91-x depended on recognizing it.

Hence, the ‘collaboration’ up to turn 17 was asymmetrical in that the students’ contributions were vastly different. Observations by Sfard (Sfard, Nesher, Streefland, Cobb & Mason, 1998) are relevant. "Being extremely demanding in terms of concentration and intellectual effort, mathematical problem solving can perhaps be best practiced in seclusion" (p. 42): 'effort can be focused on the problem without the distraction of communication, which is itself a strenuous activity' (ibid). Challenges in the task included drawing the segments to tilt realistically, which other students found difficult; and discerning the approach to calculation, where several options were possible. Perhaps giving priority to her own reasoning over listening to Katie was the only way Jenny could progress through a solution.

Another aspect of the interaction is Jenny negated Katie, by failing to listen to her and dismissing use of the 91 cm. Negation can impact on perceptions of one's own mathematical ability. That Katie quite often felt inept echoed through her conversation (see Forster, 2000a). Therefore, individual attempts at the challenging task, before discussing it, may have been preferable for both students.

Moderating the pace After Jenny had identified the measurements on the diagram (Figure 7c), the episode

proceeded with: 22. Katie Just wait. 23. Jenny Do you follow? 24. Katie Do you follow. Shut up, shut up, shut up. [pause]. Yes, you are absolutely right. It is 91 - x.

Katie’s “Just wait” disrupted Jenny’s progress. It drew Jenny out of her monologue. The

request to “wait” also explains the monologue and asymmetrical collaboration in that Katie

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might not have had time to formulate how 91 cm could be useful when Jenny rejected it, so therefore she did not defend it. Willams (2001) found that similar rates of understanding are a condition for successful collaboration.

Jenny's response "Do you follow" could be taken to indicate care for Katie and willingness to help. But any help was preempted by Katie’s “Do you follow. Shut up…”. A reason for the strong response is provided by Bateson (2002): "for all objects and experiences, there is a quantity that has optimum value. Above that quantity, the variable becomes toxic" (Bateson, 2002, p. 51). Here the variable was ‘help’. Alternatively, in saying “Shut up”, Katie might simply have wanted quietness as well as time to think.

The disruption resulted in Jenny being more inclusive of Katie, and Katie evidenced that she had the capacity to contribute, now that she had caught up:

25. Jenny So, what's this one [the height of the smaller triangle]? Surely we can work out this. 26. Katie Surely that would be 77 take y? 27. Jenny No. 28. Katie Yes. Yes. 29. Jenny Why? 30. Katie Because it is 77 take that amount [the height of the object, Figure 7b]. 31. Jenny Oh, yes. . . . Here, Katie held to her views when Jenny disagreed, and the mathematics advanced. By now though, the lesson was nearly at an end, which the students realised. The episode finished with Jenny speaking aloud her method as she wrote it, with occasional interjections by Katie and two calls to wait (“Wait, let me catch up’ and “Okay, let me catch up”). Jenny formed Pythagorean equations for each triangle, and substituted one into the other, ready for simultaneous solution. She and Katie both fed the equation into their graphics calculators and produced the same answer just as the bell rang. Discussion

Nine interactions over the ten lessons involved joint effort on questions that neither student had worked individually, so they fitted Linn and Burbules (1993) definition of collaborative endeavour. Conversation in four, including Example 3, started spontaneously and it was dialogic in nature: the students offered suggestions, accepted each other’s suggestions and advanced them. First and second turns appeared to be important for establishing interest (see Example 3). The action resembled mathematical conversation as described by Gordon Calvert (2001) and I assume it indicated hermeneutic listening. However, unlike the hermeneutic listening in Example 2 which was associated with securing help and repairing conversation, this hermeneutic listening in the context of collaboration was associated with spoken mathematical reasoning; and it was directed at calculation and syntax within large tasks.

The other five instances of collaboration, including Example 4, involved solving tasks the teacher set, for which he suggested joint work. Jenny took the lead in all of them, at the start or soon afterwards. She tended towards monologue in parts of each, with Example 4 being the extreme case. Reasons for monologue may have been that thinking through complexity required single-mindedness and relief from communication. A negative consequence was the repeated judgments and interruptions may have been detrimental to Katie’s perceptions of her mathematical competence. It was evident, though, that Katie influenced the action and sometimes sanctioned Jenny’s monologue through lack of assertiveness.

Looking at the students’ working relationship overall, the evidence indicates that they always initially attempted questions individually, except when the teacher suggested joint work. Individual attempts first meant that complexity could be addressed without any requirement to communicate, which may have the preferable approach for both students. Hence, in view of the

15

students’ ‘natural preference’, and their confusion in Example 4, if a teacher intends collaboration, individual attempts to sort out information, seem preferable to having students work complex tasks together from the start, which is also Brown and Renshaw’s (2000) recommendation.

Another issue evident in Example 4 was that the students thought at different rates, which resulted in three requests by Katie for Jenny to wait. Other calls to wait occurred on other occasions. Individual attempts first allow for students to work at different rates, albeit one may progress further in a solution than the other.

In bringing this discussion to a close, I will mention briefly the monitoring interactions, which were on task but contributed minimally to mathematical progress. They numbered 32 and involved between two and five turns. Katie initiated 22 of them. She: commented about homework and assessments, checked her progress, and made asides to Jenny during whole-class discussion. She asked questions before commencing solutions, part way through solutions, and checked solutions.

Jenny gave short responses, usually in agreement, and they signified evaluative listening—she evaluated from her own point of view. Sometimes a third turn followed in which Katie acknowledged Jenny's help (e.g., "Well, I am getting it Jen, I really am"). In Noddings (1984) and Schweickart’s (1996) terms, Katie cared for Jenny through such acknowledgements, and Jenny received what she needed most to continue answering Katie.

Once, Katie appealed to Jenny, "Jenny, I can't do these problems" and Jenny responded with her own demise, indicating sympathy and hermeneutic listening, as in Example 2. The interaction ended with Katie again telling Jenny to "Shut up". Sympathy apparently had reached a toxic level (Bateson, 2002).

Jenny initiated 10 monitoring interactions. She checked a solution with Katie ("What have you got?") and Katie replied ("Oh give me a chance"). Otherwise, she made asides to Katie during whole-class work, spoke about the tasks she was working on, or summonsed Katie to action. Katie responded with like ideas or took up Jenny's ideas, sometimes echoing her words, thereby indicating attunement to Jenny, and hermeneutic listening. Again, through her responses, Katie could be seen to care for Jenny and to sustain their relationship.

Conclusion

The rate at which Katie and Jenny initiated conversation with each other over the ten

lessons of my inquiry was high—an average of more than seven times each lesson. All instances of peer-tutoring, collaborative endeavour and monitoring have been mentioned above and the examples that I have provided show the students’ relationship certainly did not suffer from monotony! Neither did it suffer indifferent listening with careless interpretation, of which Sfard (2001) provides a classic example. Rather, the students’ responses to each other typically indicated listening with the mind, which I mapped as Davis's (1996) evaluative, interpretative and hermeneutic listening. Evaluative listening involved hearing and making judgments from personal viewpoints, interpretative listening was directed at accessing information, and hermeneutic listening was synergistic and involved relating to the other.

Peer-tutoring seemed to mainly involve evaluative listening by the tutor (who was usually Jenny) and interpretative listening by the one being tutored (Katie), and the complementary listening acts usually allowed the peer-tutoring to progress efficiently. Collaboration that the teacher instigated also evoked evaluative listening by Jenny (or minimal listening) and interpretative listening by Katie: Katie followed what Jenny said and their listening underpinned the ‘collaborative’ completion of tasks. However, in both the peering-tutoring and collaborative work, the listening stances sometimes resulted in confusion and inefficiency, particularly when Katie failed to hold to suggestions that she made. Progress was enhanced when Katie was

16

critical in her listening, voiced any disagreement, and persisted in presenting her views until she established their validity.

In the case of disagreement, evaluative listening was proposed for both students. They listened for confirmation of their own ideas, but satisfactory resolution depended on both being open to the other’s view. Evaluative listening by Jenny was also proposed in the monitoring interactions, in response to Katie’s frequent comments and questions.

Otherwise, hermeneutic listening seemed operative and was usually very temporary. It mediated expressions of demise, which secured peer tutoring for Katie, and it mediated affirmation, which repaired conversation. It was a means by which the students administered care for each other, including in the monitoring interactions. It arose in spontaneous collaboration, when it underpinned mathematical progress.

In conclusion, implications for teaching that flow out of the inquiry are as follows. First, students might benefit if encouraged towards critical listening and speaking up in a working relationship, including when being peer-tutored. As well, they could be encouraged to start with a question when approached for help (e.g., “What have you done so far?”), so they do not make assumptions about what help is needed. Encouragement of practices can include whole-class discussion on the issues and modelling of them by the teacher (see Bowers & Nickerson, 1998). Established patterns of interaction between students, such as those between Katie and Jenny, can be impervious to change, so students may benefit from working with different partners. In this way, they might experience leading and following and not be trapped in one role. Other considerations for teaching are that different rates of understanding can mediate against successful collaboration, and complex tasks might be best achieved through individual work before any formal collaboration. As well, spontaneous collaboration can be an enriching, magical experience. Students could be encouraged to share regularly their methods and thinking with their working partners so that an orientation to sharing and spontaneous collaboration develops.

Appendix 1

The diagram from which Figure 1a in the paper was extracted. The diagram included vectors for a question that followed the one described in the paper.

17

The diagram from which Figure 1b was extracted. Some angles were erased and were the ones disputed in the conversation. The angles shown match those decided on in the conversation.

Worked solution from which Figure 2 was extracted

18

Diagram relating to Figure 6

Diagram relating to Figures 7a and 7b

Diagram relating to Figure 7c

19

Detecto r

Det ector

ObjectA

B

Appendix 2

Locating an Object in No-(Wo)man's Land Graphics Calculator Activity

X-raying a tooth to determine if a filling is necessary, satellite spying on the enemy, and prospecting for underground minerals are some of the many instances where technologies use reflected radio or sound waves to get images of objects that are not immediately accessible. The following activity explains how these work. 1. Set up axes in a cleared space and place an object in

the quadrant, as shown, not close to the axes. 2. Place the detector at A, anywhere on the y axis,

and measure the distance to A from the origin. MONITOR the position of the object by turning the detector until a stable reading is obtained. Record the distance.

3. Repeat with the detector at a point B on the x axis. 4. Calculate the co-ordinates of the position of the object. While motion detectors use sonic (sound) waves for measuring distance, the Stealth Station Treatment Guidance Platform (see accompanying article), used for detecting tumors, uses electromagnetic waves travelling at 299 000 kilometers per second. Wave pulses emitted from detectors are reflected off a tumor, back to the detector. The time between emission and receiving the reflected pulse back is used to calculate distance. 5. Imagine that the diagram above represents the examination by a neurosurgeon of a

tumor located at the point marked 'object'. The detectors are placed quite close to the person's skin at A (0, 3) and B (1, 0), where measurements are in cm. For the detector at A, the time between emission and receiving back a wave pulse is 1.6019 " 10-10 seconds. For the detector at B, the time is B is 2.0940 " 10-10 seconds. Locate the position of the tumor.

In reality, most detection situations involve three-dimensional space and you need three detectors to locate position and a fourth one as a check for error. Using the GPS (Global Positioning System) requires signals from four satellites to reach a receiver in order to accurately locate position, anywhere on Earth. There are 24 satellites around the Earth that make up the GPS system.

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