At the completion of the data collection and transcription process I began the task of analysing the video data within its entirety. The analytic approach taken was “sit, look, think, look again” (Pirie, 1996, p. 556). Video of the lessons was viewed and reviewed, transcripts of the lessons were read and reread, and both were interpreted and categorised for evidence of teacher positionings. Categories of teacher positionings had not been proposed or predetermined prior to observing the video as I did not want to “blind” myself or “make it difficult to notice unanticipated” positionings (Powell et al., 2003, p. 423). Merriam (1998) suggests I was “having a conversation with the data, asking questions of it and making comments to it” (p. 181). References to the field, personal, and theoretical notes I made added to the comments and questions I was able to make.
Through a process of open-coding the units of data, I identified potential themes from real examples within the transcripts. Open-coding is the process of breaking down, examining, comparing, conceptualising, and categorising participants’ words to identify and develop themes and interpret data. Researchers are guided in the codes that they develop by their theoretical framework, their research questions, and the nexus of what they observe (Powell et al., 2003). Coding was completed in this research by attributing meaning to events within the video, identifying key concepts in the video, and labelling those that were most significant.
To increase the manageability of the volume of data available to me (72 lesson transcripts), a process of data reduction was required. “Data reduction is a form of analysis that sharpens, sorts, focuses, discards, and organises data” (Miles & Huberman, 1994, p. 11) and is therefore a necessary part of the analysis process. Before I began the analysis of teacher positionings I realised I needed to frame periods of positioning rather than each individual act of positioning as it occurred. A different act of positioning could occur with each new dialogue, so I drew on the construct of a Social Episode (Harré, 1998b), described in Chapter Four, to frame the positionings as an event. Each social episode began with an act of teacher positioning that resulted in mathematical know-how being shared and was concluded when the discussion reached a natural conclusion. This process
provided me with indicators of which chunks of dialogue to code and why, and for identifying data not relevant to this study which could be disregarded.
Data were analysed in the same order they were gathered. Pacific School first followed by Tasman School and teachers of the oldest students first, through to the youngest. In each phase of analysis I started with the transcripts of Paula, Pacific School, Years 5 and 6, lowest then highest group. The examples from Paula’s transcripts formed the basis for the initial set of codes. This set of codes was altered and added to as I worked through all the data. I then completed each phase by analysing the videos and transcripts of the six teachers at Pacific School and all Tasman School teachers. The four phases of data analysis are described below. Throughout these phases I referred to the participant observations, documents and archival records for corroborating and contradicting evidence that would reiterate, enhance, or challenge my initial findings.
For the first phase of analysis I read the lesson transcripts and looked for examples of teachers and students sharing their mathematical know-how. Examples included one student explaining to another why they were incorrect, a teacher modelling multiplicative thinking with an array showing 3 groups of 5, and a student using their written recording to defend their strategy. I identified the teacher positioning that preceded the mathematical sharing and coded the positioning as teacher talk, text, or actions. Teacher talk, text, and actions were selected as codes because discourse in positioning theory includes talk, text, and actions of any kind (Davies & Harré, 1990). Table 5.3 shows the six codes used for describing the teacher positioning as reflexive (self) or interactive (other) and whether the positioning occurred through teacher talk (T), text (Tx), or actions (A).
Examples of the phase one reflexive positioning codes include the teacher revoicing a student’s explanation (RPT), recording a student’s explanation (RPTx), and providing a model for students to discuss (RPA). Phase one interactive positioning examples include the teacher asking students to discuss their mathematical disagreements (IPT), record their own or a peer’s thinking (IPTx), and provide a model of their own or a peer’s thinking (IPA).
Table 5.3: Phase one codes
Teacher Talk Text Actions Reflexive Positioning RPT RPTx RPA Interactive Positioning IPT IPTx IPA
I reviewed the transcripts to determine if any episodes of shared know-how had been missed and as I did this I coded all episodes according to the teacher positioning and their words, texts, or actions. The teacher positionings were coded and constantly compared in order to define and refine their properties (Willis, 2006). Relationships between the codes were noted, trialled, and grouped as themes. As the themes were further considered I looked for conflicts, such as cases that negated initial concepts, negative instances, or erroneous conjectures and considered these cases as theory was created (Powell et al., 2003). Relationships amongst the codes were closely examined and codes that appeared to go together were grouped; from these groupings tentative concepts were developed. The concepts then started to build categories through which theory was being created.
In the second phase of analysis I identified the mathematical contexts in which the teachers’ positionings occurred. The individual contexts identified were grouped as: strategy errors, misconceptions, and self-corrections; difference, efficiency, and sophistication of mathematical explanations; connections and relationships; and patterns. These contexts provided me with insight into the types of triggers that prompted teachers’ positionings that led to mathematical know-how being shared.
For the third phase I placed the positioning codes and mathematical contexts onto a matrix. From here, I plotted the teachers’ positioning acts according to the codes and contexts. This process enabled me to determine each teacher’s pattern of positioning with their lowest and highest strategy group. P indicates Pacific School and T Tasman. The initial letter of the teacher’s name is the second letter of the code. The third letter L or H refers to the lowest or highest strategy stage group.
Table 5.4: Examples of teacher placement on the positioning codes and contexts matrix
At this point in the analysis it became apparent that not all teachers’ positionings resulted in mathematical know-how being shared because not all teachers and their groups were represented on the matrix. For example, there was little or no evidence of Naomi at Pacific School positioning her lowest group or Lisa from Pacific School positioning students in her highest group to share their know-how. Some teacher talk, text, and actions appeared to constrain the sharing of know- how, so I repeated phases one, two, and three of the analysis, this time looking for examples of teacher positioning that constrained mathematical know-how. I did not uncover any different positions or mathematical contexts but student behaviour was added as a non-mathematical context because for one teacher group behavioural expectations strongly influenced when know-how could be shared. Throughout phases one, two, and three field, personal, methodological, and theoretical notes were compared and contrasted, and confirmed or refuted with my on-going observations and analysis.
At phase four of the analysis I established themes to describe the positioning pattern of each teacher with their lowest and highest strategy group. These categories were derived from the codes and contexts determined in phases one and two. Category examples are provided in Table 5.5.
RPT RPTx RPA IPT IPTx IPA Errors Misconceptions Self-corrections PLL PLL Difference PLL PLL Efficiency THL PNH PNH PGL THL THH PNH THL Sophistication THL THL PLL THL Connections PGL PGH PGL PGH PGL PGL PGL Relationships Patterns THH THH
Table 5.5: Examples of themes of affording and constraining teacher positioning
Affording Teacher Positions Constraining Teacher Positions Teacher provides a model/written recording
for students to discuss
Teacher asks students to watch/listen – no discussion
Teacher requires students to solve mathematical disagreement
Teacher explains why an answer is correct/incorrect
Teacher seeks student explanation and justification
Teacher explains for students and instead of students
Teacher appropriates student difference/efficiency/sophistication explanation for discussion
Teacher limits opportunities for creative solution methods
Teacher highlights mathematical connections/relationships
Teacher protects student from mathematical/cognitive conflict Teacher expects students to help each other
correct/self-correct errors/misconceptions
Teacher praises correct answers Teacher allows students space to
experience mathematical conflict
Teacher seeks correct answers only
In the final phase of the analysis I revisited each transcript and noted evidence of each affording and constraining positioning theme. Evidence of teacher talk, text, and actions within each theme was closely examined and tested for robustness. At this point I was able to determine the patterns of positioning of each teacher with their lowest and highest strategy stage groups. The result of this grouping was that I identified seven teachers who positioned both groups in the same way and five teachers who positioned students in the lowest and highest groups differently.