Methodological implications of the interpretative paradigm and the grounded

While coding transcripts, I listened again to the recordings to note down the voice tone of the tutor’s and students’ utterances. This resulted into making changes to the transcripts, as well; usually altering utterances that I found to have been differently transcribed from what I heard while listening again. Lerman (2001) stressed that the process of formatting transcripts is “never-ending” (p.54), and comprises a layer of interpretation in data; for instance, other researchers may alter more utterances while listening again and so on. I considered my changes in transcripts, and thus my transcripts in general, to be a layer of my interpretation in data. Additional layers of interpretation in my research process were: the place of the recorder which sometimes clearer captured utterances from students close to it; my comments on the tutors’ actions in the right margin of my field notes for later discussion with the tutor; thus my field notes in general; and every account I produced after an episode in observations and associated discussions.

Holland, Lachicotte, Skinner and Cain (1998, cited in Jaworski, 2015) recognised figured worlds of interpretation, with each one to be:

a socially and culturally constructed realm of interpretation in which particular characters and actors are recognised, significance is assigned to certain acts, and particular outcomes are valued over others. (p.52, italics added)

My aforementioned considerations on layers of interpretation in my study belong in my world (i.e. the researcher’s world). Within that world, I observed SGTs while assigning significance to the tutor’s and the students’ actions. My comments on the right margin of field notes reveal that I valued some tutors’ actions over others. Yet the place of the recorder indicates that I valued those students’ actions which I captured in recordings and/or field notes.

There were also two additional worlds to mine in SGTs: that of the students, and that of the tutor. The students assigned significance to the mathematics, and valued the tutor’s teaching as well as the solution of tasks in SGTs. The tutor, in her/his effort to teach the mathematics, assigned significance to the mathematics, and valued some students’ contributions over others.

After her citation regarding figured worlds, Jaworski (2015a) stressed that “the rigour of the research lies in justifying interpretations and rooting conjectures” (p.178). My discussions with the tutors after SGTs, added to my analysis the tutors’ layer of interpretation with regard to students’ and their own actions in SGTs. That layer of interpretation, or else the tutor’s world, added rigour to my grounded analytical approach. For instance, I rooted my interpretations of each tutor’s tools and strategies in observational data. Then, I justified my interpretations with the use of data from my discussions with the tutors.

Lerman (2014) asserted that in research:

What we produce, in the end, is some form of text that reflects the researcher’s interpretation of how the data function as evidence of knowing and learning mathematics according to the theoretical position adopted. (p.16)

In this chapter, I referred to my choices about the methods I used in analysis; and the data I collected “in order to be able to call it evidence of knowing and learning” (Lerman, 2015, p.16). Furthermore, in the previous chapter (Chapter 2), I explained how I viewed the theoretical position I adopted, which is the Vygotskian approach. My consideration nevertheless is that my theoretical position is not limited to my methodological decisions and the sociocultural perspective. Rather, it also includes what teaching, meaning making and learning meant to me before participant selection, and before my grounded analytical approach to data.

I commenced my doctoral studies when I was an early career tutor with experience ranging from 1-1 teaching to teaching to 90 students. I had experiences of teaching, meaning making, and learning as a pupil and later as a student, as well as a theoretical understanding of those terms from my undergraduate and postgraduate studies. My experiences as a pupil/student indicated to me that I enjoyed teaching in which there was certain mathematical challenge. As a tutor, I felt rewarded when students were optimistic and smiled with their achievements. However, my theoretical underpinnings were those that formed my views for teaching, meaning making, and learning. In particular, having studied various theories of teaching, knowing and learning as a student, I thought of the Vygotskian theory as the most convincing. As a tutor and later as an observer, I valued opportunities provided to students to develop mathematical meaning socially in the classroom or lecture theatre. I also valued the teaching of informed tutors by mathematics education literature, and their effort to enable the students’ meaning by using that information in their everyday practice. I also admired the breadth and depth of the mathematical background of tutors who developed mathematics in their research. I started my observations without being judgmental to tutors’ teaching and without preconceptions of a better teaching than the observed one. Rather, my intention was to seek ways of teaching mathematics and the associated mathematical meaning making within my views of the Vygotskian perspective. I considered the different ways of teaching I observed and analysed as being informing for me as a tutor, as well.

In observations, I dressed neutrally in order not to be intrusive, and to avoid bias from the influence of the researcher. In discussions with the tutors after my observations, I avoided questions with mathematics education terminology, as well as leading

questions that could have implied expected answers. I was also supportive of the tutors, and developed a nice relationship with them.

Guthrie (2010) stressed that the data collection is more complete when the observer is a non-active participant,; this is because the observer’s attention can then focus only on the observation. So, the observer is less distracted by her own role, as it is restricted to a non-active participation in teaching/learning e.g. only taking field notes. My views on my role as an observer were congruent with Guthrie’s non-active participant; so, during SGTs, I was taking field notes while sitting in my chair. That was beneficial for my research, since it enabled me to capture as much of the tutorial context as I possibly could in my notes.

Additionally, in Schwandt’s interpretative terminology, there is the uninvolved observer (2000, p.194) who intends to be external, i.e. without inducing any change or disturbing during observations and at the same time, without being “literally at a distance or from behind some kind of one-way mirror” (Schwandt’s, 2000, p.207). My experience of observations in Pilot and Main study 1 unveiled that the uninvolved observer was not realistic. Rather, my role was most aligned to a reference from Maxwell:

As observers and interpreters of the world, we are inextricably part of it; we cannot step outside our own experience to obtain some observer-independent account of what we experience. (1992, p.283)

I could not be an uninvolved observer. I was inevitably part of SGTs. I was cautious not to disturb, and the tutors in the main study ensured me that their teaching did not seem to have any change due to my presence; so, the tutor or the students rarely talked to me during SGTs.

3.7 Trustworthiness of the study

Trustworthiness of a qualitative study regards evidence with which the reader trusts the findings and conclusions of the study. Miles and Huberman (1994, p.277) reviewed “26 tactics for drawing and verifying conclusions”, and produced standards for the quality of findings and conclusions of qualitative research by “pairing traditional terms with those proposed as more viable alternatives for assessing the trustworthiness and authenticity of naturalistic research [Guba & Lincoln, 1981; Lincoln, 1990; Lincoln & Guba, 1985]”. They produced five main overlapping criteria:

1. objectivity/confirmability of qualitative work [i.e. “relative neutrality and reasonable freedom from unacknowledged researcher biases−at the minimum, explicitness about the inevitable biases that exist” (p.278)];

2. reliability/dependability/auditability [i.e. “whether the process of the study is consistent, reasonably stable over time and across researchers and methods” (p.278)];

3. internal validity/credibility/authenticity [i.e. “truth value” of findings (p.278)]; 4. external validity/transferability/fittingness [i.e. “whether the conclusions of a

study have any larger import” (p.279)]; and

5. utilisation/application/action orientation [i.e. “what the study does for its participants, both researchers and researched” (p.280)].

The following sections form an account of the ways that my study met each of those issues.

3.7.1 Objectivity/confirmability

I established the objectivity/confirmability of my study by describing and explaining in detail my methods for data collection and analysis. In those descriptions and explanations, I presented the actual sequence of how data were collected and analysed towards the emerging analytical framework, ‘Teaching Knowledge-in-Practice’. I was explicit about my interpretations and theorisations, and I discussed in detail about layers of interpretation in my research process. Moreover, I reported on issues I considered to avoid bias, and ethical issues.

3.7.2 Reliability/dependability/auditability

In Chapter 2, I explained my views on the Vygotskian theory, and its connectedness with my research study. In Section 3.6 of this chapter, I described my role in the research process taking into account my methods for data collection, the interpretative paradigm of my research, and my grounded analytical approach. Furthermore, during data collection and data analysis, I worked closely with my supervisors, published articles, and delivered presentations at various national and international conferences. To conclude, the connectedness of theory with my research study, the description of my role in the research process, and the forms of peer and colleague review at conferences, supervisory meetings and publications contributed to the reliability/dependability/auditability of the study.

3.7.3 Internal validity/credibility/authenticity

I collected data of various tutors’ teachings in the tutorial setting over three academic semesters. I established the internal validity/credibility/authenticity of the study by keeping detailed records of my data collection, and by using triangulation. In particular, I applied to the study triangulation of sources (i.e. observational data and interview data with the tutors); and triangulation of methods (i.e. grounded analytical approach to the data in which research literature was embedded).

3.7.4 External validity/transferability/fittingness

Although replication of a research study for achieving the same outcomes is not applicable within the interpretative paradigm, Lincoln and Guba (1985) stressed that a detailed description of the study increases the possibility of transferability of the research findings and outcomes, since other researchers could judge whether a transfer is possible. To this end, I provided a thick description of the tutorial setting, and presented my criteria for participant selection in detail. Those criteria resulted into three cases of teaching with a variation of participants’ research expertise and communication with students, which was crucial for the findings about tutor’s knowing for teaching mathematics. Additionally, a cross-case analysis illuminated the findings about the categories for teaching strategies and tools, and gave rise to the emerging analytical framework.

External validity/transferability/fittingness also relates to the contribution of this research study. In this thesis, I made explicit that the contribution to mathematics education research literature is the analytical framework ‘Teaching Knowledge-in- Practice’. ‘Teaching Knowledge-in-Practice’ analyses university teaching practice and knowledge with regard to students’ mathematical meaning making, through a sociocultural perspective.

3.7.5 Utilisation/application/action orientation

Utilization/application/action orientation of this study is with regard to its participants and potential beneficiaries. I mentioned in Section 3.4.3 that some participants of the second pilot study discussed with enthusiasm with each other about their experience of participating in my study. In addition to this, some participants told me that they would like to read my analyses, and get informed by other tutors’ teaching. In a wider consideration, this thesis is accessible online to potential readers, such as tutors who teach mathematics at university level in various settings. It could also be of interest to researchers who analyse teaching knowledge and teaching practice with regard to students’ mathematical meaning making at university level.