The research undertaken was located in classroom settings. The complex milieu that is the classroom requires an approach to research that acknowledges that complexity; one that recognises that attempts to reduce this intricacy of relationships, and the multitude of underlying socio-cultural influences and discourses, to single constituent elements disregards the interdependence of these relationships and the multifaceted nature of human interactions. These relationships are as eclectic as the situations and environments from which they arise. Meanings that emerge from the inter-relationships between students, teacher, classroom phenomena, pedagogical media, and all the associated influences and underlying discourses, can be lost if situations are fragmented and reduced for the perceived purposes of some unattainable objectivity. The understanding of the inter-connected features, including the methodology of inquiry is impoverished by a reductionist approach (Kinchloe & Berry, 2004). The situating of learning within a social context is not the only influence that gestured towards the utilisation of an interpretive paradigm. Beck (1979) discussed the purpose of social science as being immersed in interpretive
perspectives. He contends that the purpose of social science is the comprehension of social reality through various perspectives, arguing that social sciences do not reveal ultimate truth, but negotiated human definitions of reality, and the examination of these evolving virtualities. They allow some sense making and enable clearer understanding of situations. They are concerned with explanation and clarification of the world that humanity has created around itself; a world that is multi-layered and socially constructed, with events and the people and circumstances that constitute them, uniquely situated in a particular time and context. To comprehend the reasons for particular individual interpretation, and the ensuing action it instigates, requires an insider’s viewpoint, one where the researcher is able to share, or at least understand, the individual’s experiences.
Questions concerned with understanding the process of learning, situated within classroom learning locations, and associated with mathematical understanding, evoke a qualitative methodology. As researchers have identified and investigated further aspects of the learning process as functions of the dynamic relationships and specific contexts in which the learning takes place, so their methods of research have changed. They needed to collect data that enabled them to more fully understand features while set in the appropriate context, rather than trying to be context free; that is by observations (e.g., Cohen, Manion & Morrison, 2000), case studies (e.g., Bassey, 1999), or interviews (e.g., Kvale, 1996). These approaches were hence considered significant for this study as I sought, within a classroom environment, to better understand the learning process as students engaged in mathematics investigations through the pedagogical medium of the spreadsheet.
As the dynamic relationships and situational contexts were acknowledged as significant aspects of the learning process, the requirement to collect data set within the learning environment was recognised. Naturalist approaches (e.g., ethnomethodology) are concerned with interpreting everyday phenomena. While manifest as both linguistic and situational interpretations, they are cognisant of the uniqueness pertaining to situations of occurrence, and a commitment to methodology. Burrell and Morgan (1979) have discussed the notion of typification of everyday experiences, as a way of making sense of social orders,
and synthesised phenomenology with this, through interpreting multiple realities, each constrained by its situation. Observation and description were significant in this process, and inherent in both is interpretation. As discussed in Chapter Three, hermeneutics, the study of interpretation, not only proffers a way to better understand a localised learning situation; it also allows the researcher to better understand the methodology of research. Central to this are the worlds of both the participants and the researcher. Just as the participants bring their historically situated pre-conceptions and discourses to each situation, so to does the researcher. There are multiple versions and interpretations of situations, and multiple perspectives from which these interpretations are evoked. As such, the space the researcher occupies in each version of their interpretation is as much a part of the data as the observations themselves (Brown, 2001; Mason, 2002). Much recent research in mathematics education has drawn on contemporary social science research and given greater emphasis to “the positionings, motivations, discursive formations and emotions of the researchers involved” (Brown, 2008).
As an illustration, consider a set of data collected when a group of students were using spreadsheets to investigate the patterns formed by the one hundred and one times table (see Figure 1). This was in the form of transcripts of their dialogue and the output they produced on the monitor: what they said and what they did. When it was initially analysed, it told a story of the students applying a visual lens to the number patterns that emerged for them. Underlying discourses led me to notice that aspect, to bring it to the foreground. When the same data was examined at a later stage of the study the hermeneutic circle was employed as alternative discourses were now privileged. The story that emerged most recently reflected the students’ understanding as it evolved through iterations of engaging with the tasks and the consequential repositioning of their perspectives. These modified perspectives in turn framed the subsequent re-engagement with the tasks. The data was still in the same original form, but the researcher’s viewpoint had been transformed, thus the interpretation and discussion were different. Another researcher might have privileged other perspectives.
Narrative frameworks, from which accounts are fabricated, temporarily fix these historically positioned interpretations. Interim ‘fixes’ of the phenomena that allowed the provisional interpretations at the local level are likewise providing temporary ‘fixes’ of the emerging methodology. Within this recognition of the evolving perspectives is the notion that there is no absolute truth waiting to be revealed by the appropriate methods, but rather an unveiling of fragmented perspectives that elucidate the researcher’s understanding. This echoes a post- structuralist position that perceives any meanings as partial, and the occupation of interpretation to “ keep the trembling and endless mirror play of signs and texts in play” (Caputo, cited in Gallagher, 1992, p. 278). Ways of knowing are discursive, and selves are multiple, fragmented, and constrained by their dominant discourses (MacLure, 2003). In a version of mathematical learning flavoured by this perspective, mathematics is a social construct premised on previous interpretative stances (Brown, 2001). From this perspective, an examination of the learners’ preconceptions, their interpretations (as manifest in their dialogue and actions) and how they subsequently re-engaged with the activities gave insights into the layering of meaning as their understanding evolved.
Situating meaning making as a process of consensus dependent on language, cultural conventions and metaphors, emphasises the social aspect of interpretation (Mason, 2002). Kinchloe and Berry (2004) have likewise maintained that meaning making “cannot be quarantined from where one stands or is placed in the web of social reality” (p. 82). They also advocated that interpretive research involved the connection of the subjects to their prevailing discourses, the acknowledgement of the researcher’s perspective and his/her relationship with the participants, and embedding the sense making in human experiences and interactions. Meanwhile, Brown (2001) saw the hermeneutic task as a revealing of meaning, “but an historically situated meaning dependent on the media and experiences through which it is observed” (p. 4). This indicates that the interpretation of dialogue and the associated negotiation of generalisations, are rich ingredients in the research process, particularly when accompanied by the articulation of corresponding researcher perspectives.
The emphasis on the researcher perspective, however, is precursory to the foremost critiques of the interpretive methodologies. If the notions of behaviour and interpretation are constrained only by the participants’ viewpoints, there is the danger of partiality or incompleteness, through the limitations of the construction of those viewpoints (Bernstein, 1974; Giddens, 1976, Layder, 1994). However, while the recording of phenomena is significant in the developing portrait of interpretation, the insights of participants are crucial. Other concerns with applying an interpretive lens to research involve the subjectivity of the researcher. Key elements of the research process, observation, description and analysis require selection. The researcher’s underlying preconceptions and intentions may influence those selections and if the researcher’s perspective is part of the interpretation, part of the data and analysis, objectivity is hard to reconcile (Mason, 2002). Mason also wondered, given that data is a construction by the researcher, whether they would compose the identical record in the same situation again. Likewise, he identified potential for the mingling of the data which emerges from the analysis, and the original phenomenon; “the complex interplay between story and experience” (Mason, 2002, p. 228). Language is the vehicle of the data, description, and analysis, and as such these are inherently subjective. Language is connotative by design and the interpretations it permits become constitutive of ongoing meanings. Understanding arises from consensus borne of the engagement and interpretation of phenomena, with each interpretation influenced by, and influential, in the ongoing process. Interpretation provokes possible explanations, but there is always a productive gap between interpretation and explanation that provides the space for understanding to emerge. This space allows the play between the familiar and the unfamiliar from which interpretation evokes new thinking. The principle of distanciation emphasises that all interpretation is transformative to some degree, but never in an absolute way (Gallagher, 1992). As Brown (2001) has discussed in a separate, but associated, context, “…understanding evolves continuously but is represented through tangible product, capturing the moment, such as pieces of writing, calculations, diagrams and so forth” (p. 98). It is the reconciliation of these snapshots of tangible phenomena through discourse that enhance that emerging, yet dynamic understanding. The educational researcher must likewise be concerned with the tangible and the interpretive.
Ethnographic research is concerned less with predictive generalisations, than with the formation of generalised descriptions, the interpretation of events. The researcher’s perspective is not the sole contributor. As LeCompte and Preissle (1993) contend, “...meanings are accorded to phenomena by both the researcher and the participants; the process of research, therefore is hermeneutic, uncovering meanings” (pp. 31-2). This does not mean a purely subjective, record of events fashioned through the personal filters of the participant or researcher. Methodologies have emerged that help alleviate validity and consistency concerns: models with commonalities of design (LeCompte & Preissle, 1993; Lincoln & Guba, 1985). Discourse analysis, recordings, notes made in situ, observations and interviews all have interpretive elements that give crucial insights, and if consistent and collaborative, layer the sedimentation of a valid history of events. Some researchers (e.g., Kinchloe & Berry, 2004) advocate the use of bricolage as an educational research methodology. This approach, premised on a critical hermeneutics perspective, contends that research methods should by necessity be eclectic if they are to examine the complexity of educational processes in meaningful ways. Bricoleurs actively assemble their research methods from the available, deemed-appropriate strategies that are afforded by broad social science research paradigms, including practical, theoretical and interpretive approaches. A range of methods was engaged in this study to elicit better understanding of the complexity of learning and the ways students came to their understandings, and a hermeneutic circle was enacted through the practice of research. However, for me as researcher, a moderate hermeneutic perspective emerged as being most productive, one that privileged the transformative view of education rather than the emancipatory one ascribed by critical hermeneutics and the bricolage.
There was also the need to gain understandings of the learning occurring at an individual level and the possible reasons for this, that is, the understanding of actions or implications rather than causes. This too indicated the need for elements of an interpretative paradigm. To gain insights into, and an understanding of, the learning that might occur for individuals, observations in the learning environment and interviews with participants were used to provide
important information. Clandinin and Connelly (2000) outlined an historical perspective of narrative inquiry, and demonstrated the process, by recounting what narrative inquirers do. They contend that narrative enabled the researcher to investigate experience in a way that situates change or the learning within the context it occurs, or the narrative it is derived from. Like Geertz (1995), they appeared to see understanding evolving concurrently, but not necessarily in parallel, over a range of perspectives both phenomenal and attitudinal, as change inevitably occurs and has effects. Their thesis is that education is a form of experience, and narrative is the way of representing and comprehending experience (Clandinin & Connelly, 2000). In educational research settings, narrative inquiry is usually associated with the stories of reflective practitioners (Schon, 1983), or action research where practice, closely meshed with theory, induces deliberate aspects of actuating teacher change (e.g., Somekh, 2001). The research undertaken in this study contained elements of a narrative inquiry approach in the stories told, both individually and in groups, of the participants’ perceptions of the learning process.
The researcher’s viewpoint is implicitly situated with the interpretations of the data. However, the lens that is our present state is not constant. While imbued with the social, political, and cultural influences that shape its perspectives, it also shifts in its construction over time, and with varying audiences. Geertz (1995), within an anthropology context, maintained that it is not only the phenomenon that changes over time; the onlooker’s viewpoint changes too. He identified the setting in which the phenomena occurred, its intellectual and moral justification, and the nature of the discipline the onlooker is viewing from, as also shifting. Sanger (1994) added a further view:
For the post-modernist language philosopher, data are arbitrary and are therefore vulnerable to a wide variety of analytical operations. The authorship of the data, in the form of the actor’s statements, may be denied and the entire process of data gathering, together with the data, seen to be a composite artefact regulated by arbitrary historical currents (p. 178).
While varying influences might pervade the data, methodology, and interpretative analysis, the researcher, while recognising this, can nevertheless only investigate that which is offered through his/her current lens. The sorts of spaces the researcher occupies, and the extent to which these may be delusory or illuminating depending on the story they are telling and to whom, are precursory to the data itself. The data is seen through varying sets of eyes. It is important to understand how those eyes see, and how they produced the objects they described. The researcher’s personal narrative was a vehicle for revealing those fragmented perspectives and for giving insights into how the analysis was a function of those personal viewpoints, on any particular occasion. The personal narrative and the transformative process the researcher inevitably experiences are more than illuminating; they are part of the data itself and fundamental to the methodology. An ongoing diary of reflections, the documentation of interactions with supervisors, along with the writing and presentation of papers all constructed and promulgated this personal narrative.
Mathematics education research is an ongoing, evolving process with each individual engagement in research extending its boundaries. At the individual level the researcher undergoes a transformative process (e.g., Mason, 2002; Schostak, 2002) as they initially envisage their study from preconceptions drawn from their prevailing discourses. Their interaction with the literature, data, and colleagues, with its associated reflection, modify the researcher’s perspective. The space from which they perceived the research shifts, and they re-engage from a modified position. This process is hinged to the evolution of mathematics at both the individual level (for researcher and participants) and the broader, more general understandings. The individual research process is informed by preconceptions borne of those mathematical discourses as well as the discourses in associated areas e.g., social science research. Likewise, the ongoing formation and transformation of mathematics education research is influenced by the cultural formation of mathematics as it adjusts through interpretation at the individual level, and the transformative process of the individual research trajectory. They are mutually influential of each other, and in both the individual and broader forms research methodology evolves through cycles of interpretation, as attention oscillates between engagements through the gaze of
underlying perspectives, to modification of those perspectives through that engagement.
For instance, the students in this study engaged with the tasks from the perspective of, and through their preconceptions in, the associated areas. Seeing the output of their mathematising in the visual, tabular form of the spreadsheet modified those preconceptions as they made interpretations of their interaction. In the following brief excerpt, two pupils were investigating the 101 X activity (see Figure 1).
They had produced the following output:
1 101 2 202 3 303 4 404
Tim So it’s the number, then a zero, and then the number again
Carl Yeah, yeah. 5 will be 505, 55 would be 55055. Drag down. ... … 13 1313 14 1414 15 1515 16 1616 17 1717 … …
Carl What? It’s just repeating.
Tim Like doubles, so 18 would be eighteen, eighteen and 55 would be fifty-five, fifty-five.
They continue refining their generalisation through the modification of their perceptions as they interpret the outcome of their engagement and adjust their perspective. Their generalisations are based on the number and positioning of the digits. They have used a form of visual reasoning to generalise the pattern (Presmeg, 1986). They then re-engaged with the activity from a fresh perspective
with the interpretation and understanding evolving in this ongoing manner. The broader discourse of mathematics (in this case visual reasoning) was likewise transformed (albeit slightly) by this engagement. The boundaries of mathematics per se were extended, or existing positions enriched, by that engagement. Other pupils commented in the interviews on the way the spreadsheet environment assisted their interpretation, e.g.,
Chris Columns make it easier – they separated the numbers and stopped you getting muddled. It keeps it in order, helps with ordering and patterns.
This cultural formation of mathematics evolved as the mathematics phenomenon was engaged with the subsequent interpretations influencing the way mathematics was perceived.
The individual engagements of the students were also influential on my researcher perspectives and interpretations of the data, and the research methods that were employed. The analysis of the initial data revealed this emerging story around the affordance of the spreadsheet environment to structure the output visually. This analysis of the data, in conjunction with other constitutive influences e.g., the research literature, modified my approach to a more interpretive perspective. I looked to research methods that would give alternative insights into these visual interpretations as the pupils’ attention shifted alternately from preconception to interaction. Viewing the data through this lens gave further insights into the investigation of the research questions, in particular, the ways understanding emerged for the pupils, and the ways the pedagogical medium of the spreadsheet influenced their understanding. Mathematics education research was modified simultaneously as I engaged in research practice drawn from my existing prevailing discourses in mathematics education research, engaged in the research process, and then modified my perceptions of mathematics education research. The individual transformational research trajectory resonates and modifies mathematics education research per se. In this case, the collegial dialogue, writing papers and presenting at conferences, and writing articles for journals, that indicated this visual, tabular structure and its influence on the
research process I employed to productively interpret the situations, has extended to some small extent the boundaries of mathematics education research.
In rejoinder, the mathematising at an individual level, the cultural formation of