Whaia e koe te iti kahurangi Ki te tuohu koe me maunga teitei
Seek that which is precious, If you are to bow down Let it be to a lofty mountain
Introduction
The previous chapter examined the theoretical perspectives, and associated literature, that underpins the methodology of the thesis. It provided a framework in which to situate the key elements of the research. The interpretive stance that emerged evolved through the research process itself and as such became constitutive in the data production, while also providing a lens through which the data was considered. This chapter contemplates these two aspects of the methodology. Firstly, the transformative process of the research and how the revisioning of the researcher’s approach to the analysis and associated ways of knowing, led to varying perspectives. The historical marking of the results and discussion evidenced these evolving perspectives as they emerged through a variety of analytical lenses. This illustrated a hermeneutic process, with cyclical engagements involving both the theoretical literature and interpretation of the data modifying the dominant research discourses, with iterations of the hermeneutic circle. The evolving theoretical framework emerged as these modified discourses were subsequently used to re-engage with literature and data. It is important that these perspectives were historically indexed as they evolved, as they articulated the cultural, philosophical and ideological basis for the perspective the researcher held at each particular juncture (Zevenbergen & Begg, 1999). The theoretical framework is a dynamic, formative notion that shapes the research and is shaped by the research. Guided by the literature and theoretical
viewpoints, the research framework was refined within the unique practical context.
As rehearsed in the previous chapter, mathematics is not a fixed reality beyond the scope of human influence. It is better envisaged as a socio-cultural way of thinking. It is a shared ongoing interpretation of situations and perspectives, some of which have been embedded in our traditional beliefs (e.g., five plus one is six) that we treat them as reality. They have become ‘truths’ by the repeated communal consensus of interpretation. Mathematics is an evolving set of perceptions, seeming to become more complex on its peripheries, yet more refined in its core identities, with each iteration of interaction, reflection and interpretation by its users. The elements of mathematics that are engaged transform the perceptions of the person interacting with the mathematics, but likewise those elements are transformed by their engagement with mathematicians, learners, or researchers, even if only by a minuscule amount. The boundaries of mathematics are expanding or becoming more refined through that interaction. The socio-cultural formation of mathematics can also be envisaged as a hermeneutic process, one where iterations of engagement, reflection, interpretation, then re-engagement from modified perspectives fashion those emerging theories.
In this study, for example, the affordance of the spreadsheet environment to more easily manage large amounts of data, opened up opportunities for the students to explore the activities in alternative ways to the approaches they might have employed in a typical classroom setting (i.e. one with students working at tables or groups of desks, using pen-and-paper technology, with equipment available). The Year 6 pupils, for instance, were able to generate and manipulate large amounts of numerical output within their spreadsheet models of the situations that would not be practical in the classroom setting. They could investigate and interpret the mathematical phenomena in an alternative manner hence the boundaries of their mathematical investigating, and by implication their understanding, were extended. As a consequence, the verge of what constitutes school mathematics, and mathematics itself were also extended. Each iteration of interpretation evoked by this alternative filter was simultaneously iteration in the
cultural formation of mathematics. Mathematics had become a slightly modified version of its previous self. The perceptions of mathematics associated with the way we express ourselves and make sense of our world (Radford et al., 2007) had changed. The spreadsheet environment’s facility to manage large amounts of data quickly and accurately, also allowed the students access to different types of situations and problems, and to investigate mathematics in more realistic contexts (e.g., Ridgway et al., 2006). In a similar way, this had also transformed the nature of school mathematics. The participants’ perception of what mathematics is and therefore general perceptions of mathematics have been altered. The engagement, reflection, and transformation of perspectives at an individual level resonate (no matter how slightly) in the general perspective.
This hermeneutic process echoes the viewpoint of learning in mathematics education articulated in Chapter Three. Implicit to this perspective is the inextricable linking of mathematics, learning in mathematics, and the research of mathematics learning. They are mutually formative practices, and evolve in an interactive manner. Viewed from this perspective, the reshaping of mathematics through alternative filters, the reorganisation of mathematical understanding through engaging mathematics phenomena with digital pedagogical media, and the transformative research process the researcher undergoes, also have a symbiotic relationship. As such, they were each constitutive of the methodology that could be productively employed in the investigation of the research questions. The following excerpt gives insights into that relationship.
A group of pre-service teachers was exploring the 101 X task with the spreadsheet available. They read the explanation of the task before beginning the investigation process:
Kyle I haven’t predicted. I was just going to put in A1 times 101 and drag down (does it).
The following output was produced: A B
1 101 2 202 3 303
4 404 … … 14 1414 15 1515 16 1616
The pedagogical filter through which it was engaged shaped the group’s initial interaction with the task. They have immediately used the functionality and affordances of the spreadsheet environment to explore the situation. Their approach, as demonstrated by the dialogue, was not to try an individual example as might be expected with a paper-and-pencil medium, but to form a symbolic model of the situation designed, with the spreadsheet’s functionality in mind, to create a visual model - a table of related, consecutive, numerical values. The spreadsheet medium has led them to investigate in an alternative manner, expanding the potential strategies for mathematical investigation and the scope of mathematics. Their interpretations and understandings were different, and articulated in visual terms, e.g., the type and position of the digits:
Kyle [referring to 44440, the output from 44]. Its like double the number, but with zero added on.
The ‘double the number’ comment refers to a repeating of the digits rather than doubling as a process, again accentuating the visual element to their interpretation. As well as the spreadsheet environment expanding the potential to mathematise and the types of understandings that might emerge, the pre-service teachers’ investigative processes e.g., ‘drag down’ indicated the need for alternative research approaches. For this study, the approach to collecting data required the collection of synchronous data relating what they said (the taped dialogue) with what they did (the printouts of their output). Methods for data collection will also need to evolve as the nature of mathematics, and the ways it is understood, evolve. The desire by researchers to develop ways to more accurately collect synchronous data generated in digital environments, so as to gain more insightful interpretations of the learning processes that emerge, is symptomatic of the connectivity between the evolution of mathematics, learning in mathematics, and research in mathematics education.
The second part of the methodology chapter addresses the methods by which data were collected and analysed at various points in time. It includes the initial approach taken to obtain the desired data, then the refining of the data analysis according to the modified rationales (Schostak, 2002). It is the story of how it was intended to produce the understandings and knowledge through the approaches taken. These approaches by necessity are inextricably linked to the emerging theoretical frame, as both are constituents of the hermeneutic circle that is the research process, in the version of research privileged in this study. Meanwhile, a local hermeneutic circle also occurred, with the interpretation of the data as the students engaged in the process of evolving their mathematical understanding. This chapter therefore, is the crafting of the rationale for the approach taken, and an introduction of the resources used to elicit the understandings that emerged.