1 Introduction

The intention of this paper is to illustrate the legacy David Clarke (Clarke) has left through his Complementary Accounts Methodology (Clarke, 1998), as employed in the Learner’s Perspective Study (e.g. Clarke et al., 2006). By illustrating the influences of this methodology on my (Williams, 2005) doctoral research, insights are gained into the potential for Clarke’s methodology to inform mathematics education research into the future, including theoretical perspectives additional to social learning theories.

The quality of Williams’ (2005) doctoral thesis undertaken within the Learner’s Perspective Study is demonstrated through the awards it attracted and the comments accompanying these awards, about the multi-theoretical framework conceptualised and innovative research design. These awards (2006 Australian Association of Researchers in Education Doctoral Award and 2007 University of Melbourne Chancellor’s Prize for Social Sciences), along with the theoretical framework employed suggest this research as ideal to illustrate the potential of Clarke’s methodology to support research foci beyond those for which the methodology was intended.

The Learner’s Perspective Study is one of the many collaborative international research projects Clarke has led. The longevity of the project, the increased number of participating countries over time, the number and diversity of findings about complex classroom interactions, and the publications communicating these findings (e.g. Clarke et al., 2006, 2010; Kaur et al., 2013) provide significant contributions to research internationally.

Definitions of the terms, ‘theory’, ‘methodology’, ‘research design’, and ‘data collection instruments’, used throughout this paper are drawn predominantly from Silverman (2003).

‘Theory’ is ‘a set of concepts used to define and/or explain some phenomenon’ (p. 3).

‘Methodology’ is ‘a general approach to studying research topics (p. 3)’. It includes choices we make about subjects/cases we study, methods of data collection and forms of analysis justified in terms of the theoretical framework employed.

‘Data collection instruments’ (Silverman’s ‘Methods’) are ‘specific research techniques … like observation, interviewing and audio recording (p. 4).’

‘Research design’ is an additional term added for its relevance to this paper. It is a subset of Silverman’s methodology: choices we make about subjects/cases we study, methods of data collection.

2 Historically situating Clarke’s research

Clarke’s Classroom Learning Project research design (Clarke, 2001b) and The Learner’s Perspective Study research design (Clarke et al., 2006) required data to study negotiative processes during learning, and interactions between students and the teacher and whether these interactions influence each other. Such studies require access to data to study interactions in class and, both the teacher’s and the students’ voices reflecting on this activity. As will be shown, Clarke’s Complementary Accounts Methodology (1998) met these requirements. It can be viewed as an innovative culmination of strategies to overcome limitations to previous research methodologies relevant to study of complex classroom activity. Limitations of previous methodologies, and how they informed later methodologies, including Clarke’s research methodology, are discussed below.

Research methodologies that studied activity in class in the mid twentieth century involved observation by researchers who coded a pre-determined set of behaviours for statistical analyses (e.g. Bourke, 1985) because behaviourist theory (where each stimulus was expected to always give the same response) was the predominant paradigm. When researchers became aware that students had ‘knowing minds’ that informed their responses (Piaget, 1980), cognitive constructivist theories (e.g. Piaget, 1980) began to replace behaviourist theories (Watson, 1919). Cognitive constructivist methodologies employed data collection instruments designed to study student decision making and student learning (e.g. Thornton, 1999). Student interviews that collected valid data on student decision making and meaning making through connections to specific classroom activity were crucial to Clarke’s methodology.

With the student and teacher interviews Clarke employed in his studies to gain multiple perspectives on classroom activity, he took steps to overcome previously identified limitations including the following. In interviews, students may decide not to share everything, may not remember everything, and/or may provide general responses not responses specific to student activity under study (Ericsson & Simons, 1980). Although ‘talk-out-loud’ interviews (Krutetskii, 1968/1976) in which the student described their thinking during problem-solving tasks overcame some limitations, it introduced others. These interviews could interrupt the problem-solving process or result in curtailed responses, if the student wanted to concentrate fully on their problem solving. In addition, these interviews could not be undertaken in class without disturbing the activity Clarke wanted to study. The absence of opportunities for researchers to validate interview reports had also been of concern to researchers over time (deCharms, 1976).

Video studies explored children’s development of new mathematical knowledge during interactions in class (Cobb et al., 1992) and in small groups (Wood & Yackel, 1990). An advantage of video was that it could be revisited multiple times during the analysis of classroom activity (e.g. Cobb et al., 1992). International classroom video studies began to emerge before the end of the twentieth Century. The TIMSS video study (Stigler & Hiebert, 1999) explored how mathematics was taught in Year 8 classes in the USA, Germany, and Japan. Selection of schools and teachers was intended to form a representative sample of teachers in each country. Over 230 classes were videotaped in total. A single camera focused on each teacher for a lesson. The students were not a focus of attention. The one-camera research design employed by Stigler and Hiebert (1999) was not sufficient to capture the complexities of mathematics classroom learning that Clarke wanted to explore. Clarke’s studies required not only the inclusion of the student and teacher voice in classroom videos, but also the interpretations of classroom activity by these participants. Complementary Accounts Methodology harnessed technological advances to develop new ways to undertake classroom research, new ways to analyse classroom activity. By using mixed image student and teacher lesson video as stimuli for student and teacher interviews, many of the previously perceived limitations to interview as a data collection tool were addressed (Nisbett & Wilson, 1977). Lesson video stimuli in student (and teacher) interviews focused the student (or teacher) on specific activity, aided recall, and could sometimes enable the researcher to probe activity the student (or teacher) had deliberately decided not to discuss. As perspectives from each data source were equally valued with Complementary Accounts Methodology (Clarke, 2001a), inconsistencies between data accounts were seen by Clarke as reasons to explore these inconsistencies, rather than to search for ‘what was right’. The meaning of the term ‘accounts’ as employed in this paper is elaborated in Sect. 3.

3 Classroom Learning Project

The Classroom Learning Project included the voices of lesson participants (students, teacher) and the voices of a selection of international researchers who analysed these classroom interactions, each employing a theoretical framework of their own selection. The phenomenon under study was ‘negotiative processes in mathematics and science classrooms’ (Clarke, 2001a, p. 19). The intention was ‘[t]o capture complexity in classroom situations, [so] the data included “multiple potential meanings”’ (Clarke, 2001b, p. 1).

The project illustrates the three key features of Clarke’s Complementary Accounts Methodology (1998), to:

  1. 1.

    Generate ‘“integrated data sets” combining videotape and interview data …’

  2. 2.

    Include ‘the reflective voice of participant students and teacher in the interviews …’, and

  3. 3.

    Undertake an ‘analytical approach that utilises a research team with complementary but diverse areas of expertise to carry out a multi-faceted analysis of a common body of classroom data’ (Clarke, 2001b, pp. 1, 2).

The term ‘accounts’ was employed at two different levels within Clarke’s research. Predominantly as accounts of the research team (Item 3 above) but also as accounts from different data sources. Since the interpretation of any practice will vary according to how you are positioned by it or in relation to it, different data sources will inevitably prompt the construction of ‘complementary accounts’ (Clarke, 2001b).

The sources of the data accounts included the participant voices (Item 2 above), lesson videos, mixed-image videos, field notes, and student work.

For the Classroom Learning Project, data were collected in ‘usual’ mathematics and science lessons, in a private co-educational school in outer Melbourne, 55 lessons over a 3-year period. Two video cameras captured lesson activity, one focused on the teacher and one on two to four students. A mixed video image was generated during the lesson (with students at centre screen and the teacher as an insert in the corner). This mixed image enabled simultaneous analysis of the actions of the student pair and the teacher. A researcher took field notes during the lesson to inform the post-lesson interview. Post-lesson student interviews stimulated by mixed image lesson video increased the validity of lesson reconstruction by providing memory traces of the event (Ericsson & Simons, 1980). This video stimulation also had the potential to add richness to the student responses by aiding student recall (Nisbett & Wilson, 1977). In addition, questions were asked about student thoughts during the lesson, parts of the lesson they considered important, their thinking and feeling during the lesson, their new learning, and the context in which that new learning occurred. If time, the student was also invited to discuss the parts of the lesson drawn attention to in researcher’s field notes. The teacher undertook a video-stimulated interview after the students were no longer in that teacher’s class (ethics condition). In this interview, the teacher played the lesson video, and gave a moment-by-moment account of what was happening, to provide a different perspective on the lesson.

The ‘integrated data sets’ developed (synchronised video of student pairs or groups and video of the teacher in class, interviews with students and the teacher related to different parts of the lesson, and student worksheets produced during the lesson) were sent to the international research team for analysis. The criteria employed for selection of these ‘integrated data sets’ were frequent lesson interactions, and lessons that would support analyses of negotiative processes from a variety of social theoretical perspectives of the nature of learning. Accounts from students and the teacher, and interpretation of the video data, informed researchers’ accounts of the negotiative processes. Clarke (2001c) synthesised the researchers’ complementary accounts to gain a rich multi-perspective portrayal of the phenomenon, processes of negotiation of meanings in science and mathematics classrooms, as opposed to an absolute truth about the nature of such negotiative processes.

4 The Learner’s Perspective Study

Originators of the Learner’s Perspective Study (Clarke (Australia), Keitel (Germany), and Shimizu (Japan)) questioned the usefulness (for their team) of the research design employed by the TIMSS international video study (Stigler & Hiebert, 1999). Clarke et al. (2006) considered a video study of mathematics classroom activity that focused only on the teacher disregarded an important influence on teacher actions—student actions.

The overarching questions that focused the Learner’s Perspective Study related to student and teacher practices and whether these practices were: interrelated, influenced student construction of social and mathematical meanings, varied across a sequence of lessons as a topic progressed, and/or were culturally specific.

The data this team collected (Clarke et al., 2006) captured classroom practices, and social interactions between class participants, and the learning (social and/or mathematical) that resulted. This was the phenomenon under study.

[P]ractice-oriented analysis of learning situates mathematical activity in relation to the social setting with which the project is fundamentally concerned … it allows us to interrogate those settings with respect to the practices they afford and constrain. (Clarke et al., 2006, p. 3)

The Complementary Accounts Methodology (Clarke, 1998) employed in the Learner’s Perspective Study collected ‘integrated data sets’ from each country that included both student and teacher voice. ‘Practice-oriented analysis’ (Clarke et al., 2006, p. 3) of these data sets was undertaken by research teams in each country and these complementary accounts were synthesised by Clarke to develop rich portrayals of interaction patterns.

Learner’s Perspective Study teams from each country employed a diversity of social theories of learning for their ‘practice-oriented analyses’. These theories included: the role of social interactions in the construction of knowledge (Cobb & Bauersfeld, 1995); the role of the teacher as more ‘expert other’ guiding the learner through new learning (Vygotsky, 1978); contributions of lesson contexts and tools (physical, symbolic, discursive) to the nature of the mathematical constructs developed, ‘situated cognition’ (Lave & Wenger, 1991); influences of gradual variation of mathematics on learning, variation theory (Marton and Tsui 2004); learning (social and mathematical) through increased participation of the learner in the learning community, communities of practice (Wenger, 1998); and affordances and constraints to learner participation, positioning theory (Harré & Van Langenhove, 1999).

The Learner’s Perspective Study Research Design included research teams in nine countries initially (Australia, China, Japan, South Africa, Israel, Sweden, Germany, USA, Philippines). By 2015 there were 18 participating countries (see Mok & Clarke, 2015). In each country, three mathematics teachers, from three schools in the same city, were selected to participate. Criteria for selection were teachers from diverse school contexts in each city, considered competent teachers of mathematics by their local communities. Data were collected in at least ten consecutive lessons in each classroom, generally preceded by a five-lesson familiarisation period, to give participants time to become desensitised to equipment (including video cameras) and multiple data collection team members in the classroom.

The two lesson videos, mixed image video, audio-recorded post-lesson video-stimulated student interviews, class work and lesson resources, field notes, and accounts from researchers with different theoretical perspectives, as employed in the Classroom Learning Project (see Sect. 3) were employed for the Learner’s Perspective Study, with the following changes and additions:

  • Video recorded student interview

  • Third lesson video, a whole class video in addition to the student and teacher video

  • Student controlled lesson video remote in interview, rather than researcher controlled

  • Video-stimulated teacher interview each week during research period

  • Mathematics student achievement data sourced from the school

  • Teacher questionnaire administered after the lesson sequence

  • Accounts from research teams in participating countries (different theories employed)

Post-lesson student interviews stimulated by mixed image lesson video, student work (e.g. written work, artifacts) and lesson resources (e.g. texts book pages, worksheets, and concrete aids) increased stimulation of memory traces students might draw upon. Students controlled the video remote and found the parts of the lesson that were important to them (for any reason), and discussed what they thought was happening, and what they were thinking and feeling. This provided data on what the student attended to rather than the student giving commentary on what the researcher attended to. It provided opportunity for students to reconstruct their decision making and their meaning making.

The Learner’s Perspective Study student interview protocol included a section after the video-stimulated interview section, where general questions were asked to find, whether the student considered the lesson was a typical lesson, the nature of lessons the student found most useful to their learning and why, and how the student thought they were progressing in mathematics and how they made that decision. These questions were intended to assist in determining the frequency of this this type of classroom activity for this class, identifying the students’ preferred ways of learning, and identifying how the student perceived themselves as a learner of mathematics and the sources they used to make such decisions (e.g. other students, the teacher, books, and internet, and/or their own ideas).

The teacher selected the lesson video to stimulate their weekly interview. Teachers were asked to find and discuss the parts of the lesson that were important to them. This provided data on what they attended to, reasons for their pedagogical decisions, and sometimes additional data about student activity, not available through other sources.

Data on student achievement provided evidence of students’ general performance in mathematics which could be compared with students’ perceptions of their progress. The teacher questionnaire provided information on the teachers’ academic and work history and their values and beliefs about the teaching and learning of mathematics.

Whole class, teacher, student, and mixed image video data enabled study of patterns of teacher and student participation and relationships between these patterns. Data on changes to patterns of participation, and changes to student construction of social and mathematical meanings was collected through these sources across the lesson sequence. Syntheses of analyses of these integrated data sets (by research teams from each country) enabled the development of rich portrayals of each phenomenon under study.

5 Williams’ Doctoral study

Williams’ research focused on student’s creative development of new mathematical ideas during problem-solving activity. The theoretical framework that emerged from her Master of Education thesis, and the additional elements integrated into that framework during her doctoral work are described. The scope of this paper is insufficient to illustrate how the integrated data set supported each element of the theory developed. Instead, data were selected to illustrate two elements of the framework. These elements were selected for their usefulness in highlighting differences between the theoretical frameworks and analysis processes employed by Williams and by Clarke, who utilised the same data set for different purposes.

Research that informed Williams’ Masters thesis (Williams, 2000), on the role of task complexity in problem-solving activity, included mathematical understandings (Schoenfeld, 1985), student–student interactions during problem solving (Wood & Yackel, 1990), social and cognitive elements of learning mathematics (Cobb et al., 1992), affective elements accompanying the learning of mathematics, and task features related to engagement (Henningsen & Stein, 1997). This study was situated in two final year higher-level calculus classes in the same school (15 students per class). The teacher (Williams) was also the researcher. Students worked in small groups (3–4 students) on complex problem-solving tasks to develop what were unfamiliar mathematical understandings prior to work on the problem-solving task. Activity of two small groups undertaking a problem-solving task (80-min session) on differentiation was videorecorded by two cameras. During the session, different members of each group briefly reported their interim findings to the class three to four times. It was found that groups idiosyncratically identified mathematical complexities that were not apparent to them at the start of the task and decided to explore them (Williams, 2000). High positive affect was found by Williams to accompany the development of new (to the students) conceptual understandings. Williams (2000) theorised this high positive affect during creative activity as ‘flow’ (Csikszentmihalyi & Csikszentmihalyi, 1992). Flow occurs when a person or group autonomously decides to overcome a self-set (or group-set) challenge that is almost out of reach and develop new skills to achieve this (Csikszentmihalyi & Csikszentmihalyi, 1992). Williams (2002) found that the challenges that became apparent during mathematical problem-solving activity were intellectual mathematical challenges and that in addition to developing new skills, students developed new mathematical understandings. Further study of this flow phenomenon was undertaken in Williams’ doctoral research.

5.1 Theoretically framing Williams’ doctoral research

This section commences with a brief overview of theory that informed Williams’ (2005). doctoral study then focuses on the phenomenon ‘quality of experience’ with emphasis on the mathematical thinking and accompanying affective elements that occur during flow. The theoretical conceptualisation of this phenomenon is developed and illustrated. The data Williams’ presents to illustrate elements of her theoretical framework are intended to highlight differences in Clarke’s and Williams’ analysis techniques.

Williams’ doctoral research (2005) explored the nature of flow experiences during creative mathematical problem solving, and influences upon these. Different to the social theoretical frameworks generally employed by Learner’s Perspective Study team members, where learning through interactions with ‘expert others’ (Vygotsky, 1978) was generally an underlying assumption, Williams (2005) focused on learning without mathematical input from an ‘expert other’ during creative problem solving (Vygotsky, 1933/1966) — learning during flow (Csikszentmihalyi & Csikszentmihalyi, 1992). This flow state was conceptualised as ‘high quality learning experience’ by Williams (2005) to capture the high quality of both cognitive and affective experiences during flow. This construct is referred to for the rest of this paper as ‘quality experience’ in which affective elements of the construction of knowledge (Csikszentmihalyi & Csikszentmihalyi, 1992; Williams, 2002), and cognitive elements of the creative construction of knowledge (Dreyfus et al., 2001a; Krutetskii, 1976) are each a focus.

Influences on quality experience that Williams (2005) theorised in her doctoral work included social elements of student activity and a personal element of students—their inclination to explore. Social elements of the process of Abstracting in Context (control, explanation, elaboration, query, affirmation, attention) (Dreyfus, et al., 2001b), including the sources of those social elements, informed Williams’ (2005) analyses when identifying whether an activity was creative. Williams conceptualised creative activity as occurring where social elements were internally sourced (arising from the student/group) during exploratory activity and, activity that was not creative as occurring when social elements (control, explanation, elaboration, query, affirmation, and in some forms attention) were externally sourced (input by an expert other).

The nature of mathematical constructs developed (Dreyfus et al., 2001b), and psychological characteristics of students associated with their inclination to explore (Seligman, 1995; Williams, 2005) are also elements of the broader Engaged to Learn theoretical model conceptualised by Williams (2005) but, their relation to quality experience is not a focus in this paper. Briefly, Williams linked student inclination to explore new ideas to Seligman’s (1995) ‘optimism’, which is an orientation to successes and failures experienced by students (Seligman 1995). She renamed failures and successes specific to mathematical problem solving as ‘not yet knowing’ and ‘finding out more’ respectively, because failures and successes were observable through these actions during problem-solving activity (see for example Williams, 2014) and Williams found that optimistic students did not use the term ‘failure’ in describing their activity or the activity of others. Optimistic students perceived ‘not yet knowing’ as temporary and able to be overcome by a student’s personal effort of looking into a situation to identify what was not working and what could be changed to increase the likelihood of ‘finding out more’. They perceived their successes as personal and permanent and took on their successes as attributes of self (‘I can do this’).

Creative mathematical thinking and affective elements of quality experience were selected for theoretical and empirical elaboration in this paper because Williams considered that these elements could provide greater opportunities to show similarities and differences between Clarke’s and Williams’ research. These elements are now elaborated further.

Quality experience is a state of ‘flow’ specific to mathematical problem solving. During flow, all sense of time, self, and the world around are lost, and energies are focused on the task at hand. Feelings of surprise, enjoyment, and exhilaration accompany such creative activity (Csikszentmihalyi & Csikszentmihalyi, 1992). Exclaiming in excitement, surprise or pleasure can provide initial indicators of the possibility of high positive affect accompanying insight development (Williams, 2000). Body language can assist in identifying high level engagement as part of identifying quality experience (Csikszentmihalyi & Csikszentmihalyi, 1992; Reeves & Reynolds, 2001; Schiefele & Csikszentmihalyi, 1995) as can voice articulation and the content of talk (Csikszentmihalyi & Csikszentmihalyi, 1992; Mercer et al., 1999). Scheflen (1973 in Sielski, 1979, p. 239) identified body language as ‘any nonreflexive or reflexive movement of part or all the body used by a person to communicate a message to the outside world’. Subsequently, and in line with how it informs this study, it has been realised that body language can provide insights into what a person is thinking and feeling, whether (or not) they intended this to be revealed to others (e.g. Quilliam, 2011). Body language can also provide indicators of development of new conceptual knowledge when a person uses gesture (Reeves & Reynolds, 2001) or diagrams (Ericsson & Simons, 1980) rather than language in describing an idea.

Krutetskii’s (1976), and Dreyfus et al.’s (2001b) theoretical perspectives provided lenses for Williams to examine creative cognitive and social elements of developing mathematical meanings during quality experience. The work of Dreyfus et al. (2001a) provided Williams with theoretical tools to examine social influences on these constructing processes. Mathematical thinking was conceptualised as creative by Williams if there was an absence of mathematical input from external sources during the exploratory interval and new conceptual understandings developed. Elements of Abstracting in Context (Hershkowitz, et al., 2001) and Krutetskii’s (1976) ‘mental activities’ were integrated by Williams to theoretically frame creative mathematical thinking:

Recognizing is simple analysis to identify what mathematics to use and how to use it.

Building-with includes three categories:

  • Element-analysis, isolating parts and examining them one by one

  • Synthetic-analysis, simultaneously examining several parts (e.g., representations, procedures)

  • Evaluative-analysis, synthetic-analysis for purposes of judgement

Constructing:

  • Synthesis, integrating mathematical ideas, developing insight (‘seeing’ something mathematically profound) and/or recognising generality

  • Evaluation, progressively overviewing the mathematics generated to identify inconsistencies, and/or consider further uses for the mathematics developed.

The elements of creative thinking (in italics in the categories above) can be identified through Hershkowitz et al.’s (2001) ‘visible’ cognitive elements in student talk in class, and in student interview reconstructions. Krutetskii’s mental activities are embedded within subcategories within the descriptions of these italicised categories.

5.2 Data sources to identify cognitive and affective elements of quality experience

Williams’ Doctoral Research Study sourced data from The Learner’s Perspective Study which employed data collection instruments that included minor changes to the interview protocol for Australia. All one hundred and thirty-two Australian students in the Learner’s Perspective Study were interviewed by Williams. This data set (interviews, lesson videos, and worksheets (complimentary data accounts) was found sufficient by Williams for study of quality experience. The Australian interview protocol was modified, on Williams’ suggestion, to include an additional method of communication of new mathematical ideas for students who did not yet have the language to communicate these new ideas orally. Pen and paper were made available for students to draw or write when they chose to do so. To identify whether a mathematical activity was creative, Williams expanded a question on the interview protocol to help her identify whether new mathematical ideas were introduced by the exploring student/s or an ‘expert other’ in the class or developed by the constructing student or group. She extended the question ‘Did you learn anything new today?’ to ‘Did you learn anything new today and if so, how did that happen?’ These minor changes retained the integrity of the Learner’s Perspective Study interview protocol and had the potential to add additional data about the student’s learning and thinking that would contribute richer data to each of the two projects.

Table 1 shows ways that the ‘integrated data set’ of the Learner’s Perspective Study provided multiple opportunities to identify indicators of quality experience and gain rich understandings of the quality experience phenomenon. Multiple affective and cognitive indicators of quality experience [Table 1, Columns 1, 2] can be identified through the multiple data sources [Column 3], and these indicators can be operationalised in multiple ways [Column 4].

Table 1 Indicators of quality experiences identified through Williams’ Doctoral Research employed Learner’s Perspective Study multi-source data
Full size table

For example, the indicator ‘energies focused on the task at hand’ can be elaborated as ‘being unaware of world around’, ‘participating in activity’, and/or ‘continuing to focus beyond the time allocated’ through either or both lesson video and interview data. ‘Energies focused on the task at hand’ can be identified through a combination of operationalisations including ‘body directed towards the task or task activity’, and ‘participating in the activity’. ‘Latching’ (connecting to talk of others in class) and ‘cutting’ (interrupting the talk of another in interview in eagerness to contribute to or share new ideas) during talk with others are two ways in which high positive affect during creative activity can be operationalised. These indicators might be identified in group ‘creative talk’ (Wegerif, 2006) in class, creative internal talk (Vygotsky, 1933/1966) in class and/or reconstructed in interview, or creative activity in the interview. Combinations of indicators rather than a single indicator is required for a high magnitude of positive affect.

5.3 Quality experience: illustrative data, analyses and results

A ‘case’ was defined as the creative activity undertaken by a student, and influences upon it. These cases were situated in Year 8 lessons in three different schools, Case 1 (Kerri, USA) and Cases 2 and 3 (Eden and Leon respectively, Australia). Excerpts of each case have been selected to illustrate elements of analyses undertaken. Brief descriptions of the contexts in which quality experience occurred are given. The context for each case is described up to the time when insight development was first identified (from case analysis).

Case 1 [Table 2, Row 2]: Kerri, was a Year 8 student in a class of students identified as ‘gifted’ in the USA. The teacher was demonstrating how to find the linear equation of a line when two points were given. Kerri had already worked this out. She spontaneously focused her own exploration, wondering why linear equation notation was y = ax + b not a = bc + d. She suddenly leaned over her page and began to write (see Williams, 2007 for more detail), her body motionless. When she looked up, her body relaxed, and she exclaimed.

Table 2 Indicators of high positive affect in Cases 1, 2, 3, data source, and type of indicator
Full size table

Case 2: The class worked with the computer application ‘Green Globs’, where thirteen large green dots are randomly positioned on integer co-ordinates points on a 10 × 10 axis system, and equations of linear functions were input to hit and destroy these. Higher scores were gained when more globs are hit with one linear function. For more detail, see Schoenfeld et al. (1993) where a similar application ‘Black. Blobs’ is described. Eden and Darius were friends seated side by side but not paired together. Darius generated a family of parallel lines in his attempt to hit two globs. Eden focused for several minutes on Darius’ screen as these parallel lines were generated, then asking Darius if he was using trial and error. On finding he was, Eden returned to his own computer and worked silently, motionless, for seven minutes. He then turned away from his computer, his body relaxed, and he exclaimed softly. Only part of his statement was audible. See Williams (2007) and Table 2 herein for more detail.

Case 3: Leon was ‘looking in’ (see Williams, 2006) on Pepe’s activity as Pepe tried to make a triangle with three side lengths given. In his interview, Leon reported that he had wondered why Pepe was using a compass when he was meant to be ruling straight lines to make a triangle. He asked Pepe what he was doing but Pepe continued to focus intently on the triangle. Leon insistently asked again, and Pepe abruptly told him to think for himself. Leon continued to silently watch Pepe’s work, before exclaiming.

Illustrative analyses from within these three cases are now included. In her interview, Kerri reported the excitement she experienced in class ‘It was just kinda [kind of] like “hoo hoo”’ [Column 5], the content of her insight ‘oh that’s why it is y = ax + b’ [Column 6], and pleasure she felt ‘It’s realization’. No evidence of interactions with other students was found [Columns 3, 4], but there were diverse indicators of flow and of development of insight in video and interview data once a targeted additional analysis of lesson video was undertaken, informed by the interview. From her descriptions in her interview Kerri identified the interval in the lesson when this activity occurred [Column 6]. This led to targeted additional video analyses. Without Kerri’s interview report, her class activity would not have been noticed because her actions were silent [Column 3] and her exclamation of surprise was almost inaudible ‘this (is the same) x’ [Column 4].

Eden’s [ Case 2] lesson activity, captured initially on the student camera, focused on Darius and his partner. The whole class camera captured Eden when he returned to his computer, visible towards the back of the video image. His computer screen was not visible. Eden’s question to Darius ‘How the hell did you get 19 [game score]?’, his body language in class, and his almost inaudible exclamation after further work on his computer, indicated the possibility of quality experience. Field notes captured the intensity of Eden’s actions, contributed to the selection of Eden for interview after that lesson, and informed the interview probes. In interview, when asked to discuss his lesson activity, Eden used pen and paper to show connections he had made between tabular, and graphical representations and the equation he had input on the screen. He explained the relationship he had found between the co-ordinates of different points on a line and their linear equations. He emphasised words as he shifted his pen back and forth between representations (graph, table, equation). His interview added detail about the insight he had developed and influences on its development. Eden’s reconstruction of lesson activity, in his interview, led to more targeted additional analysis of the student and whole class video. Eden increased the connections he made between representations (graphical and numerical) (Dreyfus et al., 2001b; Krutetskii, 1976) as he explored what appeared to be a pattern linking integer co-ordinates (impetus to explore, Dreyfus et al., 2001b) and checked (novel-building with, Dreyfus et al., evaluative analysis, Krutetskii) its generality (‘mental activity, constructing’ Krutetskii; ‘Abstracting in Context, constructing’ Dreyfus et al.) between this pattern and verbal representation of the symbolic representation of a line (the equation). Eden’s process of creative development of mathematical insight was supported by his social interactions with Darius and with the mathematical ideas upon which he idiosyncratically attended to on Darius’ computer screen (Dreyfus et al., 2001a). See Williams (2007) for further detail. The accounts (and the particular features of these accounts) which informed analysis of Eden’s thought processes included Eden’s interview (body language indicating high positive affect, verbal account of his thinking, diagrammatic representations generated by Eden and used by him to link representations in his explanations to the interviewer), the class video of Darius’ pair (interactions between Darius and Eden, Eden ‘looking in’ (Williams, 2006) on Darius’ computer screen, dynamic visual display on Darius’s screen stimulating Eden’s impetus to explore, and Eden’s transition to his own computer to explore further), the whole class camera showing the task set by the teacher and Eden working silently at his own computer. His creative constructing process included ‘recognising’ (Dreyfus et al., 2001b) the link. By simultaneous analysis of these accounts, a rich portrayal of Eden’s mathematical, social, and affective activity was developed. Indicators of inclination to explore were displayed by Eden when he described how he problem-solved (in his interview): ‘you just have got to sort of think out the answers in your head (pause) occasionally you have …- got to write down on paper what you are thinking about and eventually get the answer’. This statement indicates Eden perceived that not knowing was temporary and that personal effort was needed to find out more (Seligman, 1995; Williams, 2014).

Leon [Case 3a] is the focus of the third case. Pepe [Case 3b] is included because his activity is integral to Leon’s insight development. In his interview, Leon used the lesson video to provide a commentary on his and Pepe’s activity in class. He repeated parts of the interaction between Pepe and himself that were almost inaudible and reported the thinking he had undertaken when he silently watched Pepe’s work. He reported that he initially had no idea what Pepe was doing ‘What the hell are you doing …?’. He explained in interview ‘I didn’t know what he was doing with the compass … he’s supposed to be ruling straight lines’. He also reported his comment to Pepe made after he gained insight ‘oh … so you can get the angle that is sloping down’ as he held the fulcrum of the ruler (at Pepe’s demand) as Pepe swung it. Leon’s high-level engagement in class became more apparent when he discussed the lesson video in his interview. His interview report explained the question he asked Pepe in class ‘What the hell are you doing …?’ In addition, Leon’s exclamation in class, and part of what he had stated, could be heard once Leon identified this in the video. Leon’s collaborative relationship with the interviewer (repeating what was not clearly audible) strengthened the subsequent more detailed video analysis and provided additional evidence of quality experience.

5.4 Learner’s perspective data sufficient for Williams’ Quality Experience Analyses

Learner’s Perspective Study data collection instruments, as employed in Australia, were sufficient to identify quality experiences in the three cases studied. Parts of the process of knowledge construction during flow were identified in ways that would not have been possible if multiple data sources had not been linked. Stimulation of the interview with the student video generated further data from the lesson video because students identified points in the lesson that were critical to their constructing processes. Without this video stimulation, those points in the lesson could easily have gone unnoticed. In addition, without the pen and paper added to the interview protocol Eden may not have been able to explain the mathematical ideas, including the links between representations he developed.

The iterative process of shifting backwards and forwards between data sources as further information became available through another data source, progressed the richness of the portrayal of creative mathematical thinking and high positive affect, including social and personal influences upon this, within the quality experience phenomenon. Because lesson and interview data were collected asynchronously, data generated through one instrument could inform data collection through another (e.g. field notes in lesson informed interview probes for Eden). The field notes suggested Eden had developed insight. This informed the probes in his interview. The interview responses to those probes identified that watching Darius’ computer screen had stimulated Eden’s inclination to explore. The lesson video in conjunction with the interview was used to find out more about how Eden used Darius’ mathematical display.

Student voice was crucial to the analyses in various ways, not only were there instances where the student described their thought processes, reported their high positive affect, and identified important points in the lesson, there were also instances where students’ voice articulation provided data about feelings and whether a constructing process was occurring or had occurred. The diversity of indicators generated across multiple data sources added detail to the findings and raised awareness of what might be learnt about students by monitoring their body language when studying creative mathematical thinking and positive affect in class. The interlinked ways of employing the multiple data sources (complementary accounts) added to the richness of each portrayal of quality experience. This assisted the subsequent analysis of commonalities and differences between different quality experiences portrayed by the same researcher.

6 Insights

Where Clarke studied social objects, patterns of participation in class and influences upon these, Williams studied individual and group learning. processes and influences upon these. The integrated data set generated for the Learner’s Perspective Study supported research questions posed by each researcher. Clarke employed his Complementary Accounts Methodology with a focus on synthesising accounts from research teams in each country. The purpose was to develop elaborated portrayals of social phenomena like for example ‘kikan-shido’ (‘between desk walking’) (O’Keefe et al., 2006). The team found that the form of ‘kikan-shido’ (how it was enacted) was similar in each country but the purposes for which it was employed differed (e.g. monitoring student work, helping students requiring assistance). Williams (2005) on the other hand undertook a micro-analysis of data related to quality experience. Data accounts from various sources (e.g. student interview, student video, whole class video, student work) were examined through Williams’ multi-theoretical lens to find data relevant to the quality experience phenomenon. Data collection was iterative—data from one source enabled additional generation of data from another source (e.g. field notes informed interview probes, student interviews informed further analysis of lesson video data). Each employed multiple theoretical perspectives (but in different ways). See Table 3.

Table 3 Insights: Clarke’s Complementary Accounts Methodology applied differently by Williams
Full size table

Clarke synthesised researcher accounts that employed different social learning theories where Williams integrated elements of relevant theories into her theoretical framework. Each benefitted from the opportunity to simultaneously analyse data from different sources.

Table 3 displays results of the analysis of similarities and differences between how David Clarke and other members of the Learner’s Perspective Study research team employed Clarke’s (1998) Complementary Accounts Methodology and Williams’ (2005) adaptation of this methodology for her own purposes. Two elements of the Complementary Accounts Methodology (Clarke, 1998) were directly relevant to Williams’ research (integrated data sets, and inclusion of student voice). The third element, complementary multi-theoretical accounts of international researchers was not, but complementary multi-source data accounts were analysed using a multi-theoretical framework to develop an elaborated portrayal of the phenomenon. Table 3 provides insights into the usefulness of Clarke’s methodology for purposes beyond those for which it was intended. The capability to simultaneously examine activity from various sources, and to iteratively generate further data due to the asynchrony of data collection from different sources illuminates the potential for this methodology to be adapted and employed for various research foci beyond social learning theories. Opportunities to continually revisit this complex data set opens out opportunities for various forms of micro-analysis in addition to the many analyses the Learner’s Perspective Study Methodology has already supported.

It is testament to the strength of Clarke’s Complementary Accounts Methodology that Williams (2005) was able to employ this methodology for such different theoretical purposes, thus extending ways in which Complementary Accounts Methodology can be interpreted. Considering a legacy as a bequest from the past available to those in the future, David Clarke’s legacy, Complementary Accounts Methodology, as demonstrated herein is a powerful methodological tool that has the potential to support and help to shape classroom research based on social theories of learning and classroom research based on learning theories that also draw from other diverse research domains well into the future.