The findings of the data analysis are illustrated through two cases of teachers engaged in professional development, their mathematics teaching within their classrooms, and their students’ participation in the lessons. The cases provide a detailed and thick description (Barab & Squire, 2004; Gravemeijer & Cobb, 2006) of how each of these teachers developed algebra ears and eyes and then how effectively they facilitated algebraic reasoning in their classrooms. Direct quotations from the teachers are used in the findings from the study group meetings and interview data. The use of teacher voice provided a means to develop an understanding of their focus and reasoning as they worked to develop their own algebra ears and eyes and algebraic reasoning in their classroom. The discussion also draws on the teachers’ task design and implementation and use of pedagogical strategies. Vignettes and examples from the classroom are provided to illustrate the actions of the teachers and students. Findings pertaining to corresponding shifts in student engagement and participation within each of the classrooms are also included with data sourced from classroom videos and photo elicitation interviews.
The findings are reported in four distinct phases beginning with an overview of classroom observations in the school year prior to the professional development. The following phases report on the findings at the beginning, middle, and end of the study.
5.10.1 Ecological validity, generalizability, and trustworthiness
A key aim of design research is to achieve ecological validity, that is “the results should provide a basis for adaptation to other situations” (Gravemeijer & Cobb, 2006, p. 46). In this way innovative instruction which is developed in the course of design research can then be used by
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others to support student learning productively in different classroom contexts. This is similar to how Barab and Squire (2004) describe generalisability within the context of design-based research with the findings from one study being able to be replicated to other contexts. Use of thick description is highlighted as a means of making replication possible and also supporting teachers and outsiders to think how the instructional design could be adjusted to their context appropriately (Barab & Squire, 2004; Gravemeijer & Cobb, 2006; McKenney, Nieveen & van den Akker, 2006). The reporting of this study aims to provide a comprehensive description of all elements of the research. The cases include insight into teacher participants’ involvement in the professional development activities and a rich description of the teaching and learning context based on observation and participants’ (teachers and learners) perspectives. Furthermore, the inclusion of teacher participants’ feedback to develop and modify the framework of teacher actions (see Chapter Six) to facilitate algebraic reasoning significantly strengthens the ecological validity of the study (Gravemeijer & Cobb, 2006).
Trustworthiness and credibility are important elements of design research and may be seen as akin to notions of reliability and validity (Barab & Squire, 2004; Cobb et al., 2003). With the large amount of data generated in the course of a design experiment it is central that the resulting claims are trustworthy. Therefore there is strong link between trustworthiness and the credibility of the analysis. Cobb et al. (2003) argue that by explicitly describing the criteria and types of evidence used for inferences, other researchers can then “understand, monitor and critique the analysis” (p. 13). In the current study, a systematic approach was used to code the large amount of data generated through video records, interviews, and field-notes. There is a detailed description of the varied forms of data collection and also of the data analysis and coding given in this chapter.
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The credibility and trustworthiness of a research study can also be enhanced through triangulation of different data sources. This is based on the premise that by using multiple data sources, weakness in a single data source is counterbalanced by strengths in another (Barron et al., 2013; McKenney et al., 2006). The current study drew on multiple sources of data including interviews, documents, and field-notes to triangulate the primary video data source. Extended engagement by the researcher with participants in the field also enhances credibility and trustworthiness of analysis and can reduce observer effects (McKenney et al., 2006). A key feature of this study was the sustained engagement with the teacher participants acting as co- researchers and the students in their classrooms over the duration of the study.
5.11 SUMMARY
This chapter outlined the key research question and guiding sub-questions for the study. Justification was provided for the selection of the qualitative research paradigm and use of design research methodology. A summary of the key characteristics of the design research and case study methodology was given
Rich descriptions were given of the research setting and data collection methods. It was shown how the data collection drew on multiple sources which provided opportunities to investigate perspectives from different participants. The study was shown to be cyclic and iterative and the complementary role of data collection and analysis was highlighted. Detailed descriptions of the multiple levels of coding were given which supported the development of the case studies.
Findings presented in the following chapters are based on two cases. Each case details a participant teacher’s experiences during the professional development programme. They serve to
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illustrate the different pathways teachers take in developing algebra ears and eyes. They highlight the complex factors which are involved in facilitating algebraic reasoning in the classroom including task development and implementation and the development of classroom and mathematical practices which support engagement in early algebra. Subsequent shifts in student engagement and participation are documented.
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CHAPTER SIX
DEVELOPING ALGEBRAIC REASONING IN A
MATHEMATICAL COMMUNITY OF INQUIRY: MRS
STUART
6.1
INTRODUCTION
Within this chapter, the case study is organised in four distinct phases which relate to when data collection occurred. The first section reports on the classroom context prior to the beginning of the professional development. Each of the following sections reports on involvement in the professional development; how algebra was integrated into the classroom; then the developing classroom culture; and finally the student perspectives. A commentary that links the changes in the classroom to the literature accompanies each section.
Section 6.2 describes the classroom context in Mrs Stuart’s classroom prior to the professional development commencing. Section 6.3 highlights the initial steps to introduce algebraic reasoning. It begins by outlining Mrs Stuart’s active participation within the study group and her early reflection on practice. It shows how Mrs Stuart began to plan for algebraic reasoning opportunities although this was not supported by her existing classroom practices. Finally it portrays the implementation of changes to the classroom community and students’ engagement with these.
Section 6.4 shows the continuing development towards integrating algebraic reasoning in the classroom. Focusing on task implementation, it describes how Mrs Stuart shifted to using inquiry as “a way of being” (Jaworksi, 2006; 2008). It also demonstrates the emergence of
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teacher requirement for collaborative interaction and shifts in students’ learnt ways of thinking and acting mathematically.
Section 6.5 outlines the actions to embed algebraic reasoning in the classroom further. It details Mrs Stuart’s re-conceptualisation of her understanding of algebra. It shows how this supported her to design and implement tasks which provided greater affordances for algebraic reasoning. Finally it illustrates the changing practices in the classroom and shifts in student roles.