Representational fluency (RF) is the ability to create, interpret, transpose within, translate between, and connect multiple representations in doing and communicating about mathematics. The need for research on students' development of representational fluency with technology originates from the notion of deepening students' development of conceptual understanding through making connections between representations (Zbiek, Heid, Blume, & Dick, 2007). According to Heid and Blume (2008), and Kieran and Yerushalmy (2004), relatively few classroom-based research studies have been conducted to investigate how students develop RF with computer algebra systems (CAS), an important tool that can support movement between multiple representations. The motivation for this research stemmed from careful examination of such issues, which Arbaugh and colleagues (2010) purport are significant to both researchers and practitioners alike.

Purpose and Research Questions

The purpose of this research was to: (a) develop an empirically grounded instructional theory of students' development of RF in a CAS and paper and pencil (P&P) learning environment, (b) to document students' development of RF, and (c) to come to understand characteristics of the learning ecology in which this development was situated. More specifically, the research questions that guided the study were:

1.   How do ninth-grade algebra students develop RF in the context of learning to solve linear equations in a CAS and P&P environment?

2.   What means of support seem to facilitate students' development of RF in a CAS and P&P environment?

Part of the larger research project, the focus of this presentation will be limited to a discussion of the instructional theory.

Research Design and Theoretical Orientations

Various processes of learning and means of supporting that learning were investigated through design research (Gravemeijer & Cobb, 2006). From an emergent perspective on learning (Cobb & Yackel, 1996), the research design and methodology focused on both classroom mathematical activity and individual students' activity and cognition.

Conjectured Instructional Theory

A conjectured instructional theory includes both a hypothesized process of learning and means of support for that learning (Gravemeijer & Cobb, 2006). This theory is consistent with Simon's (1995) notion of a hypothetical learning trajectory that includes mathematical goals, instructional activities, and conjectures of students' thinking. The conjectured instructional theory was constructed as a result of a literature review and was developed a priori to a classroom teaching experiment in which the theory was subsequently tested and revised.

Learning Processes

Students' development of RF was conjectured to occur in the context of using tools and learning to solve linear equations. The processes of transposing, translating, and making connections among multiple representations were targeted. The main mathematical goals were to (a) develop fluency in creating and using multiple representations of linear expressions and equations to solve problems, (b) to understand the equals sign as an equivalence relation, and (c) be able to explain solving equations as a process of reasoning. The four main aspects of Kieran and Drijver's (2006) teaching experiment on equivalence and equations—(dis-)connecting the numeric and algebraic, the notion of equivalence, the notion of restrictions, and coordination of solving an equation and the notion of equivalence—were the foundation of the conjectured learning progression.

The main rationale for assuming the aforementioned learning progression is based on literature in which it is purported that in order to be successful in solving equations, students need to have a solid understanding of the meaning of the equals sign (e.g., Knuth, Stevens, McNeil, & Alibali, 2006). In the present study, an understanding of the equals sign was conjectured to develop through the notion of equivalence, through numerical, verbal, and graphical approaches before symbolic approaches (e.g., Yerushalmy, 2006).

Means Of Support

Means of support for learning span several aspects of instructional design including activities, tools, activity structure, and discourse (Cobb, 2003). The specific means of support conjectured to facilitate the aforementioned learning processes were developed out of an examination of literature on RF and the coordinated use of CAS and P&P:

á      Anticipate the results of tool-based activity (Pierce & Stacey, 2002).

á      Engage in uni-, bi-, and multi-directional translations (Janvier, 1987).

á      Reflect on and reconcile tool-based representations (Kieran & Saldanha, 2008).

á      Actively seek to coordinate information and make connections between representations (Friedlander & Tabach, 2001).

These means of support were both activity oriented and cognitively oriented, with implications for discourse and task design. Other means of support were specific to the activity structure. Kieran and Drijvers' (2006) articulation of a task-technique-theory framework guided the development of activities. First, students used CAS and P&P in studying the equivalence of expressions from numerical and graphical perspectives. Second, students created expressions and equations, and tested their equivalence with CAS commands such as equality, the with "|" operator, and other P&P techniques. Finally, students created and solved linear equations from numeric, graphic, and symbolic representations with a variety of tools and techniques.

The aforementioned instructional theory guided the development of the instructional experimentation that was carried out in the second phase of the research: a teaching experiment was conducted in collaboration with a classroom teacher (Cobb, 2000). Data from video, field notes, and classroom artifacts were collected on a daily basis for four weeks. Ongoing analyses of these data were focused on refining the instructional theory in which it served as both a lens and an object of analysis. Finally, retrospective analyses of all data were informed by Cobb and Yackel's (1996) interpretive framework in which lenses on individuals' development of RF and corresponding classroom practices were significant.

Discussion

The purpose of this research was not to give an account of "what works," but rather to offer a situated account of how the progression of developing RF might unfold (cf. Cobb, 2003). The articulation of a conjectured instructional theory has significant implications for both research and practice. The conjectured processes of learning and means of support for RF were grounded in literature on students' difficulties with particular processes (e.g., making connections) and content (e.g., understanding the equals sign as an equivalence relation), contributing to theory development that can also be used by teachers to inform instructional practice.

References

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