Analyzing students’ communication and representation of mathematical fluency during group tasks
Introduction
The development of fluency is a goal of mathematics teaching. Students who develop fluency use accurate, efficient, and flexible ways of dealing with numbers. These students are able to cope with tasks involving mathematics in everyday life (Kilpatrick, Swafford, & Findell, 2001; Watson & Sullivan, 2008). To be considered proficient in mathematics, students must be able to think mathematically, fluently make choices between strategies, and engage in mathematical discussions with others (Kilpatrick et al., 2001). Therefore, a student’s fluency is an indicator of proficiency. Fluency, often termed ‘procedural’ fluency, is defined by Kilpatrick et al. (2001) as “knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently” (p. 121). Although a helpful definition, it omits to acknowledge the understanding and reasoning required by students to truly enact fluency. Cartwright (2018) suggests that “mathematical fluency is the result [emphasis added] when students’ strategies and ability to reason are concurrent with their conceptual understanding” (p. 208). Instead of being one thread of mathematical proficiency, mathematical fluency is an outcome when students’ abilities to understand, use procedures, and apply reasoning align. ‘Mathematical fluency’ is the operational definition of fluency for this paper.
Viewing strategic competence, conceptual understanding, and reasoning collectively when discussing mathematical fluency is supported by Graven and Stott (2012). They observed that “overlap of learner methods/responses as both procedural fluency and conceptual understanding arose, especially when considering the dimensions of flexibility and efficiency in relation to procedural fluency” (p. 146). Their findings indicated that students communicate more than just basic ‘fact fluency’ when verbalizing their solutions and strategies. Students’ mathematical fluency may be observable through discussions where evidence of their conceptual understanding, reasoning related to their solutions, and connections made to prior learning or known facts are visible. The purpose of the study reported in this paper was to observe and analyze students’ characteristics of mathematical fluency through groups’ oral, written, and drawn representations.
Section snippets
Reconceptualizing fluency
A gap is present between the espoused definitions of mathematical fluency in the literature and the enacted methods used in research to observe or assess students’ mathematical fluency. Russo and Hopkins (2018) acknowledge the need to develop reliable tools to measure ‘computational’ fluency to capture “all aspects of the definition of fluency … particularly the neglected flexibility and strategy choice [emphasis added] components” (p. 664). The definition of ‘mathematical’ fluency stated by
Study context
This paper reports findings that form part of a larger study exploring characteristics of mathematical fluency, and teacher noticing and interpreting of mathematical fluency. In exploring mathematical fluency, the wider study employs a framework to guide the observation and analysis of student data. The initial Characteristics of Fluency Framework (CFF) was generated by Cartwright (2018) from primary teachers’ reported conceptions and descriptions of what students’ mathematical fluency ‘looks
Methodology
A qualitative research approach was employed to investigate the mathematical fluency characteristics students exhibited during group tasks. Within qualitative research, “all data is a symbolic representation which needs to be interpreted” (Twining, Heller, Nussbaum, & Tsai, 2017, p. A2) and thus is context dependent. Situating the research within students’ regular classroom settings was appropriate for both the study’s context and the participants’ ages. Noticing mathematical fluency was not
Results
The analysis of group work samples and audio recordings (transcripts) collected during the explore phase of the lessons are reported on in this section. It is structured in two parts. The first part presents three specific student groups or ‘cases’ and the analysis of mathematical fluency characteristics observed within each case. Cases were selected to present data from different grade levels across the school. These specific groups were selected as they included extensive oral discussions
Discussion
In this section the findings and their implications for future research and classroom practice are discussed. The investigation was guided by two research questions: What characteristics of mathematical fluency do primary students demonstrate during group tasks? How do students communicate mathematical fluency through various representations? Using the CFF and the representational domains as analytical lenses led to refinements to the framework and discovered the benefits of analyzing various
Conclusion
Representations played an important role in providing different perspectives from which to observe mathematical fluency. Some characteristics were easier to observe in written responses (strategic competence) compared with those that were easier to observe orally (adaptive reasoning), particularly in circumstances where students provided no written reasoning. Characteristics of conceptual understanding were evident across both written and oral representations, but were dependent on the amount
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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