| Abstract | Calculus examination questions can sometimes be judged as having similar mathematical content when, due to textual aspects and visual mediators, they are quite different. Computer Algebra Systems (CAS) technology can have a further impact on their nature and on students’ expected answer approaches. Little research has been done to assess the impact of CAS technology on these characteristics from a discursive perspective.
This thesis investigates the impact of CAS technology on mathematical discourse. The first research question asks how a commognitive analysis framework (Morgan & Sfard, 2016) can be applied to effectively capture the complexity and difficulty level of written answer and multiple-choice examination questions in undergraduate Calculus units where use of CAS technology is available. This includes considering how such questions position students as knowers and users of CAS technology. The second research question asks (a) how effectively undergraduate students use their CAS calculator and use and interpret CAS output, especially when it is in a different format to what they are familiar with from the methods of working by-hand that they have been taught in class; as well as (b) what we can learn from a commognitive analysis of task-based interviews of students in relation to how they reflect on their answers in this situation. Finally, the third research question asks to what extent undergraduate students use CAS and when they believe its use to be most beneficial.
Commognitive analysis was used for qualitative analysis of nine questions purposively selected from 7 of 11 undergraduate Calculus assessments (examinations and tests) collected. To capture how students approach similar questions using CAS, task-based interviews were conducted with four students taking a first- or second-year Calculus unit. Their written and spoken discourse was analysed qualitatively, using commognitive analysis, to determine the extent to which they used ritual or explorative routines, aspects of how they used CAS in their problem-solving approaches, how they navigated the classroom discourse of Calculus and that of CAS, and whether any indication of the discursive footprint of high school mathematics on their Calculus discourse was present. Evidence was sought of any intrapersonal commognitive conflicts arising from their interactions with a CAS calculator, due to its different discursive requirements. Their preferences and reasons for using CAS or working by-hand were also examined. To contextualise the findings of the task-based interviews, basic descriptive statistics were produced, showing the percentage frequency distribution of responses for questionnaire data on undergraduate students’ use of, and attitudes towards, CAS technology at the time of the study.
Analysis of the test and examination scripts indicated that, while reducing the number of procedural steps involving working by-hand, questions more logically and grammatically complex in other ways were asked when CAS technology use was expected, including some with increased symbolic and graphical CAS outputs. A finding was that complexity of questions with CAS outputs also depended on how individual students were positioned to answer them, as, in positioning students as knowers of technology, such questions can involve greater logical complexity if a student is not familiar with CAS outputs.
From the task-based interviews analysis, a finding was identifying unresolved intrapersonal commognitive conflicts encountered by students, between the discourse of CAS technology itself and the more familiar classroom discourse of Algebra and Calculus. The participants mostly used ritual routines, which could have been an artefact of tasks chosen. They typically used CAS to employ a direct procedure in a single step, but also in an inter-representational fashion and as a procedure within a more complex process. The discursive footprint of high school mathematics was evident in their written and spoken discourse, with few instances of transitioning to more scholarly, university level Calculus discourse. Where they preferred CAS, reasons included saving time, visualising graphs and not being confident in solving a problem by-hand.
Applying commognitive analysis to test and examination scripts where CAS technology is available or present as outputs, contributes to research on assessment by identifying more aspects to consider in evaluating complexity of questions and how they are positioning students to respond to them. The identification of commognitive conflicts due to the different discourse of CAS can inform teaching practice about using and interpreting results with CAS technology, and effectively integrating CAS into learning materials. |
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