We explore the transition from school to university through a commognitive (Sfard 2008) analysis of twenty-two students' examination scripts from the end of year examination of a first year, year-long module on Sets, Numbers, Proofs and Probability in a UK mathematics department. Our analysis of the scripts relies on a preliminary analysis of the tasks and the lecturers' (also exam setters') assessment practices, and focuses on manifestations of unresolved commognitive conflict in students' engagement with the tasks. Here we note four such manifestations concerning the students' identification of and consistent work with: the appropriate numerical context of the examination tasks; the visual mediators and the rules of school algebra and Set Theory discourses; the visual mediators of the Probability and Set Theory discourses; and, with the visual mediators and rules of the Probability Theory discourse. Our analysis suggests that, despite lecturers' attempts to assist students towards a smooth transition to the different discourses of university mathematics, students' errors at the final examination reveal unresolved commognitive conflicts. A pedagogical implication of our analysis is that a more explicit and systematic presentation of the distinctive differences between these discourses, along with facilitation of the flexible moves between them, is needed.
This exploratory study reports on characteristics of proof production and proof writing observed in the work of first-year university students who took part in workshops on the theorem prover LEAN (https://leanprover.github.io). These workshops were voluntary and offered alongside a transition to proof module in a UK university. Through qualitative analysis of 36 student produced proofs of an unfamiliar statement we highlight characteristics of proofs produced by students who did engaged and who did not engage with LEAN. The analysis shows two characteristics of proofs written by students who engaged with the programming language. The first concerns proof writing and includes the accurate and correct use of mathematics language and symbols, together with the use of complete sentences and punctuations in proofs. The second concerns proof structure and includes the overt break down of proofs in goals and sub-goals. We conclude by hypothesising a link between the characteristics observed and the experience of engaging with the theorem prover and we reflect on the potential that engagement with this theorem prover may have in mathematics instruction at university level.
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