Summary and conclusion on Sets, Numbers and Proofs tasks

Analysis in this chapter focuses on closed-book examinations tasks from the first half of the Numbers, Sets and Probability part of the module, the Sets, Numbers and Proofs. The analysis highlights differences between the school mathematical discourse and the university mathematical discourse and at the same time the pedagogical actions in the context of assessment, that the lecturer implements aiming to assist students in their engagement.

The majority of the visual mediators are symbols and defining the numerical context of these symbols is an important routine of the university mathematics discourse, as can be seen from the interview excerpts but also the model solutions produced by the lecturer. The use of specific words and visual mediators is part of the engagement with the university mathematics discourse. In university, much attention is given to the numerical context of the variables whereas that is not necessarily the case for secondary school as usually, the numerical context is the context of real numbers. Within these three tasks, the importance of the clarification of the numerical context is visible in (1ii), (2ii) and (3). For example, the word divisor (1ii) is based on all the variables being part of the numerical context of the integers.

During the interview, L1 uses the pronoun “we” to refer to two groups. In the occasions reported in sections (5.2.3.1 and 5.3.3.1) he refers to the students and him as a group and in sections (5.1.3.1 and 5.3.3.3) he refers to the mathematical community of which he is a participant. In the latter cases and section (5.3.3.1), L1 speaks about practices of the mathematical community that the students with their entrance to university would become familiar such as engaging with routines (e.g., defining and justifying). The routines of the mathematical discourse that are present in these tasks are mostly substantiation and recall routines. The recall routines are routines of defining an object, a theorem or recalling the steps of a ritual or a substantiation routine. Regarding, the substantiation routines, in substantiating that an object has a certain property (e.g., 2iia, 2iib, 3ia, 3ib), in substantiating an equality relationship between objects (e.g., 2i, first part) or illustrating that these objects are different (e.g., 2i, second part). The substantiation and the

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defining routines are not something that the students are used to from the secondary school.

In section (5.3.3.1) L1, comments on students’ difficulty recalling the definition of an injective function. He comments on how by trying to recall the definition the students find difficulties in the logical structure and the use of the symbolic visual mediators. The symbolic mediators, as shown in the quote below, due to their nature are forming the baseline for an abstract and autonomous mathematical discourse:

“The process-object duality of symbolic mediators is a basis for compression and the subsequent extension of mathematical discourse, and it renders this discourse independent of external, situation-specific visual means. All this ensures a very wide applicability of the discourse.” (Sfard, 2008, p. 162).

The definitions, in the university mathematics discourse, are using many symbolic mediators. So the definition of an injective function is an example of a definition that involves symbolic mediators, which the students have to have a clear meaning-making in order to be able to recall the definition using the appropriate symbolism when requested.

L1, in an excerpt in section (5.3.3.3) compares the engagement with the mathematical discourse of this module with the engagement with other modules. Specifically, this comparison concerns the justification in a substantiation routine (4ia, 4ib) in different mathematical areas namely this module and Analysis. The endorsement routines, as illustrated in Sfard’s quote below differ between discourses.

“Terms and criteria of endorsement may vary considerably from discourse to discourse, and more often than not, the issues of power relations between interlocutors may in fact play a considerable role.” (Sfard, 2008, p. 134)

In the case of the module Sets, Numbers and Proofs a sketch accompanied with a short argument can be endorsed. Whereas, in the case of Analysis for a classification of a function as injective or surjective a much more detailed narrative would be required.

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Having in mind the differences between the discourses, L1 is assisting the students in their engagement with the university mathematics discourse. There are prompts about the procedure of the routines, the justification in the expected solution and the absence of directions about procedures of routines. The latter one highlights the creativity of the procedure of the routines as a characteristic of the practices of the community. According to Sfard's theory of commognition a routine has a procedure or a routine course of action which is defined as a “set of metarules that determine (e.g., in numerical calculations) or just constrain (e.g., in proving or writing a poem) the way the routine sequence of actions can be executed” (Sfard, 2008, p. 302). There are three instances (1iib), (2i), (3ii), (3iii) where the procedure of the routine is not specified, and the students’ agency is not restricted. L1 comments on the beauty of following different procedures and how that allows him to give full marks to a response that does not follow the expected procedure.

I also note that the compulsory task (task 1) is more structured and with more directions on the procedures of the routines, compared to the other two tasks (task 2 and 3). More, specifically, as mentioned earlier in this section, there are instances where there is a direction regarding the procedure of a routine (1i, 1iib) or that specific narratives can be used to assist in the procedure of the routine (1iia, 1iib) also instructions regarding the justifications (1iic, 2ii, 3i).

The directions regarding the procedures could guide students to a ritualistic engagement with the routines. As these are routines that the students are not yet familiar with, as this is a first-year module, this engagement with rituals can be seen as a base towards building an explorative engagement with the routines of the university mathematics discourse. In the next chapter, I will be turning to students’ written responses to the same tasks in order to examine their actual engagement with the mathematical discourse. Also, I will be examining for differences between what the lecturer’s intended practice and the students’ actual engagement with the university mathematics discourse.

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Chapter 6. Sets, Numbers and Proofs: