Context and the lecturer’s model solution

Tasks like (1i) were present in the exercise sheets and the coursework sheets. In these, the students were asked to prove specific statements by induction. The procedure of the routine was specified in the wording of the task as in part (1i). Concerning (1ii), students were asked to engage with the object of the divisor. One of the tasks had slightly different wording from the one in the examination. Specifically instead of “Give the definition of what is

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meant by saying that d is a divisor of a” in this task the lecturer used the following wording “Write out in full the definition of the statement “d divides both a and b.”” (Figure 5.2). Here, I note the routine-driven use of words in the task used in the exercise sheet compared to the object-driven use of words in the examination task. Another important difference illustrated between the tasks that the students were asked to engage in the duration of the year (Figure 5.2) and the one at the final examination (Figure 5.1) is the amount of directions both about the procedure of the routines and the hints provided. The above illustrates the transition from the mathematical discourse that the students are first exposed to during the year and then the one that they are expected to be able to engage in by the end of the academic year.

Figure 5.2: Task from the first exercise sheet in Sets, Numbers and Proofs

Concerning the Euclidean algorithm, there were many tasks both in the exercise sheets and in the coursework requesting its use. The procedure of the routine is mentioned in the wording of the task, as is in the task (1iib). Students were expected to engage in substantiation routines during the year. However, this engagement was more directed compared to the one in the examinations. More evidence about students’ expected engagement with these tasks and lecturers’ expectations about their engagement with the mathematical discourse can be traced in the model solution that the lecturer produced for departmental use (Figure 5.3).

In the solution produced by the lecturer for (1i), the statement is given a name, and its symbolic realization is repeated. Then the process of the inductive proof is divided into two parts: the base and the inductive step. The base step involves the substantiation of the statement for 1. First, the statement is written and then the substantiation of the narrative. Then, for the

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inductive step, a random natural number (k) is selected for which the statement holds. Using that the statement for k holds, the same number 2k+1 is added to both sides. Then by rearranging, the statement for k+1 is achieved. The lecturer ends the solution by saying that, because of this process, the statement is true for all natural numbers. Apart from the evident structure of the two steps and the concluding sentence of the substantiation routine, I note that this solution explains the steps of the procedure of the routine and whenever a new variable is appearing the numerical context of this variable is defined (this is the case for both k and n). Also, the statement is named and used as an object in the narrative.

Similarly, in (1iia) every time notation (d, a, b, m, n, k, l) is introduced, the numerical context in which the variables belong are clarified in the text. There are operators (if, then, for all) that are linking narratives about the discursive objects. Initially, the definition of the divisor is given. Then this is used to produce narratives linking the divisor d with the numbers a and b. In creating these narratives, the integer variables (k, l) are introduced and their numerical context is defined. Then, the linear combination of a and b is written, and the narratives produced earlier about a and b are used and factorised to show that the linear combination is a product of d and an integer. The summation of products of integers is an integer and so the linear combination is divisible by the divisor d.

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Figure 5.3: Model solution to compulsory task in Sets, Numbers and Proofs

In the answer for (1iib), prior to writing the Euclidean algorithm, the lecturer mentions that the answer is “Following the method in lectures”. This procedure of performing, structuring and writing the Euclidean algorithm was presented to the students in the lectures. In the interview, the lecturer comments on the choice of this representation of the algorithm (section 5.1.3). After the algorithm, the lecturer ends the solution of this part with the conclusion that the g.c.d of 123 and 45 is 3 and provides the linear combination of the two numbers that results in 3. Then, he writes the linear combination given in the wording of the task 123m+45n and provides the values -4 and 11 for m and n accordingly.

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Finally, in (1iic) the answer provided is brief and illustrates the connection with the previous narratives constructed to answer the other parts of the task. Specifically, connecting with (1iia), the linear combination 123s+45t with the two variables s and t belonging to the integers is divisible by 3, but the other side of the given equality is 7 (which is not divisible by 3). As in the previous parts of the task, the numerical context that the variables belong in is mentioned in the solution provided by the lecturer.

At the end of the solution, the lecturer provides characterisations for the tasks. (1i) is considered easy as a basic induction proof. (1iia) is considered easy as seen in the tutorials and (1iib) as computation. Then (1iic), is considered as moderate and the similarity with the tutorial sheet tasks is noted. In the next section, I discuss excerpts of the interview with the lecturer illustrating the expectations he has from his students about their engagement with the task, and with university mathematics discourse at large.

5.1.3 Lecturer’s perspectives: a commognitive account