Commognitive conflict and discursive discontinuity

3.2.1 Commognitive conflict and discursive

discontinuity

Key ideas in my study are the notions of commognitive conflict and discursive discontinuity. In this section, I discuss these and their relation. Commognitive conflict is defined by Sfard as

“the encounter between interlocutors who use the same

mathematical signifiers (words or written symbols) in different ways or perform the same mathematical tasks according to differing rules” (Sfard, 2008, p. 161).

As noted above, commognitive conflict can occur when a symbol or a word is being used by a participant in the discourse and it is incompatible with the discourse that is being expected. In the case of responding to a

question on discrete functions, a student might be using words or symbols that relate to the discourse of continuous functions (e.g. continuous, differentiable). This use creates a conflict between the student who responds to the task and the lecturer who reads the solution and expects the answer to the task to have narratives with words relating to discrete functions. The commognitive conflict here lies in incompatible word use. The expected student engagement is with discrete functions, and the actual engagement is with the discourse of continuous functions. The student here conflates the two discourses illustrating a problematic meaning-making between the discursive object of continuous and discrete function. A similar situation could occur with the use of logical symbols, namely implications (⇒), equivalences (⇔) and quantifiers (∃, ∀). These symbols indicate relationships between mathematical objects involved in the narratives. The meaning of these symbols in the university mathematical discourses are very specific, and the students who are now becoming members of this community are learning to use the symbols and the

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students could be using them ritualistically. This could be observed in their written responses in instances where the use of the symbol illustrates problematic meaning-making of the relationships between mathematical objects. For example, a student might be using an implication instead of equivalence in the narrative they produce. This illustrates conflation

between the logical statements “if … then”, which is the implication (⇒), and “if and only if”, which is the equivalences (⇔).

Another instance, where commognitive conflict can occur is in the rules of the discourses. In this case, the students could be using routines that are incompatible with the mathematical discourses that they are asked to use in that question. For example, when proving a statement, initially the students might be asked to examine whether the statement is correct by providing an example and in some cases at secondary school they would stop there. However, at the university level, proving is much more rigorous and

providing just an example is not accepted by the lecturers as endorsable narrative.

Tabach and Nachlieli (2016) discuss meta-level learning. “Meta-level development of a discourse may be of a horizontal or vertical nature” (Tabach & Nachlieli, 2016, p. 302). Manifestations of commognitive conflict can occur when two separate discourse are combined in a new discourse (e.g., discourses about functions and discourses about integers combined to form discourses about discrete functions) signalling a commognitive conflict at a horizontal level. Commognitive conflict occurs at the vertical level when a discourse is combined with its own meta-discourse (e.g., the discourse on discrete and continuous random variables combined with the discourse on random variables). In my analysis, I explore both

manifestations through discussing the various mathematical discourses that are being combined at a horizontal level or the ones that are subsumed at the vertical level.

For commognitive conflict to occur, two discourses should be present which are incompatible either regarding the rules or the mediators used. This incompatibility signals that there is a discontinuity between the discourse that the learner is asked to engage with and the one he or she engages with. The discontinuity between discourses can occur between the

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of the discourses change, and the use of word and symbol becomes more specialised. This discontinuity can also occur between various

mathematical discourses within university mathematics. For example, at university mathematics, the students are exposed to multiple mathematical discourses and they are asked to engage with multiple ones at the same time (e.g., discourse of integers and discourse of reals). Also, various mathematical areas which to the students might seem unrelated are connected between them (e.g., the discourse of continuous random variables in Probability with the discourse of Calculus). These connections between the discourses occur both at a horizontal and vertical level depending on the nature of the discourses. However, for this meta-level learning to occur students have to go through commognitive conflicts. Students should be experiencing commognitive conflicts during their studies and their lecturers with their teaching practices are aiming to help them to overcome these. Initially, the students, when coming to university or generally when being faced with a new mathematical discourse they imitate their lecturers’ actions when engaging with the discourse. Thus, they present a ritualistic use of procedures and visual mediators of

discursive objects. However, this ritualistic use changes when the students become more experienced in the discourse and becomes an exploration. It is important to note that some rituals might remain rituals until a

commognitive conflict occurs which would illustrate to the discursant that the how or the when of the routine is no longer accepted by the other participants of the discourse.

In the next section, I discuss the characteristics of the school and university mathematical discourses.

3.2.2 University and school discourses in the context