Artículo 2

The analytical lens used through this research process adapted and evolved as the study progressed. Perhaps the most important way in which this happened was to form a more inclusive notion of ritual and exploration, instead of viewing these routines as diametrically opposed. They are in fact mutually supportive in developing a mathematical discourse. It would have been naive to transpose learning into distinct and separated compartments. Learning or growing discourse happens on multiple levels with multiple influences. Rituals mark a natural tendency of human beings emphasising the social nature of learning. They are performed with others, and for others (Sfard, 2008). It is through the reification of rituals that learners can begin to explore mathematical objects. The coded data summarised as follows:

ii All PGs showed codes R2, R 3, and R11 as most frequent rituals. iii R2 followed by R3 was most frequent in PG1 and PG2.

157 The coded data showed that rituals overwhelm learner mathematical discourses on function. The most frequent ritual code appearing across all performance groups is R2. This relates to

discourse set to goals established by others, and speaks to the agency learners feel in their participation in the mathematical discourse. R3 is a code related to the mathematics being about process and transformation, was also significant. This level of discourse depicts participation in mathematics as following a set of rules, and largely rules that are set by others. This shows that formal mathematics is poorly embedded in learner discourse, making learner contributions easily changeable, and dependent on influences that are outside of mathematical justification. A certain consequence borne out in the data is that learners processed knowledge in discrete, disconnected pockets. Section 5.3.1 highlighted the general absence of an endorsed narrative for function and how this impacted exploration routines. This section permits a view of how this absence

influences the discursive rituals learners use.

This zoom out on rituals can provide a broad picture of the how the frequent rituals across performance groups impacted the ways learners participate in the mathematical discourse:

 learners would change their particular narrative or course based an alternate opinion offered by the other learner, and seldom sought justification from their partner;

 they relied on verbal and circumstantial prompts from their environment;

 they seldom substantiated, tested or reflected on the narratives or solutions at which they would arrive;

 the emphasis on maintaining and forming social relationship meant that the mathematical object was displaced from focus; and

 poorer-performing learners placed emphasis on memory and experience when dealing with objects.

The implications of ritualised discourse summarised above raises the important question as to how the rituals which learners show on the task support the reification of processes and features towards building the discourse of function.

Not all rituals (or explorations for that matter) are equal and these categories in existing literature were too broad. The study took the broader categories of ritual and exploration,

158 dissected the rituals and explorations as they occurred in the data, for defining characteristics of learner discourse. While the most frequent rituals present in learner discourses are evidenced in this section, albeit broadly, it became equally important to note codes that are absent, or less frequent. It is also apparent that groups of codes exist within the ritualised and exploratory codes assigned, which appear to make certain realisations possible, to appear as precursors to others, or to hinder further realisations. The intention was thus to check whether these codes or groups of codes facilitate transition to higher discursive levels. In light of this, the following two questions receive investigation here: first, can existing ritual codes be grouped to describe what may be required to transition from ritual to exploration? And second, by understanding the features of the ritualised routines, how can learners be assisted to transition to higher discursive levels?

The next 6 subsections begin the zoom-in view or the characterisation of rituals in relation to these questions. The analysis begins to look at the finer threads of learner utterances coded ritual to present a description of these. Using statistics arising from codes in the data, substantiated by extracts of learner talk, a picture of how the object is defined for learners begins to emerge. Commognition provides broader categories under which this characterisation

occurred, namely: goals of routines invoked; what learners talk about; the flexibility and applicability of the routine; who the utterances are addressed to; and the reason for the acceptance of a routine. Chapter 4 provided the detail of the codes listed under each of these broader categories. These codes, descriptors of exploratory and ritualised communication, mined the data to saturation, serving the need to build detail around ritual and exploration through frequent occurrences. The development of these codes offers extension to the broader commognitive categories. Arising out of the combination of the data available on this study, discursive literature and the strong theoretical base, the codes intend to build on the work available. They are not definitive. They are, instead, a means to look at learner routines and contrast these using a performance lens.