LA INTERPRETACIÓN DEL MODELO

Based on the fact that research on conjecturing and proving is growing rapidly, reflecting the importance of conjecturing and proving in mathematics education and the relevance of knowing the processes that are assumed or reported to be crucial for the success (cf. Mariotti, 2006; Stylianides, G. J. et al., 2017), our review had two main objectives: Firstly, we systematized the literature on conjecturing and proving using a topic modelling method. The algorithm returned 17 topics and a list of words composing those topics. By examining the articles and research reports that represent these topics, we have discerned that the research clustered to one topic often shares a common perspective on conjecturing and proving. Regarding this observation, we have been able to replicate the three perspectives on proving outlined by Stylianides, G. J. et al. (2017), namely the proving as problem-solving, proving as convincing, and proving as a socially-embedded activity, within the literature on conjecturing and proving. Yet, we have identified a new perspective, the discovery perspective on conjecturing and proving, within the literature and proposed a more fine-grained categorization of the perspectives introduced by Stylianides, G. J. et al. (2017). The discovery perspective emphasizes that conjecturing and proving can be used as a means for exploring, discovering, and inventing new mathematical results (cf. Villiers, 1999), especially in the field of geometry. Our analysis has shown, that the problem-solving perspective can be sub-divided into smaller categories such as the problem-solving perspective with a specific focus on strategy use, on affective and cognitive resources, or on approaches to overcome proving impasses. Studies that share the convincing perspective appear to be embedded in a specific educational context, either in the university or in the school context. Furthermore, our topic model indicates that the proving as a socially-embedded activity perspective is less present in the literature on conjecturing and proving. The articles and research reports of our document collection consist only of a small proportion of words from the “social/ collective argumentation” topic that typifies the social perspective on conjecturing and proving. This result is in line with Balacheff’s (1988) critique that there is a too strong emphasize on the logical side of proof, disregarding its social importance as a means for communication. Stylianides, G. J. et al. (2017) also remarked that the proving as a socially-embedded activity perspective is less developed than the problem- solving and convincing perspective. Even though, we were open to consider further perspectives on conjecturing and proving, we have to confirm the observation of Stylianides, G. J. et al. (2017) that some articles and research reports do not fully fit within one of the three (in our case four) perspectives. However, our topic modeling approach has allowed us to systematize the literature on conjecturing and proving and to draw conclusions about the

Study I

presence of specific topics that have been discussed in the literature in the context of conjecturing and proving. We have summarized the methodological orientations of the underlying studies of the most representative articles and research reports for each topic. This summary indicates that existing studies provide a rich qualitative basis on conjecturing and proving, but that quantitative findings resulting from the observations of larger populations are rare.

Secondly, we have analysed the most representative articles and research reports for each topic with regard to their claims and empirical findings about promising conjecturing and proving processes. We noticed that in the literature different types of processes related to conjecturing and proving have been presented and that the ways in which they have been described varies. Terms such as “exploration” (e.g., Komatsu et al., 2014; Mejía-Ramos & Inglis, 2009; Ozgur et al., 2017), “refinement of conjecture” (e.g., Komatsu, 2011, 2016), “producing understanding” (e.g., Ellis et al., 2017; Furinghetti & Morselli, 2009; Zazkis et al., 2015), or “justification” (Mejía-Ramos & Inglis, 2009; Zaslavsky et al., 2012) that are partially unprecise and difficult to operationalize have been used in numerous studies to delineate the intermediate steps that are needed to generate conjectures and to construct proofs. In some studies, they have been described rather as latent constructs than as directly observable processes (e.g., Küchemann & Hoyles, 2006; Mejía-Ramos & Inglis, 2009). However, we regarded these processes as necessary intermediate steps, as sub-goals within conjecturing and proving processes, which themselves require further, more fine-grained processes and which may be achieved in different ways. In total, we have been able to infer eleven sub-goals from the literature on conjecturing and proving. Some of these sub-goals, such as developing a strong understanding of the statement to be proved/ estimation of the truth or finding an adequate representation for the proof and an adequate proving strategy, are comparable to the four phases outlined in Polya (1945). Others, such as inventing and formulating new conjectures or refining existing conjectures, structuring and organizing inferences, or communicating and presenting arguments, occur in the phase model of Boero (1999) in a similar way. Moreover, we identified new sub-goals within the literature on conjecturing and proving such as resolving fixations/ avoiding errors or generating a shared understanding, which are of particular importance in specific situations (e.g., when an impasse is reached) or contexts (e.g., when an argumentation is embedded in a social context). As we were interested in finding out which processes have been assumed or reported to be helpful in achieving one or more of these sub-goals, we have expanded our analysis. We searched for process characteristics of conjecturing and proving that have been considered (or may be interpreted as such) as indicators of how to successfully accomplish these sub-goals. Our search resulted in a broad set of process characteristics that reflect multiple different ways in which successful conjecturing and proving processes may be carried out and that have been perceived as

Study I

relevant from different perspectives on conjecturing and proving. Based on these results, we proposed a framework that takes both sub-goals and process characteristics of conjecturing and proving into account. By studying the literature with regard to the relationships between sub-goals and process characteristics, we found that the processes used to complete a specific sub-goal may vary widely. For instance, on one extreme, doing tasks unrelated to mathematics, taking a break, or going for lunch are processes that are assumed to be helpful in recovering from fixations (Savic, 2015a). At the other extreme, overcoming impasses may involve attempting to construct a counterexample, reflecting upon various proving techniques (Savic, 2015b), or testing the limitations of a conjecture (Ellis et al., 2013). However, we have derived from the literature that anticipatory and structural thinking (including the existence of a goal in mind when choosing and employing a specific process) are the two key aspects when attempting to achieve a sub-goal, as they can facilitate the achievement of a sub-goal or, if they lack, may hinder its achievement.