ANÁLISIS MULTIVARIADO

We close this paper by proposing some implications and recommendations for future research and teaching based on our results.

Even though conjecturing and proving processes often take place in mathematics classrooms (e.g., Vidakovic & Martin, 2004; Yackel & Cobb, 1996) or in form of mathematical debates (Alibert & Thomas, 1991), the results of our topic modeling approach indicate that studies that conceptualize conjecturing and proving as socially-embedded activities are rare. Stylianides, G. J. et al. (2017) critically remarked that this perspective on conjecturing and proving is still poorly evolved and not yet coherent. Research on how individual proving ideas develop in a social context appear to be underrepresented. We claim that research in this area has to be extended and that an alternative perspective that takes individual and social characteristics of conjecturing and proving into account should be adopted.

Most of the studies described in this review have used exploratory research approaches (such as grounded theory) to understand the processes that are needed to successfully construct conjectures or generate proofs. These studies provide a fruitful qualitative basis for future research directions and for the formulation of hypotheses about promising conjecturing and proving processes. For instance, based on the findings of the study by Ellis et al. (2017) and of the study by Weber and Alcock (2004), it can be hypothesized that students who choose and use examples purposefully (in the way that they systematically vary examples, search for

Study I

similar mathematical structures, and build formal representations out of the examples) are more likely to be successful in producing semantic proofs than students who only pick some random examples. However, confirmatory research about conjecturing and proving processes is still missing. We claim that a quantitative validation of the results found in the qualitative studies would facilitate a more generalizable picture of conjecturing and proving processes. Furthermore, the findings of our in-depth analysis support the observation by Dawkins and Karunakaran (2016) that there exists a large number of studies on proof-orientated behaviour that make content- and context-independent claims about promising conjecturing and proving processes. Discussing conjecturing and proving processes in a content- and context- independent way can be criticised on the grounds that the framing of research questions and the methodological choice of collecting and analysing data may non-trivially influence the nature of the phenomena observed and thus the studies’ findings. For instance, we assume that students or mathematicians who receive support (cf. Komatsu, 2011), who have no time restrictions (cf. Herlina & Batusangkar, 2015; Savic, 2015a), who work on geometry tasks (cf. Küchemann & Hoyles, 2006), or who are allowed to use lecture notes (cf. Selden, A. et al., 2010) may behave differently than those who get no assistance, who work with time-limits (cf. Lin, F.-L. et al., 2004), or who have to solve analysis tasks (cf. Kidron & Dreyfus, 2014).We intend to sensitize the research community on the role that particular measurement methods or other context factors may have on the studies’ results about conjecturing and proving processes.

By distinguishing separate but related categories of sub-goals and process characteristics of conjecturing and proving, our framework not only highlights the complexities associated with conjecture generation and proof construction, but also offers a way to understand how and why the occurrence of specific process characteristics may increase the probability of being successful. The sub-goals describe the intermediate steps that are considered to be necessary for generating interesting conjectures and constructing valid proofs. The process characteristics are observable and assumed to be potential indictors for the success within a certain intermediate step. As the intermediate steps such as exploring the problem situation or producing understanding of the proof have been listed as (latent) sub-goals in several studies, mathematics educators and researchers will need to operationalize these constructs. The process characteristics we identified may be used to operationalize the sub-goals within conjecturing and proving, and therefore may be valuable for analysing and assessing students’ conjecturing and proving processes. Furthermore, the proposed framework may be adapted for teaching purposes. We suggest that teachers and lecturers should introduce the process characteristics in combination with the associated intended sub-goals that might be accomplished by employing them. Pointing to the different types of process characteristics can give students insights into how the sub-goals considered necessary for success may be the

Study I

consequences of the appearance of certain process characteristics during the conjecturing and proving processes. Even though it is hard to determine the relative importance of each of these sub-goals respectively process characteristics of conjecturing and proving, as the types of processes successful provers engage in may vary according to different contexts, we claim that our framework may provide guidance for enhancing the learning and teaching of conjecturing and proving in (undergraduate) mathematics classes and for systematically analysing students’ proving behaviour.

A rating scheme for assessing process characteristics of collaborative conjecturing and proving 6 A rating scheme for assessing process characteristics of collaborative

conjecturing and proving

The summary and findings of our literature review show a comparatively detailed picture of potential relationships between (collaborative) conjecturing and proving processes and proof performance. We observe a large body of qualitative research offering diverse hypotheses about relevant sub-processes and process characteristics of conjecturing and proving, but little systematically generated evidence about the importance of the hypothesized sub-processes and process characteristics. Therefore, we see the need for empirically investigating the impact of a set of process characteristics of collaborative conjecturing and proving on the quality of the resulting product.

One of the central goals of this dissertation was develop an instrument to describe and analyse collaborative conjecturing and proving processes along several theory-based process characteristics inferred from the mathematics educational as well as the psychological and the Learning Sciences literature. A high inference coding scheme was designed, based on existing guidelines and an extensive literature search. But beforehand, a theoretical excurse is presented to illustrate the considerations that guided the development of this coding scheme. This chapter describes the decisions that had been made before the data collection and data coding was realised. We demonstrate how process characteristics were chosen and operationalized, present the whole rating scheme, and provide an overview about the rater- training. We complete this chapter by illustrating how the rating scheme was applied in our empirical studies and which further instruments have been used to assess undergraduate students’ mathematical argumentation skills.

6.1 Why using “real-time” recordings and high inference coding strategies to