This study extends prior research in several ways: (1) It systematizes several individual- mathematical and social-discursive process characteristics from previous studies that have been reported in combination with successful conjecturing and proving, or argumentation outcomes. (2) It provides a rating scheme for assessing students’ collaborative conjecturing and proving processes. High inference rating strategies that worked effectively in the medical or educational domain (Gartmeier et al., 2015; Seidel, 2005) were successfully adapted and transferred. (3) Substantial standard deviations, the high inter-rater agreement as well as the consistency within the two dimensions showed that our approach is feasible in principle: The developed rating-scheme comprises seven process characteristics of collaborative conjecturing and proving that could be reliably measured and analysed with regard to the within-dyad-similarities and the internal structure of the observed characteristics. Furthermore, our rating scheme allows for a direct assessment of collaborative conjecturing and proving processes, since it does not require the transcription of students’ dialogues. It is time efficient and, from an educational perspective, it may be used for instructional purposes. Tutors may use this rating scheme, if they are taught how to apply it to identify where support is needed and which aspects they have to encourage most. As the rating scheme is largely content- neutral, it may be adapted to a variety of tasks. (4) Regarding the within-cluster variance we observed relatively high values for some process characteristics pointing out that it is necessary to consider each individual’s contributions separately, not only at the dyad level, when evaluating collaborative conjecturing and proving processes. (5) On the other hand, for most of the process characteristics, especially at the second half of the collaborative working process, the within-cluster variance was quite low. Analyses that do not take this clustering into account may result in underestimation of standard errors and overestimation of statistical significance (cf. Lee, V. E., 2000). (6) Finally, this study investigated the empirical structure of individual-mathematical and social-discursive process characteristics of collaborative conjecturing and proving. Results indicated that collaborative conjecturing and proving processes can be conceptualized as a two-dimensional construct. As a practical consequence, in order to encourage students’ collaborative conjecturing and proving skills, it would be useful to design learning environments that provide support for both components (cf. Vogel, et al., 2017).
Study II
The findings of this study delivered empirical evidence that our rating-scheme constitutes a valuable instrument for analysing students’ collaborative conjecturing and proving processes from an individual-mathematical and social-discursive perspective. It has potential for further usage in research and teaching. Since the validity criterion plays an important role in the development of instruments (e.g., Blömeke, et al., 2015), future studies should investigate which of the process characteristics we have extracted from the literature actually predict the quality of the resulting product of collaborative conjecturing and proving processes. Moreover, the rating scheme may be applied to detect effects of interventions or scaffolds that systematically foster one of the two quality facets. Making it useable for practitioners could also be a next step, as the rating scheme may support instructors and tutors while monitoring and supporting students’ proof processes. It may be used to help tutors or lecturers learn to notice and interpret important characteristics of students’ collaborative conjecturing and proving processes and thus, to enhance their “professional vision” for these processes (Goodwin, 1994; van Es & Sherin, 2002).
Study III 8 Study III
Generating structurally sound and accurate arguments: Key characteristics of successful collaborative conjecturing and proving processes
8.1 Abstract
There seems to be general consensus on the assumption that specific process characteristics of conjecturing and proving are crucial for their success, even though they may not emerge directly in the final product (the formulated conjecture and the constructed proof). Based on the literature, we have selected four individual-mathematical and three social-discursive process characteristics of collaborative conjecturing and proving that are considered essential for the successful production of conjectures respectively proofs. We empirically investigated to which extent these characteristics predicted the quality of the final conjecture and proof. Furthermore, we were interested in studying their relations to students’ prior knowledge on proof. Therefore, we examined the interaction of N=98 prospective mathematics university students working collaboratively on a conjecturing and proving task. Results indicate that generating structurally sound and accurate arguments during the collaborative discourse are key characteristics of successful conjecturing and proving processes. Furthermore, this study shows that individual-mathematical process characteristics mediate the relation between students’ prior knowledge on proof and the quality of their resulting conjectures and proofs. We present more detailed analyses of the process characteristics and their effects on the specific quality aspects of the final product and discuss implications for teaching and research.
8.2 Introduction
Inquiring mathematical conjectures, finding supporting arguments, and discussing them with peers are the daily work of mathematicians and hence considered as a substantial objective of mathematics education (e.g., Mariotti, 2006; Stylianides, A. J., Bieda, & Morselli, 2016). However, developing these complex skills is challenging for most students (Heinze et al., 2005; e.g., Heinze & Reiss, 2004; Moore, 1994; Selden, A. & Selden, 2008). In other disciplines such as politics, psychology, or philosophy, students also have to be able to generate evidence- based arguments.
This might be one reason why there is a widespread trend towards describing and analysing students’ behaviour when solving argumentation tasks. Researchers from several disciplines focus their attention on investigating what the crucial aspects of scientific reasoning and argumentation are (Fischer et al., 2014). Learning more about this may help to discover the causes for students’ main difficulties and to design adequate scaffolds. Of course, it is possible to come up with characteristics of “good” argumentation processes by theoretical analyses or
Study III
by observing students’ argumentative activities. However, the success of a good conjecturing and proving process is determined by its outcome, the final conjecture respectively proof. In this contribution, we are interested in the relations between characteristics of the processes, the final conjecture and proof, and the students’ prior knowledge on proof (representing their individual (learning-) prerequisites). Along with other researchers, we conceptualized conjecturing and proving processes as specific types of argumentative activities (e.g., Selden, A. & Selden, 2013b) and differentiate between an individual-mathematical and social- discursive component of argumentation (Kollar et al., 2014). We reviewed literature from mathematics education and educational psychology as well as the Learning Sciences to find potentially predictive process characteristics of collaborative conjecturing and proving. There are several motivations behind our approach to investigate the relations between the processes, the final product, and students’ prerequisites. First, analysing the relations between process characteristics and the resulting product may allow to uncover key characteristics for success. Secondly, it might help to detect specific challenges of students’ and thereby to find out where support is required. Based on this knowledge, new scaffolds may be designed. If the relation is due to general (learning-) prerequisites, developing scaffolds with the focus on the processes may become less important. Furthermore, we expect to receive information that may be used for diagnostic purposes to identify dyads that will run into trouble. One may also be interested in studying the relations between process characteristics and prerequisites in order to be able to diagnose more general traits such as prior knowledge. However, the main objective of this study was to investigate the effects of individual-mathematical and social- discursive process characteristics on the quality of the resulting conjecture respectively proof by controlling for prior knowledge on proof.
The first part of this paper describes different process characteristics of collaborative conjecturing and proving and hence provides an overview about what counts as good collaborative conjecturing and proving processes from a theoretical point of view. In the second part, we will present the results of an empirical analysis capturing the relations between different individual-mathematical and social-discursive process characteristics, the quality of the resulting conjecture and proof (as the outcome of students’ collaborative conjecturing and proving processes), and their prior knowledge on proof.