Mathematical conjecturing and proving

Most university courses in mathematics programs are characterized by a strong focus on the axiomatic nature of mathematics, and thus also on proof as the central scientific method of mathematics (Selden, A. & Selden, 2008). Lecturers write proofs on the blackboard, students attempt to demonstrate their understanding and skills by proving theorems on their own or in collaboration with others. However, there is often little systematic discussion in these courses on how new mathematical conjectures can be generated and on how proofs are constructed (Alcock, 2010). Students’ experiences with conjecturing and proving in schools or in university mathematics courses often lead them to “consider proof as a static product rather than a negotiated process that can help students justify and make sense of mathematical ideas” (Otten, Bleiler-Baxter, & Engledowl, 2017, p. 112). Yet, several authors (e.g., Epp, 2003; Savic, 2015a; Selden, A. & Selden, 2008) have hypothesized that often only little time can be devoted to illustrate students which strategies and processes may help to step through the proof construction process and to recover from proving impasses. Furthermore, the knowledge about what characterizes proof processes that lead to a successful outcome (i.e., an acceptable mathematical proof [according to local acceptance criteria]) is rare. To approach this issue, an extensive systematic literature search was conducted to summarize common claims and empirical findings about promising conjecturing and proving processes. 126 articles that focussed on conjecturing and proving were clustered using a topic modeling method. The algorithm identified 17 different topics. The most representative papers for each topic, in total 45 papers, were qualitatively analysed with regard to their research perspectives on which they were based and their claims and findings about the processes that are needed to successfully generate conjectures and construct proofs. This combination of statistical clustering and qualitative analyses allowed a systematic categorization of claims and empirical findings about successful conjecturing and proving processes in the literature. Based on this review, a set of characteristics of conjecturing and proving processes, that are assumed or reported to be crucial for success, is proposed. For the further analysis of such process characteristics, we started from a model differentiating students’ prerequisites they bring to bear on the proving situation, the conjecturing and proving processes they engage in, and the quality of the resulting product. The main question of the empirical work in this dissertation was, which process characteristics influence the quality of the final product (the formulated conjecture and constructed proof), and in which way they mediate the impact of students’ prerequisites on this product. Specifically, we distinguished between individual-mathematical and social-discursive process characteristics of conjecturing and proving. These process characteristics were extracted from prior research in mathematics education or in educational psychology or in the Learning Sciences. The central aim of this dissertation was to develop an instrument for assessing (prospective undergraduate) mathematics students’ conjecturing and proving processes in collaborative situations. A high-inference rating scheme with seven scales, based on theoretical considerations and on rating guidelines adapted from educational research was designed. The rating scheme was evaluated in a study with N=98 prospective undergraduate students working in dyads on an open-ended conjecturing and proving task. The results of the empirical study with regard to the basic analyses showed that collaborative conjecturing and proving processes could be rated with sufficient reliability and that the structure of the data corresponded to the underlying theoretical assumption that two dimensions, one related to individual-mathematical and one related to social-discursive process characteristics can be distinguished. The in-depth analyses pointed out that individual-mathematical process characteristics were predictive for the quality of the resulting product and mediated the relation between prerequisites (students’ prior knowledge on proof) and the quality of the product. In this way, the dissertation contributes to the scientific debate on how to assess (mathematical argumentation) skills (e.g., Blömeke, Gustafsson, & Shavelson, 2015; Koeppen, Hartig, Klieme, & Leutner, 2008) and provides theoretical and empirical insights on individual-mathematical and social-discursive process characteristics that describe the quality of collaborative conjecturing and proving processes

The Lantern Vol. 76, No. 1, Fall 2008

• Cruel • A Night in Three Parts • The Moment I Said It • To Know • I Will Never Skipskipskip a Rock • The Ravine • Untitled • Skeleton • Midnight Letter • Where Children Come From • Orphan of War • Ciega / Mezquita • The Other Side • Those Dancing Days are Gone • Cycling • The 2nd of July • The Tantric Semantics of Studying Abroad • A Three-Part Study in Musical Relations • Amway Man • Hard Luck Investigator • Spring • Interview With Poet Eleanor Wilnerhttps://digitalcommons.ursinus.edu/lantern/1173/thumbnail.jp

How to combine collaboration scripts and heuristic worked examples to foster mathematical argumentation – when working memory matters

Mathematical argumentation skills (MAS) are considered an important outcome of mathematics learning, particularly in secondary and tertiary education. As MAS are complex, an effective way of supporting their acquisition may require combining different scaffolds. However, how to combine different scaffolds is a delicate issue, as providing learners with more than one scaffold may be overwhelming, especially when these scaffolds are presented at the same time in the learning process and when learners’ individual learning prerequisites are suboptimal. The present study therefore investigated the effects of the presentation sequence of introducing two scaffolds (collaboration script first vs. heuristic worked examples first) and the fading of the primarily presented scaffold (fading vs. no fading) on the acquisition of dialogic and dialectic MAS of participants of a preparatory mathematics course at university. In addition, we explored how prior knowledge and working memory capacity moderated the effects. Overall, 108 university freshmen worked in dyads on mathematical proof tasks in four treatment sessions. Results showed no effects of the presentation sequence of the collaboration script and heuristic worked examples on dialogic and dialectic MAS. Yet, fading of the initially introduced scaffold had a positive main effect on dialogic MAS. Concerning dialectic MAS, fading the collaboration script when it was presented first was most effective for learners with low working memory capacity. The collaboration script might be appropriate to initially support dialectic MAS, but might be overwhelming for learners with lower working memory capacity when combined with heuristic worked examples later on