Forschung zum Mathematischen Argumentieren – Ein deskriptiver Review von PME Beiträgen

Die Mathematik ist eine beweisende Wissenschaft. Mathematisches Argumentieren und Beweisen (MA&B) sind zentrale Aktivitäten der Mathematik und gehören zu den wichtigsten zu erlernenden Fähigkeiten im schulischen und universitären Bereich (Brunner, 2014). Gerade in der Sekundarstufe wurde der Fokus auf Argumentieren in den letzten Jahren weltweit durch curriculare Änderungen verstärkt, entsprechend ist MA&B auch innerhalb der Didaktik der Mathematik wieder zunehmend in den Forschungsmittelpunkt gerückt. In diesem Beitrag wird Argumentieren im Sinne Toulmins relativ offen verstanden, insbesondere werden auch nicht deduktives Schlussfolgern und Beweisen als Spezialfälle des Argumentierens verstanden (Reiss & Ufer, 2009)

Students’ knowledge about proof and handling proof

Research has highlighted that students’ have problems regarding mathematical proof. In part, these have been connected to deficits in their knowledge about proof and handling proof. However, empirical data on students’ knowledge about proof and handling proof throughout secondary education is so far missing. Further, it is unclear if there is a connection between concept- and action-oriented knowledge about proof and handling proof. To address these gaps, an empirical study was conducted, investigating the knowledge about proof and handling proof concerning proof principles of N = 456 students in grade 8 to 11. Results indicate that (i) only concept-oriented knowledge significantly increases throughout secondary education and (ii) there is only little relation between concept- and action-oriented knowledge

Supporting Mathematical Argumentation and Proof Skills: Comparing the Effectiveness of a Sequential and a Concurrent Instructional Approach to Support Resource-Based Cognitive Skills

An increasing number of learning goals refer to the acquisition of cognitive skills that can be described as 'resource-based,' as they require the availability, coordination, and integration of multiple underlying resources such as skills and knowledge facets. However, research on the support of cognitive skills rarely takes this resource-based nature explicitly into account. This is mirrored in prior research on mathematical argumentation and proof skills: Although repeatedly highlighted as resource-based, for example relying on mathematical topic knowledge, methodological knowledge, mathematical strategic knowledge, and problem-solving skills, little evidence exists on how to support mathematical argumentation and proof skills based on its resources. To address this gap, a quasi-experimental intervention study with undergraduate mathematics students examined the effectiveness of different approaches to support both mathematical argumentation and proof skills and four of its resources. Based on the part-/whole-task debate from instructional design, two approaches were implemented during students' work on proof construction tasks: (i) a sequential approach focusing and supporting each resource of mathematical argumentation and proof skills sequentially after each other and (ii) a concurrent approach focusing and supporting multiple resources concurrently. Empirical analyses show pronounced effects of both approaches regarding the resources underlying mathematical argumentation and proof skills. However, the effects of both approaches are mostly comparable, and only mathematical strategic knowledge benefits significantly more from the concurrent approach. Regarding mathematical argumentation and proof skills, short-term effects of both approaches are at best mixed and show differing effects based on prior attainment, possibly indicating an expertise reversal effect of the relatively short intervention. Data suggests that students with low prior attainment benefited most from the intervention, specifically from the concurrent approach. A supplementary qualitative analysis showcases how supporting multiple resources concurrently alongside mathematical argumentation and proof skills can lead to a synergistic integration of these during proof construction and can be beneficial yet demanding for students. Although results require further empirical underpinning, both approaches appear promising to support the resources underlying mathematical argumentation and proof skills and likely also show positive long-term effects on mathematical argumentation and proof skills, especially for initially weaker students

Students’ learning growth in mental addition and subtraction: results from a learning progress monitoring approach

The purpose of this study was to measure and describe students’ learning development in mental computation of mixed addition and subtraction tasks up to 100. We used a learning progress monitoring (LPM) approach with multiple repeated measurements to examine the learning curves of second-and third-grade primary school students in mental computation over a period of 17 biweekly measurement intervals in the school year 2020/2021. Moreover, we investigated how homogeneous students’ learning curves were and how sociodemographic variables (gender, grade level, the assignment of special educational needs) affected students’ learning growth. Therefore, 348 German students from six schools and 20 classes (10.9% students with special educational needs) worked on systematically, but randomly mixed addition and subtraction tasks at regular intervals with an online LPM tool. We collected learning progress data for 12 measurement intervals during the survey period that was impacted by the COVID-19 pandemic. Technical results show that the employed LPM tool for mental computation met the criteria of LPM research stages 1 and 2. Focusing on the learning curves, results from latent growth curve modeling showed significant differences in the intercept and in the slope based on the background variables. The results illustrate that one-size-fits-all instruction is not appropriate, thus highlighting the value of LPM or other means that allow individualized, adaptive teaching. The study provides a first quantitative overview over the learning curves for mental computation in second and third grade. Furthermore, it offers a validated tool for the empirical analysis of learning curves regarding mental computation and strong reference data against which individual learning growth can be compared to identify students with unfavorable learning curves and provide targeted support as part of an adaptive, evidence-based teaching approach. Implications for further research and school practice are discussed