510 Mathematik
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We consider groups defined by non-empty balanced presentations with the property that each relator is of the form, where x and y are distinct generators and is determined by some fixed cyclically reduced word that involves both a and b. To every such presentation we associate a directed graph whose vertices correspond to the generators and whose arcs correspond to the relators. Under the hypothesis that the girth of the underlying undirected graph is at least 4, we show that the resulting groups are non-trivial and cannot be finite of rank 3 or higher. Without the hypothesis on the girth it is well known that both the trivial group and finite groups of rank 3 can arise.
We report on a study on syllogistic reasoning conceived with the idea that subjects' performance in experiments is highly dependent on the communicative situations in which the particular task is framed. From this perspective, we describe the results of Experiment 1 comparing the performance of undergraduate students in 5 different tasks. This between-subjects comparison inspires a within-subject intervention design (Experiment 2). The variations introduced on traditional experimental tasks and settings include two main dimensions. The first one focuses on reshaping the context (the pragmatics of the communication situations faced) along the dimension of cooperative vs. adversarial attitudes. The second one consists of rendering explicit the construction/representation of counterexamples, a crucial aspect in the definition of deduction (in the classical semantic sense). We obtain evidence on the possibility of a significant switch in students' performance and the strategies they follow. Syllogistic reasoning is seen here as a controlled microcosm informative enough to provide insights and we suggest strategies for wider contexts of reasoning, argumentation and proof.
Sprache kommt im Mathematikunterricht eine tragende Rolle zu. Schülerinnen und Schüler mit und ohne sonderpädagogischen Förderbedarf können jedoch Schwierigkeiten beim Erfassen von in Schriftsprache dargebotener Informationen haben. Das Ziel dieser Studie besteht darin, herauszufinden, wie solche Lesebarrieren reduziert werden können. Ein erleichterter Zugang zu Arbeitsaufträgen könnte durch den Einsatz von Leichter Sprache und Piktogrammen erreicht werden. Auch die Visualisierung kompletter Sätze durch Fotos könnte hilfreich sein. Zentrale Fragestellung dieser Studie ist, inwieweit die Verwendung von Leichter Sprache bzw. Leichter Sprache und Piktogrammen oder Fotos die Performanz bei der Bearbeitung mathematischer Aufgaben verbessert. Die Stichprobe bestand aus Schülerinnen und Schülern mit einem sonderpädagogischen Förderbedarf im Bereich Lernen (N = 144) und Lernenden ohne sonderpädagogischen Förderbedarf (N = 159). Die Schülerinnen und Schüler bearbeiteten Aufgaben, in welchen es um die Einführung des Bruchzahlbegriffs ging, in einer der folgenden Versionen: Leichte Sprache (EG 1), Leichte Sprache +Piktogramme (EG 2), Leichte Sprache +Fotos (EG 3) oder keine Unterstützungsmaßnahme (EG 4). Die Lesefertigkeit und der IQ der Lernenden wurde vor der Bearbeitung der Aufgaben erhoben, um vergleichbare Experimentalgruppen bilden zu können. Es zeigte sich ein signifikanter Effekt der verschiedenen Bedingungen auf die Aufgabenbearbeitung. Eine Post-Hoc-Analyse verdeutlichte, dass die Signifikanz aus dem Unterschied zwischen EG 3 und EG 4 resultierte. Die Schülerinnen und Schüler in EG 3 bearbeiteten die Aufgaben erfolgreicher als die Lernenden in EG 4.
Although raised in the early days of research on teacher noticing, the question of context specificity has remained largely unanswered to this day. In this study, we build on our prior research on a specific aspect of noticing, namely teachers’ analysis of how representations are dealt with in mathematics classroom situations. For the purpose of such analysis, we examined the role of context on the levels of mathematical content area and classroom situation. Using a vignette-based test instrument with 12 classroom situations from the content areas of fractions and functions, we investigated how teachers’ analyses regarding the use of representations are related concerning these two mathematical content areas. Beyond content areas, we were interested in the question of whether an overarching unidimensional competence construct can be inferred from the participants’ analyses of the different individual classroom situations. The 12 vignettes were analysed by N = 175 secondary mathematics teachers with different degrees of teaching experience and their written answers provided the data for this study. Our findings show that the data fit the Rasch model and that all classroom situations contributed in a meaningful way to the competence under investigation. There was no significant effect of the mathematical content area on the participants’ analyses regarding the use of multiple representations. The results of the study indicate that explicitly considering questions of context can strengthen research into teacher noticing.
This thesis presents the results of a series of studies (on syllogisms, on the interpretation of mathematical statements and on probabilistic thinking) conducted with the idea that different, legitimate kinds of reasoning are used by humans in a contextual way, and that therefore no single logic (e.g., classical logic) can be expected to account for this diversity.
The crucial role of interpretation is highlighted, showing how intensional and extensional reasoning may be mobilized according to it. In particular, in communication settings, this depends on our adoption of a cooperative, credulous disposition, or on the contrary, of an adversarial, sceptical one.
In reasoning about mathematics in an educational setting, students (and teachers) may be enrolled in a back and forth between believing, doubting, making sense, giving arguments and proving. These changes in dispositions imply changes in the logics used. All the studies presented show, in different ways, evidence for cooperative, intensional reasoning and, in some cases, the possibility of a shift towards the acquisition of an extensional view. This suggest that if we expect as educators the adoption of specific norms and the development of reasoning skills from students, we need first to know well what the point of departure is where they are, and that it is often not at all “irrational”.