Wie viel Grenzwert braucht der Mensch? – Unendlichkeit dynamisch und statisch begreifen

Im Alltag bezeichnet ein Grenzwert eine real messbare Größe, die aus rechtlichen Gründen nicht überschritten werden sollte (z. B. !- oder Feinstaub-Grenzwert). Demgegenüber zeichnet sich der mathematische Grenzwertbegriff gerade dadurch aus, ein theoretisches Gedankenkonstrukt zu sein, welches einem unendlichen Prozess ein idealisiertes Ergebnis zuordnet. Aus didaktischer Perspektive kann somit die Frage aufgeworfen werden, inwiefern eine rein theoretische Auseinandersetzung mit Grenzwerten im Mathematikunterricht überhaupt legitimiert werden kann

Binnendifferenzierung und pädagogisches Handeln – Entwicklungsforschung an einer Brennpunktschule

In diesem Beitrag stellen wir ein Unterrichts- und Schulentwicklungsprojekt an einer sogenannten Brennpunktschule einer Großstadt vor, das sich zum Ziel setzt, gemeinsam mit den Lehrpersonen aus der Praxis ein Konzept der Binnendifferenzierung zu erarbeiten, welches zugleich das pädagogische Handeln stärker berücksichtigt als herkömmliche Ansätze. Im Zentrum unserer bisherigen 1,5-jährigen Arbeit steht dabei die Entwicklung der Schule bezogen auf den Mathematikunterricht in stark heterogenen Gruppen. In der (mathematik-)didaktischen Forschungslandschaft sind einige praxisbezogene Ansätze und Konzepte binnendifferenzierenden Unterrichts entstanden (z. B. Leuders & Prediger 2012). Eine derartige Schule ist allerdings in ihrer (Problem-)Struktur so komplex, dass Konzepte für Unterricht an die spezifischen Strukturen anknüpfen müssen, wenn sie erfolgreich sein sollen. Um dies zu erreichen, stellen wir uns der Komplexität in einem systemischen Zugang, welcher in einem Handlungsforschungsansatz auf evolutionäre Innovation (Reinmann 2005) ausgerichtet ist. In der praktischen Arbeit äußert sich dies in einer Form des fachspezifischen Unterrichtscoachings (Staub & Kreis 2013). Dabei übernehmen die Forscherinnen im Entwicklungs- und Unterrichtsprozess Mitverantwortung für die Gestaltungsarbeit. Die Datenerhebung erfolgt durch ethnographische Feldnotizen, dokumentarische Methoden und Interviews. Dabei liegt der Fokus auf der Erarbeitung eines Konzeptes zum binnendifferenzierten Mathematikunterricht unter den spezifischen Gegebenheiten und der Dokumentation des Entwicklungsprozesses aus verschiedenen Perspektiven – vornehmlich einer mathematikdidaktischen, pädagogischen und soziologischen Perspektive. Ziel des Artikels ist es, die Intention und Richtung der bisherigen Entwicklungsforschungsarbeit aufzuzeigen

Measuring Pre-service Primary Teachers’ Shame in Mathematics—a Comprehensive Validation Study

Emotions play an essential role in educational processes. Previous research has mainly dealt with achievement emotions which are experienced in specific situations such as exams or learning situations in mathematics (e.g. enjoyment or anxiety). Some achievement emotions are rather experienced in social contexts in mathematics and are closely related to the self. These emotions such as shame are assumed to be also relevant for mathematics achievement. However, a reliable and valid instrument is missing to measure shame in mathematics. Validity evidence for the newly developed Shame in Mathematics Questionnaire (SHAME-Q) was collected in three studies with pre-service primary teachers. Study 1 investigated the content validity by conducting a systematic expert panel study. Study 2 and study 3 examined with two different samples the factorial structure and relations to other constructs in terms of discriminant (enjoyment) and convergent (anxiety) validity as well as to pre-service teachers’ grade in school mathematics, their intention to teach mathematics at school, and gender. The data supported strongly the validity assumptions as well as reliability and parsimony of the instrument. Psychometric limitations of SHAME-Q and applicability of the questionnaire are discussed.Peer Reviewe

Towards a research base for implementation strategies addressing mathematics teachers and facilitators

Implementing innovations in mathematics education relies on scaling up professional development (PD) for mathematics teachers. Within the German Center of Mathematics Teacher Education (DZLM), the complexities involved in scaling-up processes are addressed with three implementation strategies: a material, a personnel, and a systemic strategy. For each of them, the research base is empirically established by design research and intervention studies. The personnel strategy focuses on preparing teachers for innovative teaching approaches and on preparing PD facilitators for conducting PDs on the content involved. The empirical research required for creating a research base for the personnel strategy investigates what teachers and facilitators need to know about their learners (students on the classroom level, teachers on the PD level) with regard to a specific (PD) content. The main focus of the material strategy is to provide support for teachers and facilitators by adaptable curriculum materials for classrooms and PD sessions. The empirical research on the material strategies focuses on how such material needs to be constructed to ensure both flexibility and fidelity for impactful quality implementation. The systemic strategy involves considering the systemic contexts on each level and cooperating with the respective stakeholders (e.g., school principals or regional PD authorities). The corresponding empirical research focuses on the interplay of processes and systemic conditions. In this paper, we present the implementation strategies on different levels, and typical research approaches and results for strengthening their empirical and theoretical bases.Projekt DEALPeer Reviewe

Pre-service primary teachers’ shame experiences during their schooling time: characteristics and effects on their subject-choices at university

Emotions play an essential role in pre-service teachers’ competence development, particularly in mathematics. However, the emotion of shame in mathematics has been largely neglected so far. This article deals with shameful experiences of pre-service primary school teachers during their mathematical education at school and the various effects of shame on their decision to study mathematics as a subject at university. The research consists of a qualitative and a quantitative study with 311 prospective primary school teachers who responded to a survey about their experiences of shame in mathematics at school when they were students. Results of the qualitative study emphasize the different experiences in mathematics during the school years and reveal the characteristics of these situations, for example, social exposure or competition games. In the quantitative study, pre-service primary teachers’ subject choice was analyzed in relation to their experienced shame in mathematics at school. Results reveal that shame experienced at school has effects on the initial choice in favor of mathematics at university. Implications for primary teacher education are finally discussed.Peer Reviewe

Conceptualizing content-related PD facilitator expertise

Studies of facilitators of professional development (PD) for mathematics teachers have been increasing in order to improve their preparation for conducting PD. However, specifications of what facilitators should learn often lack a conceptualization that captures facilitators’ expertise for different PD content. In this article, we provide a framework for facilitator expertise that is in line with current conceptualizations but makes explicit the content-related aspects of such expertise. The framework for content-related facilitator expertise combines cognitive and situated perspectives and allows unpacking different components at the PD level and the classroom level. Using two illustrative cases of different PD content (probability education in primary school and language-responsive mathematics teaching in secondary school), we exemplify how the framework can help to analyze facilitators’ practices in content-related ways in a descriptive mode. This analysis reveals valuable insights that support designers of facilitator preparation programs to specify what facilitators should learn in a prescriptive mode. We particularly emphasize the importance of working on content-related aspects, unpacking the PD content goals into the content knowledge and pedagogical content knowledge elements on the classroom level and developing facilitators’ pedagogical content knowledge on the PD level (PCK-PD), which includes curricular knowledge, as well as knowledge about teachers’ typical thinking about a specific PD content. Situated learning opportunities in facilitator preparation programs can support facilitators to activate these knowledge elements for managing typical situational demands in PD.German-Israeli Foundation for Scientific Research and Development http://dx.doi.org/10.13039/501100001736Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659Deutsche Telekom Stiftung http://dx.doi.org/10.13039/501100002990IPN – Leibniz-Institut für die Pädagogik der Naturwissenschaften und Mathematik an der Universität Kiel (3469)Peer Reviewe

Validating an assessment of pre-service primary teachers’ mathematics pedagogical content knowledge

Mathematikdidaktisches Wissen stellt eine zentrale Komponente der professionellen Kompetenz von (angehenden) Primarstufenlehrkräften im Bereich Mathematik dar, die mit situationsspezifischen Fertigkeiten zusammenhängt und, vermittelt über das professionelle Handeln, Effekte auf die mathematischen Leistungen von Schüler:innen haben kann. Um mathematikdidaktisches Wissen im Zusammenhang mit anderen Komponenten professioneller Kompetenz (z. B. Emotionen, Überzeugungen, Handlungsplanung, Instruktionsqualität) untersuchen zu können, bedarf es eines ökonomischen und frei verfügbaren Tests, welcher gängige Gütekriterien hinreichend erfüllt. Der vorliegende Beitrag stellt die Validierung eines solchen Tests vor. Entsprechend der Teststandards werden Grundnahmen in den Evidenzkategorien Inhalt, Struktur und Beziehungen zu anderen Variablen untersucht, um zu prüfen, ob die Testscores Schlussfolgerungen zum Konstrukt mathematikdidaktisches Wissen von Lehramtsstudierenden der Primarstufe zulassen. Die Ergebnisse liefern Validitätsargumente, die auf eine hinreichend hohe Reliabilität des Tests hinweisen und für theoriekonforme Schlussfolgerungen basierend auf den Testwerten sprechen. Der Beitrag schließt ab mit einer integrativen Betrachtung der Validierungsevidenzen, die für den Test bisher vorliegen.Peer Reviewe

Compiling video cases to support PD facilitators in noticing productive teacher learning

Abstract Background Research studies on facilitating professional development describe the knowledge and skills facilitators need to effectively attend to teachers’ learning processes. In this context, some studies gave rise to the question how to design learning opportunities to prepare facilitators to support teachers’ learning during professional development. The approaches share putting mathematics specialized content knowledge, pedagogical content knowledge on the professional development level and facilitation moves in the center. Research findings also highlight the role of video-based learning opportunities to qualify facilitators respectively. In particular, structured approaches that guide facilitators noticing toward advancing teacher learning have to be proven effective. Although research highlights using video-based material as a training tool for facilitators, what exactly the learning opportunities should consist of is up for discussion. In this article, we report on a validation study concerned with designing video cases, taken from a videotaped teacher professional development. We explored how eight experienced facilitators perceived the representativeness and the quality of the video cases for using them in facilitator professional development with respect to specialized content knowledge, pedagogical content knowledge on the professional development level, and facilitation moves. Additionally, we investigated what facilitation moves are noticeable in the video cases, and what noticing prompts can enhance these learning opportunities. Results Our results indicate that all video cases present vital opportunities to discuss how to support teacher learning in terms of specialized content knowledge, pedagogical content knowledge on the professional development level and facilitation moves. Our analyses provide insights into facilitators thinking processes, thus enriching their ratings by ideas and concerns considering the final implementation of the video cases in facilitator professional development. On the basis of our validation study, we were able to carefully check the learning opportunities that the chosen video cases can provide for facilitator professional development. Conclusions This paper provides information on how to design video-based learning opportunities for facilitator professional development. Although our results are of course bound to the six video cases we selected, our approach can be transferred to other contexts. The procedure of compiling the video cases and validating them takes up the most prominent research findings on designing facilitator professional development and pays careful attention to experiences facilitators thinking. We exemplarily show how the empirical evidence can be used to prepare facilitator PD

Aspects obstructing or facilitating examination success for first year engineering students

International audienceWhen analyzing influences on examination success in reference to learning strategies, we found that Effort played an important role. This supports the importance of affective and motivational aspects in the teaching of mathematics

Conceptualizing content-related PD facilitator expertise

Studies of facilitators of professional development (PD) for mathematics teachers have been increasing in order to improve their preparation for conducting PD. However, specifications of what facilitators should learn often lack a conceptualization that captures facilitators' expertise for different PD content. In this article, we provide a framework for facilitator expertise that is in line with current conceptualizations but makes explicit the content-related aspects of such expertise. The framework for content-related facilitator expertise combines cognitive and situated perspectives and allows unpacking different components at the PD level and the classroom level. Using two illustrative cases of different PD content (probability education in primary school and language-responsive mathematics teaching in secondary school), we exemplify how the framework can help to analyze facilitators' practices in content-related ways in a descriptive mode. This analysis reveals valuable insights that support designers of facilitator preparation programs to specify what facilitators should learn in a prescriptive mode. We particularly emphasize the importance of working on content-related aspects, unpacking the PD content goals into the content knowledge and pedagogical content knowledge elements on the classroom level and developing facilitators' pedagogical content knowledge on the PD level (PCK-PD), which includes curricular knowledge, as well as knowledge about teachers' typical thinking about a specific PD content. Situated learning opportunities in facilitator preparation programs can support facilitators to activate these knowledge elements for managing typical situational demands in PD