Papers by Ingolf Schäfer
On the one hand, we present the conception of a chain of reflections with a study-accompanying ep... more On the one hand, we present the conception of a chain of reflections with a study-accompanying eportfolio system with increasing reflection occasions in university mathematics pre-service teacher education. On the other hand, we illustrate problems and chances of this approach based on empirical data from the previous implementations of this chain of reflections. Our reflections are designed to support pre-service teachers in habitualising reflective actions and in interlinking subject matter and subject matter didactical knowledge in mathematics.
Das Y-Modell im Bereich der fachlichen Lehrerbildung in Mathematik
Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik, 2022
Le Centre pour la Communication Scientifique Directe - HAL - Archive ouverte HAL, Apr 5, 2018
in teaching complex analysis, we split the last part of the lecture into two branches according t... more in teaching complex analysis, we split the last part of the lecture into two branches according to profession: While future mathematicians deepen their understanding in a branch for them, future teachers take a branch of the lecture where they prepare a task with dynamical geometry software for pupils which is based on phenomena of complex analysis. Here, we describe the design of the course, some general aims and first results obtained from the branch for future teachers.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 1, 2017
The transition from school to university mathematics is known to present a major challenge to man... more The transition from school to university mathematics is known to present a major challenge to many students, resulting in poor performances and high dropout rates during the first semesters. In this paper we present the preliminary results on summer courses in Linear Algebra and Analysis after the first semester, which were designed based on the Abstraction in Context framework (Dreyfus & Kidron, 2014) and on the self-determination theory (Deci & Ryan, 2000). In particular, we investigate the potential of this course design to contribute to the motivation of the students and to their ability to engage in concept construction. These qualitative preliminary results will be used in further research to quantitatively assess the effect of these courses as well.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 1, 2017
We investigate university students' mental images of continuity of real-valued functions by analy... more We investigate university students' mental images of continuity of real-valued functions by analyzing the answers of a questionnaire administered to Bachelor students at the University of Bremen. Our conception of mental images is based on concept images in the spirit of and the Grundvorstellungen (basic ideas) present in German subject-matter didactics . For this purpose, we introduce the notion of "communicative simulacra." Furthermore, we catalog students' mental images of continuity that appear within this and preceding studies and demonstrate results on their acceptance in the study group. The used taxonomy and results are part of the first author's master's thesis .
The Epistemological Gap: A Case Study on Networking of APC and IDS
Advances in Mathematics Education, 2014
The case study of the epistemological gap involves two theoretical approaches, APC and IDS. It de... more The case study of the epistemological gap involves two theoretical approaches, APC and IDS. It describes a networking case that starts from a situation of seemingly contradictory analyses, develops a common methodology, and leads finally to conceptualizing and locally integrating the new concept of the epistemological gap into both theories.
Qualitätsoffensive Lehrerbildung-Projekt: "Schnittstellen gestalten - das Zukunftskonzept für die Lehrerbildung an der Universität Bremen" : Abschlussbericht : Laufzeit des Projekts: 1.01.2016-30.06.2019
Die vorliegende Publikation versteht sich als Ideengeber für die universitäre Lehre in der Lehrer... more Die vorliegende Publikation versteht sich als Ideengeber für die universitäre Lehre in der Lehrerbildung in Gestalt eines sogenannten Transferpakets. Sie berichtet über Ergebnisse des Teilprojekts 'Spotlights-Lehre' des Projekts 'Schnittstellen gestalten – Lehrerbildung entlang des Leitbildes des "Reflective Practitioner" an der Universität Bremen' im BMBF-Programm 'Qualitätsoffensive Lehrerbildung'. Inhaltlich geht es um innovative Lehrprojekte zur Verzahnung und Vernetzung von Fachwissenschaft und Fachdidaktik, die das Ziel verfolgen, Fragmentierungserfahrungen von Lehramtsstudierenden in den Fächern Mathematik, Englisch, Romanistik, Geschichte und Inklusive Didaktik zu reduzieren. Zentrales Anliegen dieses Transferpakets ist es, Transferstrategien bereitzustellen, an Beispielen zu illustrieren und in Lehrbeschreibungen einzubetten. Umfassend wird gezeigt, wie die Designprozesse zur Lehre in den beiden zentralen Modellprojekten der Fächer Englisch...
Das Y-Modell in der Stochastik – Erfahrungen aus dem ersten Transferversuch eines Design Research Projekts
Comparing the Construction of Mathematical Knowledge Between Low-Achieving and High-Achieving Students – a Case Study
The case study reported in this paper investigates whether there is a difference in the way high-... more The case study reported in this paper investigates whether there is a difference in the way high-achieving and low-achieving students construct mathematical knowledge and, if there is, how it might look. Furthermore, it investigates possible differences in metacognitive actions between these groups. For this, we study how pairs of highachieving and pairs of low-achieving students deal with a problem-solving task about the divisibility of sums, using the theory of abstraction in context and a category scheme for metacognitive activities. This paper is part of a larger ongoing project that compares knowledge construction of high-achieving and low-achieving students using different tasks.
We give a general construction for the classical limit of a quantum system defined in terms of ge... more We give a general construction for the classical limit of a quantum system defined in terms of generators of an arbitrary compact semisimple Lie algebra, generalizing known results for the su2 and su3 cases. The classical limit depends on the physical problem in question and is determined by the sequence of representations by which it is reached. Only in the simplest cases it is unique. We present explicit formulae useful in determining the classical limit in all important cases.
Eine typische Vorstellung zur Stetigkeit, die in der Schule erworben wird, ist sicher die eines G... more Eine typische Vorstellung zur Stetigkeit, die in der Schule erworben wird, ist sicher die eines Graphen, den man in einem Zug durchzeichnen kann ohne den Stift abzusetzen. In diesem Beitrag geht es um die Frage, welche zusätzlichen Vorstellungen es gibt und inwieweit zukünftige Lehrkräfte diese selbst haben oder mit ihnen umgehen können.
Quantum systems in finite dimensional Hilbert spaces proved to be a good laboratory for investiga... more Quantum systems in finite dimensional Hilbert spaces proved to be a good laboratory for investigations of connections between quantum and classical characteristics of chaotic and integrable dynamics. In particular statistical properties of quantum spectra are well established signatures of integrability on the classical level [Haa00]. Recently we investigated a general procedure of attaining the classical limit for quantum systems with Hamiltonians defined as polynomials in generators of finite-dimensional compact Lie algebras [SK06]. We generalized to an arbitrary algebra g the procedure proposed in [GK98] and [GHK00], where a particular example of the su3 algebra was used to illustrate connection between spectral statistics and properties of the limiting classical Hamiltonian system. In the above cited papers we discuss at length the physical relevance of model Hamiltonians defined in terms of generators of an arbitrary g, as well as the meaning of (various) classical limits of th...
Von explorierenden Aufgaben bis zur Mitarbeit im Forschungsteam – Forschungsgelegenheiten im Bachelorstudiengang Mathematik
Der Schwerpunkt der universitaren Mathematik liegt in der Forschung, die daher als Merkmal in die... more Der Schwerpunkt der universitaren Mathematik liegt in der Forschung, die daher als Merkmal in die Lehrveranstaltungen eingebracht werden sollte. Dies kann durch Elemente des Forschenden Lernens umgesetzt werden und zeichnet sich besonders dadurch aus, dass die Studierenden den Prozess von der Entwicklung einer Frage uber die Auswahl geeigneter Methoden und deren Ausfuhrung bis zur Darstellung der Ergebnisse in eigenstandiger Arbeit oder innerhalb einer Gruppe aktiv mitgestalten. Demzufolge werden erste Erfahrungen in forschungsartigen Prozessen gesammelt und das Reflektieren eigener Ergebnisse erlernt. Im vorliegenden Beitrag werden verschiedene Elemente dargestellt, die alle zusammen dazu beitragen, Forschendes Lernen im Lehrbetrieb im Vollfachstudium Mathematik zu etablieren und zudem wird uber erste Erfahrungen berichtet. Insbesondere wird die Gestaltung der Ubungen zur Anregung einer forschenden Haltung sowie ein neues Konzept der intensiven Forderung von Forschungsprojekten erl...
ion in context – the RBC model Hershkowitz, Schwarz & Dreyfus (2001, p. 202) regard abstraction a... more ion in context – the RBC model Hershkowitz, Schwarz & Dreyfus (2001, p. 202) regard abstraction as “an activity of vertically reorganizing previously constructed mathematics into a new mathematical structure”. This means that abstraction is an activity in the sense of Leont'ew's activity theory that comprises actions. Hershkowitz et al. identify three characteristic epistemic actions, namely recognising (R), building-with (B), and constructing (C). Recognising is described as an action in which a student becomes aware of a familiar mathematical structure in the situation, and building-with as “combining structural elements to achieve a given goal” without gaining new complex knowledge about the situation. When this happens constructing takes place. These epistemic actions are observable in social interaction and provide evidence that a process of abstraction is taking place. The actions are nested, e.g. constructing requires that the subject has already recognised and built ...
arXiv: Representation Theory, 2006
Starting with an operator in the universal enveloping algebra of a semi-simple, complex Lie group... more Starting with an operator in the universal enveloping algebra of a semi-simple, complex Lie group the nearest neighbor statistics of the spectra of this operator along a sequence of representations are discussed. After a short introduction in chapter 1 this problem is motivated by a general construction of the classical limit for quantum mechanical systems, which is adopted to this setting, in chapter 2. In chapter 3 it is shown that for simple operators, i.e., operators of the Lie algebra the nearest neighbor statistics along a sequence of irreducible representations converge to the Dirac measure. After a suitable completion of the universal enveloping algebra the convergence to Poisson statistics is proved in chapter 4 for the exponentials of generic operators. The proof makes use of a combinatorial inequality of the Katz-Sarnak type for tori, which is proved in chapter 5. In the appendix the necessary facts from group theory and the theory of nearest neighbor distributions are ga...
der "schwachen" Schülerinnen und Schüler in der Sekundarstufe 1 gerät immer mehr in den Fokus der... more der "schwachen" Schülerinnen und Schüler in der Sekundarstufe 1 gerät immer mehr in den Fokus der mathematikdidaktischen Forschung. Dabei ist der untersuchte Personenkreis nicht sehr scharf definiert und wird meist im Kontext der "Rechenschwäche" diskutiert (vgl. (Moser Opitz 2007) für einen Überblick über die verschiedenen Richtungen und Ansätze). Wenig erforscht ist allerdings die Tatsache, ob und, falls ja, in welcher Weise sich die Wissenskonstruktion solcher "schwacher" Lernender von der "normaler" oder gar "begabter" unterscheidet.
Inquiry-Based Learning – Undergraduate Research, 2019
As Huber (2009) states, inquiry-based learning as a higher education didactic method is subject t... more As Huber (2009) states, inquiry-based learning as a higher education didactic method is subject to strong criticism from many quarters. The issue of the unity of research and teaching, which serves as a central pillar for the educational-theoretical justification of inquiry-based learning, is called into question both in principle and pragmatically. This article will present inquiry-based learning in mathematics as an idealized process on the one hand, and in a real implementation on the other. This will be done against the backdrop of cultural-historical activity theory according to Roth and Radford (2011), which more precisely defines the theory for mathematics learning based on Leontiev (1978).
In diesem Vortrag geht es darum, welches Handlungspotenzial im Bezug auf den Gegenstand Mathemati... more In diesem Vortrag geht es darum, welches Handlungspotenzial im Bezug auf den Gegenstand Mathematik Schülerinnen und Schüler haben, die im gewöhnlichen Unterricht als leistungsschwach gelten, und wie dieses Handlungspotenzial in den jeweiligen Situationen aufgebaut wird. Dabei soll ausgehend von einer psychologischen Handlungstheorie der Begriff des Handlungspotenzials erläutert und in Beispielen seine Entstehung in Fördersituationen rekonstruiert werden.
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Papers by Ingolf Schäfer