@article{63433,
  author       = {{Hoffmann, Max}},
  journal      = {{mathematik lehren}},
  number       = {{253}},
  pages        = {{39--44}},
  title        = {{{Digitale Perspektiven auf das Heron-Verfahren}}},
  doi          = {{https://doi.org/10.5555/ml-253-2025_07}},
  year         = {{2025}},
}

@article{56131,
  abstract     = {{This article provides a comprehensive mathematical-didactic analysis of how the highly relevant topic symmetry can be prepared for the university education of PSTs. Methodologically, the analysis is embedded in a design research cycle and serves as preparation for the actual design of learning activities. The procedure of "specifying and structuring" learning objects is used and adapted in such a way that, in addition to mathematical aspects, profession-oriented references to school mathematics are also considered. An essential result of the analysis is the formulation of so-called interface aspects to symmetry, which prove to be helpful in establishing such references. }},
  author       = {{Hoffmann, Max}},
  journal      = {{Recherches en Didactique des Mathématiques}},
  number       = {{2}},
  pages        = {{85--120}},
  title        = {{{Symmetry as a Topic for the University Education of Pre-Service Teachers}}},
  doi          = {{10.46298/rdm.14256}},
  volume       = {{45}},
  year         = {{2025}},
}

@article{51841,
  abstract     = {{athematische Kompetenzen digital zu fördern und digitale Kompetenzen mathematisch zu fördern – dies ist eine Forderung der neuen Bildungsstandards mit Blick auf eine Bildung in der digitalen Welt. Gerade das Potenzial digitaler Medien für das fachliche Lernen wurde in vielen Studien bestätigt. Eine sinnvoll gestaltete Einbettung digitaler Medien bietet die Chance, allen fünf Prinzipien eines guten Unterrichts gerecht zu werden: Verstehensorientierung, Durchgängigkeit, kognitive Aktivierung, Lernendenorientierung & Adaptivität und Kommunikationsförderung. Die flächendeckende Nutzung digitaler Medien etabliert sich bislang nur zögerlich. Aber wie können wir Lehrkräfte stärken, digitale Medien sinnvoll einzusetzen? Wir möchten hier die Bandbreite der Möglichkeiten an Beispielen verdeutlichen, ihren Einsatz motivieren und Wege für einen guten Unterricht aufzeigen.}},
  author       = {{Barzel, Bärbel and Greefrath, Gilbert and Nagel, Mareike and Hoffmann, Max}},
  journal      = {{mathematik lehren}},
  pages        = {{42 -- 47}},
  title        = {{{Digitalisierung als Chance für alle Prinzipien guten Unterrichts}}},
  volume       = {{242}},
  year         = {{2024}},
}

@book{55193,
  author       = {{Hoffmann, Max and Hilgert, Joachim and Weich, Tobias}},
  isbn         = {{9783662673560}},
  publisher    = {{Springer Berlin Heidelberg}},
  title        = {{{Ebene euklidische Geometrie. Algebraisierung, Axiomatisierung und Schnittstellen zur Schulmathematik}}},
  doi          = {{10.1007/978-3-662-67357-7}},
  year         = {{2024}},
}

@article{56016,
  abstract     = {{<jats:title>Abstract</jats:title><jats:p>Special tasks for pre-service teachers (PSTs) in university mathematics courses (“interface tasks”) are a common innovation in recent years to overcome the second discontinuity. By this, we mean tasks that are situated by typical everyday challenges of mathematics teaching and in which PSTs must use their mathematical knowledge and skills in a professionally relevant way. In this paper, we analyze answers that PSTs have created to an interface task on symmetry. The PSTs were asked to clarify a student’s question from a mathematical perspective and then give a suitable elementarized answer. We situate these two steps theoretically and reconstruct the mathematical reasoning in PSTs' answers. Through qualitative content analysis, we examined how PSTs justify figures' symmetries from a university mathematics perspective and when responding to the fictitious student. The scenario of a student questioning the existence of 100° rotationally symmetrical figures elicited rich and varied responses, proving suitable for an interface task. We compared PSTs' reasoning related to mathematical clarification with the reasoning related to elementarization. In many cases, this revealed a productive use of course content. An interesting result is that there is no uniform picture as to whether the arguments are more detailed in the mathematical clarification or in the elementarization.</jats:p>}},
  author       = {{Hoffmann, Max and Biehler, Rolf}},
  issn         = {{1863-9690}},
  journal      = {{ZDM – Mathematics Education}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Using academic mathematical knowledge when working on interface tasks–analyses of pre-service teachers’ arguments about rotationally symmetric figures}}},
  doi          = {{10.1007/s11858-024-01633-4}},
  year         = {{2024}},
}

@article{56197,
  abstract     = {{<jats:title>Abstract</jats:title><jats:p>Special tasks for pre-service teachers (PSTs) in university mathematics courses (“interface tasks”) are a common innovation in recent years to overcome the second discontinuity. By this, we mean tasks that are situated by typical everyday challenges of mathematics teaching and in which PSTs must use their mathematical knowledge and skills in a professionally relevant way. In this paper, we analyze answers that PSTs have created to an interface task on symmetry. The PSTs were asked to clarify a student’s question from a mathematical perspective and then give a suitable elementarized answer. We situate these two steps theoretically and reconstruct the mathematical reasoning in PSTs' answers. Through qualitative content analysis, we examined how PSTs justify figures' symmetries from a university mathematics perspective and when responding to the fictitious student. The scenario of a student questioning the existence of 100° rotationally symmetrical figures elicited rich and varied responses, proving suitable for an interface task. We compared PSTs' reasoning related to mathematical clarification with the reasoning related to elementarization. In many cases, this revealed a productive use of course content. An interesting result is that there is no uniform picture as to whether the arguments are more detailed in the mathematical clarification or in the elementarization.</jats:p>}},
  author       = {{Hoffmann, Max and Biehler, Rolf}},
  issn         = {{1863-9690}},
  journal      = {{ZDM – Mathematics Education}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Using academic mathematical knowledge when working on interface tasks–analyses of pre-service teachers’ arguments about rotationally symmetric figures}}},
  doi          = {{10.1007/s11858-024-01633-4}},
  year         = {{2024}},
}

@inproceedings{57895,
  abstract     = {{In our paper, we present a study in which we investigate which strategies pre-service teachers (PSTs) use to find and, if necessary, reject possible candidates for congruence theorems for quadrilaterals. This study was conducted before the PTSs attended a university geometry course. In this way, statements about learning prerequisites can be made. For the study, we analyzed group discussions of PSTs to identify typical approaches and evaluate them from a mathematical perspective. The results can be considered for the further development of courses for PSTs and generate hypotheses
for further research.}},
  author       = {{Hoffmann, Max and Schlüter, Sarah}},
  booktitle    = {{Proceedings of the Fifth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2024, 10-14 June 2024)}},
  editor       = {{González-Martín, Alejandro S. and Gueudet, Ghislaine and Florensa, Ignasi and Lombard, Nathan}},
  keywords     = {{Teachers’ and students’ practices at university level, Transition to, across and from university mathematics, Teaching and learning of specific topics in university mathematics, Congruence, Quadrilaterals}},
  publisher    = {{Escola Univerist`aria Salesiana de Sarri`a – Univ. Aut`onoma de Barcelona and INDRUM}},
  title        = {{{How Do Advanced Pre-Service Teachers Develop Congruence Theorems for Quadrilaterals?}}},
  year         = {{2024}},
}

@article{45786,
  abstract     = {{Intending to counteract Klein’s second discontinuity in teacher education, we explored and applied the innovation of “interface ePortfolio” in the context of a geometry course for preservice teachers (PSTs). The tool offers the possibility of implementing the design principle of profession orientation. In the article, we theoretically clarify what we understand by this principle and locate our innovative concept against this theoretical background. We empirically investigate the extent to which counteraction against the second discontinuity is successful by analyzing reflection texts created in the interface ePortfolio, focusing on PSTs’ perspectives. Our qualitative content analysis shows that most of them perceive the innovation as helpful in the intended sense and indicates that the course concept, in general, and the interface ePortfolio, in particular, have helped establish relevant links between the course content and their later work as teachers.}},
  author       = {{Hoffmann, Max and Biehler, Rolf}},
  issn         = {{1863-9690}},
  journal      = {{ZDM – Mathematics Education}},
  keywords     = {{General Mathematics, Education}},
  publisher    = {{Springer}},
  title        = {{{Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course}}},
  doi          = {{10.1007/s11858-023-01505-3}},
  year         = {{2023}},
}

@inproceedings{31849,
  author       = {{Hoffmann, Max and Biehler, Rolf}},
  booktitle    = {{Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022)}},
  editor       = {{Trigueros, Marı́a and Barquero, Berta and Hochmuth, Reinhard and Peters, Jana}},
  keywords     = {{Teaching and learning of specific topics in university mathematics, Transition to, across and from university mathematics, Student Teachers, Geometry, Congruence, Double Discontinuity.}},
  publisher    = {{University of Hannover and INDRUM.}},
  title        = {{{Student Teachers ’ Knowledge of Congruence before a University Course on Geometry}}},
  year         = {{2023}},
}

@inproceedings{43097,
  author       = {{Florensa, Ignasio and Hoffmann, Max and Romo Vázquez, Avenilde and Zandieh, Michelle and Martínez-Planell, Rafael}},
  booktitle    = {{Proceedings of the Fourth Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2022, 19-22 October 2022)}},
  editor       = {{Trigueros, Marı́a and Barquero, Berta and Hochmuth, Reinhard and Peters, Jana}},
  title        = {{{Innovations in university teaching based on mathematic education research}}},
  year         = {{2023}},
}

@article{56200,
  abstract     = {{<jats:title>Abstract</jats:title><jats:p>Intending to counteract Klein’s second discontinuity in teacher education, we explored and applied the innovation of “<jats:italic>interface ePortfolio</jats:italic>” in the context of a geometry course for preservice teachers (PSTs). The tool offers the possibility of implementing the design principle of <jats:italic>profession orientation</jats:italic>. In the article, we theoretically clarify what we understand by this principle and locate our innovative concept against this theoretical background. We empirically investigate the extent to which counteraction against the second discontinuity is successful by analyzing reflection texts created in the interface ePortfolio, focusing on PSTs’ perspectives. Our qualitative content analysis shows that most of them perceive the innovation as helpful in the intended sense and indicates that the course concept, in general, and the interface ePortfolio, in particular, have helped establish relevant links between the course content and their later work as teachers.</jats:p>}},
  author       = {{Hoffmann, Max and Biehler, Rolf}},
  issn         = {{1863-9690}},
  journal      = {{ZDM – Mathematics Education}},
  number       = {{4}},
  pages        = {{737--751}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Implementing profession orientation as a design principle for overcoming Klein’s second discontinuity – preservice teacher’s perspectives on interface activities in the context of a geometry course}}},
  doi          = {{10.1007/s11858-023-01505-3}},
  volume       = {{55}},
  year         = {{2023}},
}

@inproceedings{31367,
  author       = {{Hoffmann, Max and Biehler, Rolf}},
  booktitle    = {{ Bedarfsgerechte fachmathematische Lehramtsausbildung. Analyse, Zielsetzungen und Konzepte unter heterogenen Voraussetzungen }},
  editor       = {{Halverscheid, Stefan and Kersten, Ina and Schmidt-Thieme, Barbara}},
  pages        = {{351--368}},
  publisher    = {{Springer Spektrum}},
  title        = {{{Schnittstellenaufgaben in der Analysis I zur Verknüpfung von Schul- und Hochschulmathematik - Aufgabenbeispiele und Ergebnisse einer Evaluationsstudie.}}},
  year         = {{2022}},
}

@inproceedings{31365,
  author       = {{Biehler, Rolf and Hoffmann, Max}},
  booktitle    = {{ Professionsorientierte Fachwissenschaft. Kohärenzstiftende Lerngelegenheiten für das Lehramtsstudium Mathematik}},
  editor       = {{Isaev, Viktor and Eichler, Andreas and Loose, Frank}},
  isbn         = {{9783662639474}},
  issn         = {{2197-8751}},
  pages        = {{ 49–72}},
  publisher    = {{Springer}},
  title        = {{{Fachwissen als Grundlage fachdidaktischer Urteilskompetenz – Beispiele für die Herstellung konzeptueller Bezüge zwischen fachwissenschaftlicher und fachdidaktischer Lehre im gymnasialen Lehramtsstudium}}},
  doi          = {{10.1007/978-3-662-63948-1_4}},
  year         = {{2022}},
}

@phdthesis{31363,
  abstract     = {{Vorgestellt wird ein Entwicklungsforschungsprojekt zur Konzeption und Durchführung einer Veranstaltung "Geometrie für Lehramtsstudierende". Die Schwerpunkte des Projekts sind zum einen die inhaltliche Gestaltung der Veranstaltung und zum anderen die Umsetzung von Professionsorientierung. Bezogen auf den inhaltlichen Aufbau wird das auf metrischen Räumen aufbauende Axiomensystem der "Saccheri-Ebene" vorgestellt und mit alternativen axiomatischen Zugängen zur ebenen Geometrie verglichen. Die Frage nach der Umsetzung von Professionsorientierung in Fachveranstaltungen ist eng mit der Problematik der zweiten Diskontinuität verbunden. In der Arbeit wird dieses Problem auf Grundlage der Synthese von theoretischen Hintergründen zur Bedeutung von mathematischem Wissen und Können für professionelle Handlungskompetenz von Mathematiklehrkräften diskutiert und darauf aufbauend werden theoriebasierte Entwurfsprinzipien für professionsorientierte Fachveranstaltungen entworfen. Zentrale Elemente der methodischen Gestaltung sind die sogenannten "Schnittstellenwochen" zu den Themen Kongruenz und Symmetrie sowie das begleitende Schnittstellen-ePortfolio. Das zentrale Ergebnis der Arbeit ist ein theoretisch fundiertes und empirisch evaluiertes ganzheitliches Veranstaltungskonzept für eine professionsorientierte Geometrie-Veranstaltung für Lehramtsstudierende, dessen Konzeption auf andere Fachveranstaltungen übertragbar ist. Darüber hinaus ergeben sich im Rahmen der durchgeführten Entwicklungsforschung verschiedene neue Beiträge zur Geometriedidaktik in Schule- und Hochschule.}},
  author       = {{Hoffmann, Max}},
  pages        = {{410}},
  title        = {{{Von der Axiomatik bis zur Schnittstellenaufgabe: Entwicklung und Erforschung eines ganzheitlichen Lehrkonzepts für eine Veranstaltung Geometrie für Lehramtsstudierende}}},
  doi          = {{10.17619/UNIPB/1-1313}},
  year         = {{2022}},
}

@inbook{56201,
  author       = {{Biehler, Rolf and Hoffmann, Max}},
  booktitle    = {{Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik}},
  isbn         = {{9783662639474}},
  issn         = {{2197-8751}},
  publisher    = {{Springer Berlin Heidelberg}},
  title        = {{{Fachwissen als Grundlage fachdidaktischer Urteilskompetenz – Beispiele für die Herstellung konzeptueller Bezüge zwischen fachwissenschaftlicher und fachdidaktischer Lehre im gymnasialen Lehramtsstudium}}},
  doi          = {{10.1007/978-3-662-63948-1_4}},
  year         = {{2022}},
}

@inbook{56203,
  author       = {{Hoffmann, Max and Biehler, Rolf}},
  booktitle    = {{Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik}},
  isbn         = {{9783658340667}},
  issn         = {{2197-8751}},
  publisher    = {{Springer Fachmedien Wiesbaden}},
  title        = {{{Schnittstellenaufgaben in der Analysis I zur Verknüpfung von Schul- und Hochschulmathematik – Aufgabenbeispiel und Ergebnisse einer Evaluationsstudie}}},
  doi          = {{10.1007/978-3-658-34067-4_20}},
  year         = {{2022}},
}

@misc{31385,
  author       = {{Hoffmann, Max}},
  booktitle    = {{Mathematische Semesterberichte}},
  pages        = {{295–297}},
  title        = {{{Rezension: Hendrik Kasten und Denis Vogel: Grundlagen der ebenen Geometrie – Eine zugängliche aber exakte Einführung in die ebene Geometrie}}},
  doi          = {{10.1007/s00591-021-00299-3}},
  volume       = {{68}},
  year         = {{2021}},
}

@inbook{31364,
  author       = {{Hoffmann, Max}},
  booktitle    = {{ Lehrinnovationen in der Hochschulmathematik.  praxisrelevant – didaktisch fundiert – forschungsbasiert}},
  editor       = {{Biehler, Rolf and Eichler, Andreas and Hochmuth, Reinhard and Rach, Stefanie and Schaper, Niclas}},
  isbn         = {{9783662628539}},
  issn         = {{2197-8751}},
  pages        = {{179–204}},
  publisher    = {{Springer Berlin Heidelberg}},
  title        = {{{Einsatz von Schnittstellenaufgaben in Mathematikveranstaltungen – Praxisbeispiele aus der Universität Paderborn}}},
  doi          = {{10.1007/978-3-662-62854-6_9}},
  year         = {{2021}},
}

@inproceedings{31372,
  author       = {{Hoffmann, Max}},
  booktitle    = {{Beiträge zum Mathematikunterricht 2020}},
  editor       = {{Siller, Hans-Stefan and Weigel, Wolfgang and Wörler, Jan Franz}},
  pages        = {{1353--1356}},
  publisher    = {{WTM-Verlag}},
  title        = {{{Schnittstellenaktivitäten zum Kongruenzsatz WSW}}},
  doi          = {{10.17877/DE290R-21368}},
  year         = {{2020}},
}

@misc{31386,
  author       = {{Hoffmann, Max}},
  booktitle    = {{Mathematische Semesterberichte}},
  pages        = {{119–121}},
  title        = {{{Rezension: Andrew Granville und Jenniver Granville: Prime Supects: The Anatomy of Integers and Permutations}}},
  doi          = {{10.1007/s00591-019-00269-w}},
  volume       = {{67}},
  year         = {{2020}},
}