Schwarzkopf, R. (2019). Productive communication occasions in primary school mathematics lessons: On the learning-theoretical function of argumentation In: A.S. Steinweg (Ed.): Darstellen und Kommunizieren - Tagungsband des AK Grundschule in der GDM. Bamberg, 55-68.
Nührenbörger, M. & Schwarzkopf, R. (2019). Argumentative arithmetic: Algebraic learning opportunities in primary school arithmetic lessons. In B. Brandt & K. Tiedemann (Eds.), Interpretative classroom research (15-35). Münster: Waxmann.
Schwarzkopf, R., Nührenbörger, M. & Mayer, C. (2018). Algebraic understanding of equalities in primary classes. In C. Kieran (Eds.), Early Algebra (195-212). Rotterdam: Springer.
Nührenbörger, M. & Schwarzkopf, R. (2017). Algebraic thinking in primary school arithmetic lessons. In U. Kortenkamp & A. Kuzle (Eds.), Contributions to mathematics education (713-716). Münster: WTM-Verlag.
Hußmann, S. & Schwarzkopf, R. (2017). Functional context. In M. Abshagen et al. (Eds.), Basic knowledge of teacher education. Basic knowledge for successful maths teaching. Seelze: Kallmeyer, 113-130.
Schwarzkopf, R. (2017). Don't children have to learn arithmetic first? In K. Akinwunmi (ed.) Thinking algebraically - exploring arithmetic. Die Grundschulzeitschrift, 306. Seelze: Friedrich, 18-23.
Nührenbörger, M., Meyer, C., Schwarzkopf, R. (2017). Algebraic Understanding of Equalities in Primary Class. In. C. Kieran (Ed.), Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds: The Global Evolution of an Emerging Field of Research and Practice. Rotterdam: Springer, 195-212.
Neumann, A.-L., Schwarzkopf, R. (2017). Game contexts for chance. In M. Nührenbörger & U. Häsel-Weide (Eds.) Learning maths together - calculating with all children. Frankfurt am Main: Grundschulverband, 220-229.
Nührenbörger, M., Rösken-Winter, B., Ip Fung, Ch., Schwarzkopf, R. & Wittmann, E. (2016). Design Science and its Importance in the German Mathematics Educational Discussion. Cham: Springer.
Nührenbörger, M. & Schwarzkopf, R. (2016). Processes of Mathematical Reasoning of Equations in Primary Mathematics Lessons. In N. Vondrová (Ed.): Proceedings of the 9th Congress of the European Society for Research in Mathematics Education (CERME 9) (316-323). Prague, ERME.
Schwarzkopf, R. (2016). The Project Pendel-M: Design Science Between Normative and Descriptive Theories. ICME
Nührenbörger, M. & Schwarzkopf, R. (2016). Algebraic Understandings of Equalities in Primary Classes. ICME
Schwarzkopf, R. (2015). Argumentation processes in primary school mathematics lessons: An insight. In Budke, A. et al. (eds.): Learning to argue professionally. Didactic research on argumentation in the subjects. Münster: Waxmann, pp. 31-45.
Nührenbörger, M., & Schwarzkopf, R. (2014). Mathematics as a game. mathematik lehren (168), 10-11.
Nührenbörger, M. & R. Schwarzkopf (2014). Mediating tasks. Interplay between symbolism and visualisation. Mathematik differenziert, 5(4), 20-26.
Nührenbörger, M., & Schwarzkopf, R. (2014). Mathematics as a game. mathematik lehren (168), 10-11.
Schwarzkopf, R. (2013): Sizes: Measuring, calculating, estimating - fundamental insights in primary school. In: Grundschulmagazin H.5, 7-12.
Nührenbörger, M. & R. Schwarzkopf (2013): Equations between "calculating" and "converting". In G. Greffrath, F. Käpnick & M. Stein (eds.): Beiträge zum Mathematikunterricht (Volume 1), 716-719.
Nührenbörger, M. & R. Schwarzkopf (2013): Equalities in operational exercises: Discoveries on plus arrows. In: Mathematik differenziert, H. 1, 23-28.
Schwarzkopf, R. (2012) Who wins? - On the trail of chance. In: G. Müller, C. Selter & E. Ch. Wittmann (Eds.): Numbers, patterns and structures. Scope for active learning and practising. Stuttgart: Ernst Klett Verlag, 183-187.
Böttinger, C., Bräuning, K., Nührenbörger, M., Schwarzkopf, R. & Söbbeke, E. (eds.) (2010). Mathematics in the minds of children. Suggestions for mathematics didactic reflection. Seelze: Klett-Kallmeyer, including the following contributions:
- Nührenbörger, M., & Schwarzkopf, R. (2010). Mathematical thinking processes of children (8-16).
- Nührenbörger, M., & Schwarzkopf, R. (2010). The development of mathematical knowledge in social-interactive contexts (73-81).
- Nührenbörger, M., & Schwarzkopf, R. (2010). Discourses on mathematical contexts (169-215).
Schwarzkopf, R. (2009): Invented representatives as an occasion for discussions in maths lessons. In: Die Grundschulzeitschrift 23 (222/223), 46-48.
Schwarzkopf, R. (2007): Elementary Modelling in Mathematics Lessons: The Interplay between "Real-World" Knowledge and "Mathematical Structures". In: W. Blum & P. Galbraith & H.-W. Henn & M. Niss (Eds.): Modelling and Applications in Mathematics Education. The 14th ICMI Study. New York: Springer, 209-216.
Schwarzkopf, R. (2006): Elementary modelling in primary school. In: A. Büchter et al. (eds.): Realistic maths teaching - from the subject and for practice. Hildesheim: Franzbecker, 95-105.
Schwarzkopf, R. & A.S. Steinweg (2006): La formazione degli insegnanti di matematica in Germania (Mathematics Teacher Training in Germany). In: Bollettino della Unione Matematica Italiana. Sezione A, La Matematica nella Società e nella Cultura. Series VIII, Vol. IX-A Dicembre 2006/1: 499 - 532
Schwarzkopf, R. (2004): Elementary Modelling in Mathematics Lessons: The Interplay between "Real-World" Knowledge and "Mathematical Structures". In: W. Henn & W. Blum (Eds.): ICMI Study 14: Applications and Modelling in Mathematics Education. Pre-Conference Volume of the Study Conference in Dortmund (Germany), Feb. 13-17, 241-248.
Schwarzkopf, R (2003): Justifications and new knowledge: The span between empirical and structural arguments in primary school mathematical learning processes. In: Journal für Mathematik-Didaktik 24 H. 3/4, pp. 211-235.
Schwarzkopf, R. (2003): Solving puzzles with corners and edges: Drawing without taking off. In: The primary school magazine. Issue 164/April, 44-48.
Schwarzkopf, R. (2003): Analysing Processes of Solving Word Problems in Mathematics Lessons Interaction: Framings and Reference Contexts between Real World and Mathematics. Paper submitted and accepted for CERME 3: Third Conference of the European Society for Research in Mathematics Education, 28 February - 3 March 2003 in Bellaria, Italy
Schwarzkopf, R. (2002): Analysing interaction processes in factual arithmetic in fourth grade mathematics lessons. In: W. Peschek (ed.): Contributions to mathematics education. Hildesheim: Franzbecker, 451-454.
Guder, K. & R. Schwarzkopf: (2001) Wie lange sind wir in diesem Jahr in der Schule? - ein Unterrichtsversuch zur Reflexion über Modellbildung in der Grundschule. In: C. Selter & G. Walther (Eds.) : Learning maths and common sense. Festschrift for Gerhard Norbert Müller. Leipzig; Stuttgart; Düsseldorf: Klett-Grundschulverlag, 75-82.
Schwarzkopf, R. (2001): Argumentation analyses in lessons in the early grades - independent reasoning with exceptions. In: Journal für Mathematik-Didaktik 22 H. 3/4, 253-276.
Schwarzkopf, R. (2001): "Wir haben es herausgefunden" - argumentative relationships between new and old knowledge. In: Kaiser, G. (ed.): Contributions to mathematics education. Hildesheim: Franzbecker, 564-567.
Schwarzkopf, R (2000): Argumentation Processes in Mathematics Classrooms - Social Regularities in Argumentation Processes. In GDM (Eds.): Developments in Mathematics Education in Germany - Selected Papers from the Annual Conference on Didactics of Mathematics Potsdam, 139-151.
Schwarzkopf, R. (2000): Argumentation as an interactive process. In: Neubrand, M. (ed.): Contributions to mathematics education. Hildesheim: Franzbecker, 587-590.
Schwarzkopf, R. (2000): Argumentation processes in mathematics lessons - theoretical foundations and case studies (Diss.). Hildesheim: Franzbecker.
Schwarzkopf, R. (1999): Argumentation Processes in Mathematics Classrooms - Functional Analysis of Arguments - A Method to Describe Orally Developed Arguments. In GDM (Eds.): Developments in Mathematics Education in Germany - Selected Papers from the Annual Conference on Didactics of Mathematics Bern, 89-100.
Schwarzkopf, R. (1999): Argumentation processes in mathematics education. In: Neubrand, M. (ed.): Contributions to mathematics education. Hildesheim: Franzbecker, 461-464.

Materials for practice

Heß, B., Nührenbörger, M., Schwarzkopf, R., Tubach, D. (2018). 1-1 cards. Sorting and organising tasks, developing calculation strategies. Leipzig: Klett.
Heß, B., Nührenbörger, M., Schwarzkopf, R., Tubach, D. (2018). 1-1 cards. Sorting and organising tasks, developing calculation strategies. Leipzig: Klett.
Heß, B., Nührenbörger, M., Schwarzkopf, R., Tubach, D. (2018). 1+1 cards. Sorting and organising tasks, developing calculation strategies. Leipzig: Klett.
Nührenbörger, M. & Schwarzkopf, R. (Eds.) The number book. Leipzig: Klett; of which:
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2019). Das Zahlenbuch 4. student book and workbook. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. & Hunscheidt, D. (2018). Das Zahlenbuch 3. student book and workbook. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2017). Das Zahlenbuch 2. student book and workbook. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2017). Das Zahlenbuch 1. student book and workbook. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2019). The number book 4th teacher's book. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. & Hunscheidt, D. (2018). The number book 3. teacher's volume. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2017). The number book 2. teacher's volume. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2017). The number book 1. teacher's volume. Leipzig: Klett.
- Häsel-Weide, U., Nührenbörger, M. & Reinold, M. (2019). Support booklet for number book 4. Leipzig: Klett.
- Häsel-Weide, U., Nührenbörger, M. & Reinold, M. (2018). Support booklet for the number book 3. Leipzig: Klett.
- Häsel-Weide, U., Meier, S. & Nührenbörger, M. (2017). Support booklet for the number book 2. Leipzig: Klett.
- Breucker, T., Häsel-Weide, U. & Nührenbörger, M. (2017). Support booklet for the number book 1. Leipzig: Klett.
- Häsel-Weide, U. , Nührenbörger, M. & Reinold, M. (2019). Support commentary on learning for the number book 4. Leipzig: Klett.
- Häsel-Weide, U. , Nührenbörger, M. & Reinold, M. (2018). Förderkommentar Lernen zum Zahlenbuch 3. Leipzig: Klett.
- Häsel-Weide, U. , Meier, S. & Nührenbörger, M. (2017). Support commentary for learning the number book 2. Leipzig: Klett.
- Häsel-Weide, U. & Nührenbörger, M. (2017). Förderkommentar Lernen zum Zahlenbuch 1. Leipzig: Klett.
- Berg, M., Götze, D., Maske-Look, M. (2019.). Förderkommentar Sprache zum Zahlenbuch 3. Leipzig: Klett.
- Berg, M., Götze, D., Maske-Look, M. (2018.). Support commentary on language for the number book 3. Leipzig: Klett.
- Götze, D., Hang, E. (2017.). Förderkommentar Sprache zum Zahlenbuch 2. Leipzig: Klett.
- Götze, D., Hang, E. (2017.). Förderkommentar Sprache zum Zahlenbuch 1. Leipzig: Klett.
Nührenbörger, M., Schwarzkopf, R. & Tubach, D. (2016). Playing with numbers. Leipzig: Klett.

Development and research interest

Pendulum (practical development projects in dialogue with educators and teachers) (since 2015)
(sponsored by "Ernst Klett Verlag")

The development of mathematical skills is one of the central goals of maths lessons in primary school. Essentially, the aim is not just to teach children to carry out mathematical procedures and determine results according to rules. Rather, they should learn to understand and flexibly utilise relationships between numbers and terms. In this sense, the children develop elementary but also theoretical insights into the subject of maths. The aim of the project is to encourage the children to engage in productive argumentative discussions with basic mathematical structures. To this end, established teaching scenarios for mathematics lessons will be further developed in co-operation with practitioners, tested and reflected upon against the background of everyday lesson organisation. The analyses of teaching episodes are intended to show how children can gain mathematical insights, articulate them in the classroom and develop them further.

Co-operation: Marcus Nührenbörger (Dortmund)

Co-operation: Anna-Lena Barkley, André Köhler

About the person

Studies, doctorate, academic appointment

Since 2014	Professor of Mathematics Education (W2) at the University of Oldenburg
2004-2014	Senior Academic Councillor at the Institute for Development and Research in Mathematics Education at TU Dortmund University
2000-2004	Research assistant at the Institute for Development and Research in Mathematics Education at the University of Dortmund, Head of the regional working group "Mathematics in primary schools" as part of the "mathe 2000" project
2000	Doctorate from the Westfälische-Wilhelms-Universität Münster
1996-1999	Research assistant at the Institute of Mathematics Education at the Westfälische-Wilhelms-Universität Münster
1996	Diplom in Mathematics with a minor in Computing Science at the Christian Albrechts University of Kiel

University functions

Chair of the 2-subject Bachelor's examining board of School V,
Deputy member of the doctorate committee of School V,
Member of the examining board of the 2-subject Bachelor's programme of Faculty V,

Programme director of the subject Elementary Mathematics in the dual-subject Bachelor, Master Ed. Primary School, Master Ed. Secondary School and Master Ed. Special Education degree programmes,

Chair of the Study Committee of the Institute of Mathematics,

Member of the faculty development group of School V.

(Changed: 11 Feb 2026) | Kurz-URL:Shortlink: https://uol.de/p43721en

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