Schwarzkopf, R. (2019). Productive opportunities for communication in primary school mathematics lessons: On the learning-theoretical function of reasoning. In: A.S. Steinweg (Ed.): Representation and Communication – Proceedings of the Primary School Working Group of the GDM. Bamberg, 55–68.
Nührenbörger, M. & Schwarzkopf, R. (2019). Reasoning through calculation: Opportunities for learning algebra in primary school arithmetic lessons. In B. Brandt & K. Tiedemann (Eds.), Interpretative Unterrichtsforschung (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 relationships. In M. Abshagen et al. (Eds.), Basic Knowledge for Teacher Training. Basic Knowledge for Successful Mathematics Teaching. Seelze: Kallmeyer, 113–130.
Schwarzkopf, R. (2017). Don’t children need to learn arithmetic first? In K. Akinwunmi (ed.), Algebraic Thinking – 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-based contexts for probability. In M. Nührenbörger & U. Häsel-Weide (Eds.) Learning Maths Together – Teaching Maths to 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 with 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 in Subject-Specific Contexts. Didactic Research on Argumentation in School 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). The teacher’s role: Interplay between symbolism and visualisation. Differentiated Mathematics, 5(4), 20–26.
Nührenbörger, M., & Schwarzkopf, R. (2014). Mathematics as a game. mathematik lehren (168), 10–11.
Schwarzkopf, R. (2013): Quantities: Measuring, Calculating, Estimating – Fundamental Insights in Primary School. In: Grundschulmagazin No. 5, 7–12.
Nührenbörger, M. & R. Schwarzkopf (2013): Equations between ‘Calculating’ and ‘Converting’. In G. Greffrath, F. Käpnick & M. Stein (eds.): Contributions to Mathematics Teaching (Volume 1), 716–719.
Nührenbörger, M. & R. Schwarzkopf (2013): Equations in operational exercises: Discoveries involving plus signs. In: Differentiated Mathematics, Issue 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. Opportunities for active learning and practice. Stuttgart: Ernst Klett Verlag, 183–187.
Böttinger, C., Bräuning, K., Nührenbörger, M., Schwarzkopf, R. & Söbbeke, E. (eds.) (2010). Mathematics in Children’s Thinking. Ideas for Reflection on Mathematics Education. Seelze: Klett-Kallmeyer, including the following contributions:
- Nührenbörger, M., & Schwarzkopf, R. (2010). Children’s Mathematical Thinking Processes (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 relationships (169–215).
Schwarzkopf, R. (2009): Invented representatives as a catalyst for discussion in mathematics 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.): Real-life Mathematics 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: Bulletin of the Italian Mathematical Union. Section A, Mathematics in Society and Culture. Series VIII, Vol. IX-A December 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), 13–17 February, 241–248.
Schwarzkopf, R (2003): Justifications and new knowledge: The gap between empirical and structural arguments in primary school mathematics learning processes. In: Journal für Mathematik-Didaktik 24, no. 3/4, pp. 211–235.
Schwarzkopf, R. (2003): Puzzling with Corners and Edges: Drawing Without Lifting the Pen. In: The Primary School Journal. Issue 164/April, 44–48.
Schwarzkopf, R. (2003): Analysing Processes of Solving Word Problems in Mathematics Lessons: Interaction, Framings and Reference Contexts between the 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): Analysis of interaction processes in practical arithmetic in Year 4 mathematics lessons. In: W. Peschek (ed.): Contributions to Mathematics Teaching. Hildesheim: Franzbecker, 451–454.
Guder, K. & R. Schwarzkopf: (2001) How long are we at school this year? – a teaching experiment on reflection regarding modelling in primary school. In: C. Selter & G. Walther (eds.): Learning Mathematics and Common Sense. Festschrift for Gerhard Norbert Müller. Leipzig; Stuttgart; Düsseldorf: Klett–Grundschulverlag, 75–82.
Schwarzkopf, R. (2001): Analyses of reasoning in early-year classroom teaching – independent reasoning with exceptions. In: Journal für Mathematik-Didaktik 22, nos. 3/4, 253–276.
Schwarzkopf, R. (2001): “We’ve worked it out” – argumentative relationships between new and existing knowledge. In: Kaiser, G. (ed.): Contributions to Mathematics Teaching. 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 Teaching. Hildesheim: Franzbecker, 587–590.
Schwarzkopf, R. (2000): Argumentation Processes in Mathematics Teaching – Theoretical Foundations and Case Studies (PhD thesis). 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 lessons. In: Neubrand, M. (ed.): Contributions to Mathematics Teaching. Hildesheim: Franzbecker, 461–464.

Resources for classroom use

Heß, B., Nührenbörger, M., Schwarzkopf, R., Tubach, D. (2018). 1·1 Cards. Sorting and organising tasks, further developing calculation strategies. Leipzig: Klett.
Heß, B., Nührenbörger, M., Schwarzkopf, R., Tubach, D. (2018). 1-1 Cards. Sorting and organising problems, developing calculation strategies. Leipzig: Klett.
Heß, B., Nührenbörger, M., Schwarzkopf, R., Tubach, D. (2018). 1+1 Cards. Sorting and organising problems, developing calculation strategies. Leipzig: Klett.
Nührenbörger, M. & Schwarzkopf, R. (Eds.) The Number Book. Leipzig: Klett; including:
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2019). The Number Book 4. Pupil’s Book and Workbook. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D., Heß, B. & Hunscheidt, D. (2018). *Das Zahlenbuch 3*. Pupil’s book and workbook. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2017). The Numbers Book 2. Pupil’s Book and Workbook. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2017). The Numbers Book 1. Pupil’s Book and Workbook. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2019). The Numbers Book 4. Teacher’s Guide. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D., Heß, B. & Hunscheidt, D. (2018). The Number Book 3. Teacher’s Guide. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2017). The Number Book 2. Teacher’s Guide. Leipzig: Klett.
- Nührenbörger, M., Schwarzkopf, R., Bischoff, M., Götze, D. & Heß, B. (2017). The Number Book 1. Teacher’s Guide. Leipzig: Klett.
- Häsel-Weide, U., Nührenbörger, M. & Reinold, M. (2019). Support Workbook for *Das Zahlenbuch 4*. Leipzig: Klett.
- Häsel-Weide, U., Nührenbörger, M. & Reinold, M. (2018). Support Workbook for ‘Zahlenbuch 3’. Leipzig: Klett.
- Häsel-Weide, U., Meier, S. & Nührenbörger, M. (2017). Support workbook for ‘Zahlenbuch 2’. Leipzig: Klett.
- Breucker, T., Häsel-Weide, U. & Nührenbörger, M. (2017). Support Workbook for ‘Zahlenbuch 1’. Leipzig: Klett.
- Häsel-Weide, U., Nührenbörger, M. & Reinold, M. (2019). Support Commentary: Learning for ‘Zahlenbuch 4’. Leipzig: Klett.
- Häsel-Weide, U., Nührenbörger, M. & Reinold, M. (2018). Support Guide for ‘Lernen zum Zahlenbuch 3’. Leipzig: Klett.
- Häsel-Weide, U., Meier, S. & Nührenbörger, M. (2017). Support Guide for ‘Lernen zum Zahlenbuch 2’. Leipzig: Klett.
- Häsel-Weide, U. & Nührenbörger, M. (2017). Support Guide: Learning with the Number Book 1. Leipzig: Klett.
- Berg, M., Götze, D., Maske-Look, M. (2019). Support Guide: Language for *Zahlenbuch 3*. Leipzig: Klett.
- Berg, M., Götze, D., Maske-Look, M. (2018). Language Support Guide for ‘Zahlenbuch 3’. Leipzig: Klett.
- Götze, D., Hang, E. (2017). Language Support Guide for the Number Book 2. Leipzig: Klett.
- Götze, D., Hang, E. (2017). Language Support Guide for ‘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: 24 Jun 2026) | Kurz-URL:Shortlink: https://uol.de/p43721en

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