The mathematician and mathematics education enthusiast Felix Klein saw the reason for the problem of double discontinuity in the fragmentation of school and university mathematics, the origin of which lies in the difference in the appearance of mathematics in school and university (Klein, 2016/1924). The mathematician and mathematics education enthusiast David Tall (2008) even describes school and university mathematics as different worlds ("worlds of mathematics"), contrasting the descriptive-symbolic mathematics of school with the formal-axiomatic mathematics of university. And also mathematician and mathematics education enthusiast Hans Freudenthal expressed himself similarly: "The definition of mathematics varies. Each generation and each subtle mathematician within each generation formulates a definition that corresponds to his or her skills and insights." (in Davis & Hersh, 1985, p. 4, citet in Käpnick, 1998, p. 53; translated from German) At the same time, school pedagogical models for didactic analysis (e.g., Klafki, 1985; Köker & Störtländer, 2017) point out the fundamental importance of situation and condition analyses for lesson planning processes (Przybilla, J., Vinerean-Bernhoff, M., Brandl, M., & Liljekvist, Y., 2021).

The notion or the idea of mathematics is thus a decisive and fundamental factor for all further didactic-pedagogical considerations. At the same time, this results in the school- and university-didactic task to bring different notions/ideas/views/beliefs/perceptions/worlds into relation and harmony with each other.

Since worldwide educational institutions are also subject to the cultural sovereignty of the respective countries, the cultural context also contributes to the prevailing notion of mathematics.

Since 2010, we have been collecting and analyzing learners' and teachers' notions on mathematics in different institutional and cultural international contexts.

There exists relevance in particular for our following problem or research areas:

• Defragmentation/Interconnectedness & Double Discontinuity (DD):
  • First DD: transition from school to university.
  • Second DD: transition from university to school (e.g. DIMM-project).
• Mathematics Giftedness:
  • Structural systemic coupling of a viable defining construct to its meaningful, contextual environment (e.g., Brandl, 2010); Anthropological Approach (Sternberg, 1996).
  • Identification and inclusive fostering/support in heterogeneous classes.
• International cultural heterogeneity in mathematics education

References:

Brandl, M. (2011). A Constructive Approach to the Concept of Mathematical Giftedness based on Systems Theory. In M. Avotiņa, D. Bonka, H. Meissner, L. Ramāna, L. Sheffield & E. Velikova (Eds.), Proceedings of the 6th International conference on Creativity in Mathematics Education and the Education of Gifted Students (pp. 35–39). University of Latvia, Angel Kanchev University of Ruse. https://drive.google.com/file/d/1swHpzztybDHjdGDGFWyZECXTOukpZ552/view

Käpnick, F. (1998): Mathematisch begabte Kinder. Modelle, empirische Studien und Förderungsprojekte für das Grundschulalter. Peter Lang. ISBN 3-631-33395-1

Klafki, W. (1985). Neue Studien zur Bildungstheorie und Didaktik: Beiträge zur kritisch-konstruktiven Didaktik. [New Studies on Educational Theory and Didactics: Contributions to Critical-Constructive Didactics] Beltz: Weinheim.

Klein, F. (2016/1924). Elementary Mathematics from higher standpoint. Volume I: Arithmetik Algebra Analysis. (G. Schubring, Trans.) Springer. (Original work published 1924). https://doi.org/10.1007/978-3-662-49442-4

Köker, A., & Störtländer, J. C. (Eds.) (2017). Kritische und konstruktive Anschlüsse an das Werk Wolfgang Klafkis. [Critical and constructive connections to the work of Wolfgang Klafki] Beltz Verlag: Weinheim, Basel.

Przybilla, J., Vinerean-Bernhoff, M., Brandl, M., & Liljekvist, Y. (2021). Rooms of Learning – A conceptual framework for student-centered teaching development in a digital era. Working Papers in Mathematics Education, 2021(2), 1–40.https://www.diva-portal.org/smash/get/diva2:1609427/FULLTEXT01.pdf

Sternberg, R. J. (1996): What is Mathematical Thinking? In R. J. Sternberg & T. Ben-Zeev (1996) (Eds.), The nature of mathematical thinking (pp. 303–318). Lawrence Erlbaum Ass. Publishers.

Tall, D. (2008). The Transition to Formal Thinking in Mathematics. Mathematics Education Research Journal, 20(2), 5–24.