Journal of Advanced Mathematical Modeling

Journal of Advanced Mathematical Modeling

The Dialectic of Generalization and Abstraction in Mathematics Education

Document Type : Original Paper

Authors
Mohamad Sabzi 1
Reza Behmani 2
Mohammad Ali Siavoshi 3
1 Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz,, Iran
2 School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran,
3 Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz
10.22055/jamm.2026.51267.2443
Abstract
The theory of didactical situations (TDS) proposed by Brousseau is an influential framework for engineering learning environments in mathematics classrooms. However, this theory does not fully account for the learner’s internal cognitive processes when facing a crisis in an adidactical situation. In this conceptual paper, we introduce the theory of Mathematical Knowledge Architecture (MKA) as a complementary cognitive framework to TDS for addressing the learner’s internal cognitive processes. We argue that the dialectic of generalization and abstraction is the driving engine of this architecture; this dialectic, in response to cognitive crises, activates the fundamental mechanisms of MKA (such as translation, compression and reconfiguration). To demonstrate the analytical power of this framework, two instructional vignettes in analysis (complex numbers) and linear algebra (eigenvectors) are examined. The analyses show how the TDS environment stimulates the dialectic of generalization and abstraction, which in turn requires the activation of MKA mechanisms. This approach provides a deeper understanding of the teacher’s role as an architect of productive cognitive crises and offers important implications for instructional design in advanced mathematics.
Keywords
The Didactic Situation
Milieu
The Dialectic of Generalization and Abstraction
Cognitive Mechanisms
Mathematics Education
Subjects
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  • Receive Date 28 November 2025
  • Revise Date 13 February 2026
  • Accept Date 06 May 2026