Mathematical Knowledge Architecture: From Historical Discovery to Individual Learning

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics and Computer Sciences

2 School of Mathematics,, Statistics and Computer Science, Collage of Science, University of Tehran, Tehran, Iran.

Abstract

This paper introduces a new theoretical framework titled Mathematical Knowledge Architecture (MKA). We argue that the true engine of knowledge growth is the dialectical cycle between two fundamental epistemic drives: the drive toward expansion (generalization) and the drive toward coherence (abstraction). We show that this dialectic is enacted through a cognitive grammar consisting of five foundational mechanisms- translation, compression, reconfiguration, coordination, and epistemic shift. Grounded in both historical and cognitive analyses, this framework is scalable: it explains large-scale transformations in the history of mathematics as well as individual learning processes. The key innovation of the theory is the identification of a critical epistemic gap in conventional instruction- namely, the breaking of this dialectic, where knowledge becomes an fixed, museum-like product. Consequently, we argue that the ultimate goal of mathematics education is to enable epistemic alignment- that is, restoring this generative dialectic within the learning environment. MKA has far-reaching implications: it reconceptualizes Advanced Mathematical Thinking as architectural agency- the ability to act upon the structure of one’s own knowledge- and provides a structural grammar for understanding mathematical creativity and its pathology, algorithmic rigidity. The theory thus offers a dynamic model for analyzing the construction of knowledge and a practical roadmap for designing meaningful learning experiences.

Keywords

Main Subjects

Mathematics and Society

Articles in Press, Accepted Manuscript
Available Online from 15 July 2026

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History

  • Receive Date: 10 May 2026
  • Revise Date: 02 July 2026
  • Accept Date: 15 July 2026
  • Publish Date: 15 July 2026

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