Abstract
In its day-to-day regime the mathematics classroom is mainly focused on students’ mastery of specific knowledge and skills currently at hand. But do they see the bigger picture? Do they get an appropriate idea of what mathematics is essentially about? Fundamental ideas have been a regularly proposed way to outline the bigger picture. That is, to provide both mathematics educators and students with several central themes that interconnect the different areas of mathematics and its applications. Such ideas should be able to guide the selection, organization and presentation of curriculum content and subsequently make mathematics more understandable for students. This article aims to offer the English-speaking reader an overview of important stages in the specific development of the understanding of fundamental ideas within the German-speaking community of mathematics education. It embeds this line of research within the subject matter didactics tradition that this volume is dedicated to and it draws comparisons to and discusses relations between “Grundvorstellungen” and fundamental ideas.
Highlights
This article aims to offer the English-speaking reader an overview of important stages in the specific development of the understanding of fundamental ideas within the German-speaking community of mathematics education
The inclusion of patterns of mathematization may be understood as a countermeasure to early interpretations of fundamental ideas which tended to reduce them to the basic structures of pure mathematics, whereas patterns of mathematization refer to applied mathematics or applications of mathematics in general
Both the distinction of central concepts, subject-specific strategies and patterns of mathematization and the attempt to identify fundamental ideas of mathematics inductively by starting with the ideas of specific areas of mathematics may be seen as a double-edged sword: Tietze (1994) both justifies the distinction and the inductive approach with the otherwise too general nature and the lack of influence that universal ideas along the lines of Schreiber have on mathematical instruction
Summary
Linking the didactical principle of guidance by fundamental ideas back to the legacy of Felix Klein may in a stricter sense be considered an anachronism. This connection is claimed in respect to “educating functional thinking” as a guiding principle of the Meran reform, whereas “functional thinking” acts as a prototype of what will later on be called a “fundamental idea”. Another connection lies in similarities between Meran reform and “New Math” as the background against which Klein and Bruner develop their approaches to mathematics education, respectively
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