On the Importance of Experiencing Coherence and Differences in the Context of Fundamental Ideas—Five Theses

Abstract

Zusammenfassung Als fachdidaktisches Prinzip ist ‚Orientierung an fundamentalen Ideen' keine neue, gleichwohl eine Forderung, die in ihren curricularen und unterrichtlichen Implikationen z.T. unklar erscheint. Ausgehend von einer Kontrastierung mit Konzeptionen lokaler Ideen wird eine Neuinterpretation einer Orientierung an fundamentalen Ideen vorgeschlagen. In dieser Neuinterpretation wird dem Finden einer geeigneten Balance in doppelter Hinsicht eine zentrale Rolle für die fachdidaktische Forschung und den Mathematikunterricht der gymnasialen Oberstufe zugewiesen; einerseits zwischen dem ‚Vollzug' und der Reflexion mathematischer Handlungen und Tätigkeiten, andererseits zwischen der Gewährung von Kohärenzerfahrun-gen und dem Zulassen offener Differenzerlebnisse.Schlüsselwörter Fundamentale Ideen · Bildungstheorie · Kohärenzstiftung · Mathematikunterricht in der Oberstufe Mathematics Subject Classification (2000) 97-D20 · 97-D30 · 97-B40 · 97-C99 · 97-H60 Abstract Using fundamental ideas as guidelines in mathematics education is a rather old didactical principle. However, up to now its implications for curriculum planning and classroom activity haven't been specified very precisely. By contrasting fundamental ideas with local ideas a revised interpretation of 'guidance by fundamental ideas' will be outlined as an alternative. This alternative emphasizes the importance A. Vohns ( ) 228 A. Vohns of finding a balance within mathematics education research and classroom practice in the upper secondary school; both between 'executing' mathematics and reflecting mathematics and between experiencing coherence and undergoing non-sequitorial moments and alienation.

Fundamental Ideas as a Guiding Category in Mathematics Education—Early Understandings, Developments in German-Speaking Countries and Relations to Subject Matter Didactics

“…If one considers fundamental ideas relevant in this respect, strictly decomposing fundamental ideas to a set of GVs and local subject-specific strategies and afterwards just involving students with these local representations has to be considered insufficient. In Vohns (2010) it is therefore further argued that in addition to designing mathematical courses in a way that proposed fundamental ideas can be experienced by means of GVs and local subject-specific strategies, opportunities to discuss, negotiate and reflect on fundamental ideas themselves are as well needed, i.e. opportunities to think and talk about what is (or better yet: was in retrospect) central to a particular unit and how it relates to earlier mathematical experiences.…”

“…This process clearly should involve a communicative structure and could take place in seminars, during in-service education, and very important during casual discussions" (Schweiger 2005, p. 71). In Vohns (2010) it is suggested to consider proposed fundamental ideas opportunities to experience coherence and continuity, which students should negotiate about and which have to be thought of as an offer they actually can refuse on grounds of reflection upon their own experience of socially constructing mathematical knowledge in the classroom.…”

Subject Matter Didactics in Mathematics Education

“…Since then a lot of different lists of big ideas have been proposed (for the German mathematics education community see e. g. Schweiger 2006;Schwill 1993;Tietze et al 1997). However, a proof for the actual relevance for learning is still missing (Vohns 2010). In this special issue Vohns gives an overview of important stages in the specific development of the understanding of big ideas.…”

Geometrie zwischen Grundbegriffen und Grundvorstellungen

“…Der Bezeichner >>Be-griffsbild<< ist angelehnt an Talls und Vinners Concept Image, der Bezeichner >>Be-griffskonvention<< weist darauf hin, dass eine Definition notwendigerweise eine Konvention ist. bei Vohns 2005 [247], 2010 [248]). Insgesamt wird der Grundvorstellungsbegriff von den jeweiligen Autoren auch kontextabhängig verwendet, und Arbeiten zu Grundvorstellungen differieren vor allem mit Bezug auf die Frage, inwieweit Grundvorstellungen ein normativer Begriff sind, und inwieweit sie deskriptiv und konstruktiv zu nutzen sind.…”