Dissertation project of Stefan Schumacher

WG Mathematics Education (Secundary Levels)

Representational skills - Using the Example of Core Concepts of Fractions and the Fractional Calculus

"Intellectual growing up entails that one is more and more able to communicate to oneself or to others by words or symbols what one has done or is going to do." (Bruner 1974, p. 12)

The combination of the ability to deal with given external representations as well as the ability to independently generate external representations of mathematical objects is called representational competence. The focus of the thesis is on the ability to generate external representations with the goal of recording findings from independence-oriented learning processes in the form of "protocols" (Dörfler 2000, p. 111).

The dissertation project aims to investigate, among other things, the influence and optimization of the use of so-called "self-explanation prompts" (cf. among others Rau, Aleven, Rummel 2009) with respect to the development of representational competence and learning success in a self-directed learning environment on basic concepts of fractions and fractional arithmetic. In order to measure the construct representational competence in the sense of this work, a novel test instrument, so-called video items, was developed


Bruner, J. S. (1974). Entwurf einer Unterrichtstheorie. Berlin; Düsseldorf: Berlin-Verlag; Pädagogischer Verlag Schwann.

Dörfler, W. (2000): Mens for Meaning. In: Cobb, P., Yackel, E., Mc Clain, K. (Hrsg.): Symbolizing and communicating in mathematics classrooms: perspectives on discourse, tools, and instructional design. Mahwah, N.J: Lawrence Erlbaum Associates, S. 99-132.

Rau, M.; Aleven, V.; Rummel, N. (2009): Intelligent Tutoring Systems with Multiple Representations and Self-Explanation Prompts Support Learning of Fractions. Artikel online abrufbar unter:



[This dissertation is only available in german. Our summary was translated by our website team. Possible quotations have been translated in accordance with scientific regularia.]