Theoretical and empirical differentiations of phases in the modelling process

The reconstruction of pupils modelling processes can be found in many empirical studies within the literature on modelling. The empirical differentiations of the phases, which includes putting statements and actions of the pupils in the right phase, has not been reconstructed from a cognitive psychological point of view on a micro level thus far. In this article different modelling cycles are discussed with attention to distinctions in the various phases. The “modelling cycle under cognitive psychological aspects” is specifically emphasized in contrast to the other cycles. On the basis of the results of the COM2-project (Cognitive psychological analysis of modelling processes in mathematics lessons, Borromeo Ferri) the phases of the modelling process are described empirically. Some difficulties in the process of distinguishing the various phases are also pointed out. Kurzreferat: Die Rekonstruktion von Modellierungsprozessen bei Lernenden ist in vielen empirischen Studien innerhalb der Literatur zum Modellieren zu finden. Die empirische Unterscheidung der Phasen, was die Einordnung von Aussagen und Handlungen der Lernenden miteinbezieht, wurde bisher noch nicht aus kognitionspsychologischer Sicht auf einer Mikroebene rekonstruiert. In diesem Artikel wird nach einer Übersicht ausgewählter Modellierungskreisläufe der „Modellierungskreislauf unter kognitionspsychologischen Aspekten im Vergleich hervorgehoben. Somit liegt der Fokus auf einer kognitionspsychologischen Perspektive hinsichtlich des Modellierungskreislaufes. Auf der Basis von Ergebnissen des KOM2-Projekt (Kognitionspsychologische Analysen von Modellierungsprozessen im Mathematikunterricht, Borromeo Ferri) werden die Phasen empirisch beschrieben und auch Schwierigkeiten bei der Einordnung in diese Phasen verdeutlicht. ZDM-Classification: C30, D10 1. Different modelling cycles different approaches and aims Looking at the literature on modelling and applications one can find different modelling cycles. These cycles are different, because they are dependent on various directions and approaches of how modelling is understood and in some cases, if complex or non complex tasks are used. Kaiser (1995) gave a historical overview about different directions in the modelling discussion. In a new attempt (Kaiser et al., in press; Sriraman & Kaiser, 2006) to classify approaches and backgrounds of the national and international modelling discussion, it became much more evident, that different standpoints and views on modelling exist. I will not go into detail concerning the latter aspects, but discuss the various modelling cycles found in the literature. Most approaches, which will be presented belong to the German and Anglo-Saxon speaking countries. Current views of modelling in Romance speaking countries especially regarding modelling cycles could not be found. However I will not give a whole historical overview about the discussion regarding modelling cycles itself. In the following the focus lies on the view of various modelling cycles with respect to the aspects of the differentiation of real situation (RS), situation model (SM) respectively mental representation of the situation (MRS), real model (RM) and mathematical model (MM). The terms situation model (SM) and mental representation of the situation (MRS) are used synonymously in this article. In the COM2-project the latter one is more suitable (see section 2.2). In the modelling discussion in Germany there are arguments, whether or not it is necessary to distinguish between these phases mentioned before. Besides these differences it is obvious, that within the various cycles also the phase of validation and the phase of interpretation are sometimes understood as mixed or separately. This aspect is recognized but will not be discussed here in depth. So I divide four groups of modelling cycles concerning the first three phases and illustrate them as follows: Group 1: Distinction between situation model (SM)/mental representation of the situation (MRS) and real model (RM).