Research on the learning effectiveness of concrete learning settings often shows high variance in terms of effectiveness and – given the heterogeneity in the classroom – differential effects, i.e. one group of students benefits from the learning setting studied, another less so. Against the background of this state of research, the present project develops an adaptive learning environment based on assumptions about theory-based ATI effects (aptitude-treatment-interaction, Cronbach & Snow 1969) and empirically validates the ATI assumptions. In this research project, the learning object of operation comprehension is considered as a basic mathematical competence.
Studies show that a substantial proportion of students enter lower secondary school with fundamental gaps in basic mathematical skills. In particular, operational understanding is not sufficiently developed in the area of multiplication and division (Schulz et al. 2017; Ehlert et al. 2013), although this is curricularly completed by 4th grade and plays an elementary role for further learning (Baroody et al. 2006). A central aspect of operational understanding is considered to be the ability to translate situations (real, verbally described, figuratively given) into a mathematical (symbolic) term and vice versa (Schulz et al. 2019). If – as planned in the present study – a textual task is to be mathematized, mathematics didactic research generally proceeds from two main steps: first, the understanding of the described situation and the construction of a situation model. This is a mental representation of the content that integrates inferences and other relevant experiences. The next step is then mathematization, i.e. the translation of the situation model into a mathematical form (Verschaffel et al. 2000, 169). This transition from the situation described in the written task to the mathematical model (e.g., calculation) is described as the activation and application of “basic ideas” (Prediger 2008; Vom Hofe & Blum 2016).
Lack of operational understanding in text tasks suggests inappropriate or faulty mental representations. In this case, illustrative graphical representations (e.g. point fields) can support the task processing, provided that the translation between text, representation and calculation is successful. They represent an intermediate stage between concrete actions and abstract ideas that take place only in the mind (Lorenz 2019). Point patterns are thus already simplified situation models for the mathematical structure. They promote basic ideas about multiplication and division and flexible arithmetic as students switch back and forth between the dot pictures and matching calculations. Thus, these representations provide very good opportunities to illustrate the principle of multiplication (and division as its inverse), to build conceptual understanding of these operations, and also to clarify the computational laws associated with it (Schulz 2017; Barmby et al. 2009; Hurst & Hurrell 2016; Young-Loveridge 2005; Izsák 2004).
However, various studies also show that students differ in the way they use representations and that these differences, in turn, can have a major effect on learning outcomes. Accordingly, some students require different or more extensive instruction in order to be able to use external representations – such as point fields – profitably (Schnotz et al. 1994; Maichle 1994; Peeck 1994).
The aim of the project is to investigate these differential effects and to test whether an adaptive learning setting using graphical representations to promote operational understanding for multiplication and division is superior to a non-adaptive one.