world (Blum et al. 2007). Modelling is of high importance for students' current and future life and is, for example, an important part of the NCTM principles and standards for school mathematics (National Council of Teachers of Mathematics 2000). Students from different countries all over the world are required to learn how to solve modelling problems. However, a number of empirical studies show that many students have an insufficient level of modelling competency by the end of lower secondary education (Blum 2011). Thus, research on mathematical modelling should be focused on instructional methods that can support the acquisition of mathematical modelling competency from primary through to secondary school. In several empirical studies it was found that support of students' strategy use is a promising approach to improve student learning in different domains (Collins et al. 1989) and that strategy use is connected to students' modelling performance (Stillman and Galbraith 1998). In this paper we propose a strategic instrument, called "solution plan", as the focus of an instructional method for improving students' mathematical modelling competency. We report here on the effects of scaffolding mathematical modelling, using this "solution plan" as a scaffold, on students' strategies and modelling competency, including its sub-competencies.
Strategies, scaffolding, and mathematical modelling 2.1 Cognitive and metacognitive strategies
Definition of strategies and effects of strategy useStrategies can be defined as behaviors and thoughts that learners engage in and that are intended to influence their Abstract In the study presented in this paper, we examined the possibility to scaffold mathematical modelling with strategies. The strategies were prompted using an instrument called "solution plan" as a scaffold. The effects of this step by step instrument on mathematical modelling competency and on self-reported strategies were tested using nineth grade students (N = 91) at a German middle track school (Realschule) in a quasi-experimental design. Six classes were randomly assigned to the experimental group, in which students used the solution plan, or to the control group. The quantitative data analysis using ANOVAs reveals that (1) in the posttest the experimental group students reported more frequently about planning, rehearsal, elaboration and organizing strategies while solving modelling problems than the control group; (2) the "solution plan" as a scaffold supports the development of students' modelling competency, including its sub-competencies. The students who used the solution plan outperformed the other students in solving modelling problems concerning the topic "Pythagorean theorem".
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