Scaffolding mathematical modelling with a solution plan

Abstract

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".

Teaching and Learning Mathematical Modelling: Approaches and Developments from German Speaking Countries

“…The results show the potential of the solution plan as guideline: The students using the solution plan while working on the modelling problem reported that they used strategies more frequently than those of the control group. Furthermore, students using the solution plan showed higher achievement than those in the other group (Schukajlow et al , 2015a.…”

Scaffolding and dialogic teaching in mathematics education: introduction and review

“…Several studies report promising qualitative or observational results that provide proof of principle. Only a few have experimental results that show the effectiveness of their approach (e.g., Fisher et al, 2013;Kajamies et al, 2010;Roll et al, 2012;Schukajlow et al, 2012Schukajlow et al, , 2015. In our view it is therefore too early for a systematic quantitative review study such as meta-analysis of the scaffolding research (cf.…”

“…Hard scaffolds are static ones given beforehand. Among the successful hard scaffolds are solution plans (Schukajlow et al, 2015), proof flow charts (Miyazaki et al, this issue) and worked-out examples (Tropper et al, 2015). Soft scaffolds are dynamic and used on-the-fly.…”

Evaluating the suitability of mathematical thinking problems for senior high‐school students by including mathematical sense making and global planning

“…This appears to be particularly so regarding the phases of orientation and organisation, in that students often find it difficult to thoroughly understand the problem situation (cf. Goos, Galbraith & Renshaw, 2002;Stylianou & Silver, 2004) and to plan how to solve a problem (Pugalee, 2001;Schukajlow, Kolter & Blum, 2015). Research on this subject now focuses in particular on developing useful tools (e.g.…”