Fostering modeling competencies: benefits of worked examples, problems to be solved, and fading procedures

The application of mathematics to real-world problems is moving more and more in the focus of attention of mathematics education; however, many learners experience huge difficulties in relating “pure” mathematics to everyday contents. In order to solve “modeling problems”, it is first necessary to find a transition from a real-world description to a mathematical model; second, intra-mathematical calculations have to be performed; and third, the result has to be interpreted with respect to the described real situation. In the present work, the effectivity of learning with worked examples and with fading procedures was tested experimentally. A design with 4 groups was implemented: “forward fading”, “backward fading”, “example-problem pairs”, and “problem-example pairs”. The results show that translation competencies were fostered best by way of example-problem pairs, whereas intra-mathematical skills were fostered most effectively by a backward fading procedure. Subjective assessments were especially positive when fading procedures or example-problem pairs were implemented.