Abstract
As problem posing has been shown to foster students’ problem-solving abilities, problem posing might serve as an innovative teaching approach for improving students’ modelling performance. However, there is little research on problem posing regarding real-world situations. The present paper addresses this research gap by using a modelling perspective to examine (1) what types of problems students pose (e.g., modelling vs. word problems) and (2) how students solve different types of self-generated problems. To answer these questions, we recruited 82 ninth- and tenth-graders from German high schools and middle schools to participate in this study. We presented students with different real-world situations. Then we asked them to pose problems that referred to these situations and to solve the problems they posed. We analyzed students’ self-generated problems and their solutions using criteria from research on modelling. Our analysis revealed that students posed problems that were related to reality and required the application of mathematical methods. Therefore, problem posing with respect to given real-world situations can be a beneficial approach for fostering modelling abilities. However, students showed a strong tendency to generate word problems for which important modelling activities (e.g., making assumptions) are not needed. Of the students who generated modelling problems, a few either neglected to make assumptions or made assumptions but were not able to integrate them adequately into their mathematical models, and therefore failed to solve those problems. We conclude that students should be taught to pose problems, in order to benefit more from this powerful teaching approach in the area of modelling.
Highlights
Mathematical modelling is important for students’ lives, as it enables them to solve problems in the real world with the help of mathematics (Blum et al 2007)
This work contributes to the theory of problem posing, as it indicates that assumptions about the importance of students’ prior knowledge and prior experience in problem solving for problem posing (Stoyanova 1997) hold for the authentic and rich situations that were offered in our study
A main conclusion of our work is that students are able to pose and solve mathematical problems that are related to reality
Summary
Mathematical modelling is important for students’ lives, as it enables them to solve problems in the real world with the help of mathematics (Blum et al 2007). Modelling problems are real-world problems that can be characterized by diverse features such as their authentic connection to reality and their openness (see, e.g., Maaß 2010). Prior research has demonstrated that students often encounter difficulties in solving modelling problems (Galbraith and Stillman 2006; Blum and Borromeo Ferri 2009; Kaiser 2017). A promising approach for fostering modelling might be to prompt students to pose problems that are based on real-world situations. Prior research has indicated that problem posing can be a successful teaching approach for fostering students’ problem-solving abilities (Chen et al 2013). There is almost no research on modelling through problem posing
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