# Mathematical Modeling in Mathematics Education: Basic Concepts and Approaches*

## Article excerpt

In the last two decades, mathematical modeling has been increasingly viewed as an educational approach to mathematics education from elementary levels to higher education. In educational settings, mathematical modeling has been considered a way of improving students' ability to solve problems in real life (Gravemeijer & Stephan, 2002; Lesh & Doerr, 2003a). In recent years, many studies have been conducted on modeling at various educational levels (e.g., Delice & Kertil, 2014; Kertil, 2008), and more emphasis has been given to mathematical modeling in school curricula (Department for Education [DFE], 1997; National Council of Teachers of Mathematics [NCTM], 1989, 2000; Talim ve Terbiye Kurulu Baskanligi [TTKB], 2011, 2013).

The term "modeling" takes a variety of meanings (Kaiser, Blomhoj, & Sriraman, 2006; Niss, Blum, & Galbraith, 2007). It is important for readers who want to study modeling to be cognizant of these differences. Therefore, the purpose of this study is twofold: (i) Presenting basic concepts and issues related to mathematical modeling in mathematics education and (ii) discussing the two main approaches in modeling, namely "modeling for the learning of mathematics" and "learning mathematics for modeling." The following background information is crucial for understanding the characterization of modeling, its theoretical background, and the nature of modeling problems.

Mathematical Modeling and Basic Concepts

Model and Mathematical Model: According to Lesh and Doerr (2003a), a model consists of both conceptual systems in learners' minds and the external notation systems of these systems (e.g., ideas, representations, rules, and materials). A model is used to understand and interpret complex systems in nature. Lehrer and Schauble (2003) describe a model as an attempt to construct an analogy between an unfamiliar system and a previously known or familiar system. Accordingly, people make sense of real-life situations and interpret them by using models. Lehrer and Schauble (2007) describe this process as model- based thinking and emphasize its developmental nature. They also characterize the levels of model- based thinking as hierarchical.

Mathematical models focus on structural features and functional principles of objects or situations in real life (Lehrer & Schauble, 2003, 2007; Lesh & Doerr, 2003a). In Lehrer and Schauble's hierarchy, mathematical models do not include all features of real-life situations to be modeled. Also, mathematical models comprise a range of representations, operations, and relations, rather than just one, to help make sense of real-life situations (Lehrer & Schauble, 2003).

Mathematical Models and Concrete Materials: In elementary education, the terms mathematical model and modeling are usually reserved for concrete materials (Lesh, Cramer, Doerr, Post, & Zawojewski, 2003). Although the use of concrete materials is useful for helping children develop abstract mathematical thinking, according to Dienes (1960) (as cited in Lesh et al., 2003), in this study, mathematical modeling is used to refer to a more comprehensive and dynamic process than just the use of concrete materials.

Mathematical Modeling: Haines and Crouch (2007) characterize mathematical modeling as a cyclical process in which real-life problems are translated into mathematical language, solved within a symbolic system, and the solutions tested back within the real-life system. According to Verschaffel, Greer, and De Corte (2002), mathematical modeling is a process in which real- life situations and relations in these situations are expressed by using mathematics. Both perspectives emphasize going beyond the physical characteristics of a real-life situation to examine its structural features through mathematics.

Lesh and Doerr (2003a) describe mathematical modeling as a process in which existing conceptual systems and models are used to create and develop new models in new contexts. …

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