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Developing Students’ Reflections on the Function and Status of Mathematical Modeling in Different Scientific Practices: History as a Provider of Cases

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Abstract

Mathematical models and mathematical modeling play different roles in the different areas and problems in which they are used. The function and status of mathematical modeling and models in the different areas depend on the scientific practice as well as the underlying philosophical and theoretical position held by the modeler(s) and the practitioners in the extra-mathematical domain. For students to experience the significance of different scientific practices and cultures for the function and status of mathematical modeling in other sciences, students need to be placed in didactical situations where such differences are exposed and made into explicit objects of their reflections. It can be difficult to create such situations in the teaching of contemporary science in which modeling is part of the culture. In this paper we show how history can serve as a means for students to be engaged in situations in which they can experience and be challenged to reflect upon and criticize, the use of modeling and the significance of the context for the function and status of modeling and models in scientific practices. We present Nicolas Rashevsky’s model of cell division from the 1930s together with a discussion of disagreement between him and some biologists as one such episode from the past. We illustrate how a group of science students at Roskilde University, through their work with this historical case, experienced that different scientific cultures have different opinions of the value of a model as an instrument for gaining scientific knowledge; that the explanatory power of a model is linked not only to the context of its use, but also to the underlying philosophical and theoretical position held by the modeler(s) and the scientists discussing the model and its use. The episode’s potential to challenge students to reflect upon and criticize the modeling process and the function of models in an extra mathematical domain is discussed with respect to the notions of internal and external reflections.

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Notes

  1. See e.g. Keller (2002), Morgan and Morrison (1999), Loettgers (2007).

  2. “What is the Value of History of Science?” was the FOCUS theme in Isis, volume 99, 2008. Several of the FOCUS papers deal with the benefits of history of science for (future) scientists. For the issues raised in the present paper, the FOCUS paper by Maienschein et al. (2008, p. 347) is especially relevant because it addresses how historical perspectives can help clarify the assumptions and concepts used in model design, and exemplifies how such clarifications contributed to ongoing research in synthetic biology.

  3. In the current example, historians of mathematics and science might object to the use of Rashevsky’s talk to teach students about modeling and models and to have them reflect upon the function and status of modeling in science, since Rashevsky did not use these terms. However, Rashevsky’s work can be seen as an early attempt to use mathematical models to understand biological phenomena (Keller 2002). As will be seen in the section describing the students’ work, Rashevsky’s approach was to investigate whether a consistent explanation of cell division could be found that “follows logically and mathematically from a well defined set of general principles.” (Rashevsky 1934, p. 88).

  4. For discussions of history and nature of science in science teaching, see (Klassen 2006; Abd-El-Khalick 2012). For discussions in mathematics teaching see (Fauvel and van Maanen 2000; Fried 2007; Jankvist 2009; Jankvist and Kjeldsen 2011; Kjeldsen and Blomhøj 2009, 2012; Kjeldsen 2012a).

  5. This point will not be elaborated on here. Interested readers are referred to (Kjeldsen 2012a, b).

  6. For an English version of some of the ideas behind and within the KOMproject, see (Niss 2004).

  7. See (Niss et al. 2007).

  8. For discussions of the significance of the role of reflection in mathematics teaching and learning in general, we refer to (Chamoso and Cáceres 2009, p. 199). The concept of reflection has not attracted much attention in educational research on mathematical modeling. Ideas similar to ours can be found in Greer and Verschaffel (2007) and Henning and Keune (2007). However, they do not differentiate between students’ reflections related to the modeling process and reflections related to the context of application.

  9. Translated and paraphrased by the authors of the present paper.

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Authors and Affiliations

  1. IMFUFA, Department of Science, Systems and Models, Roskilde University, Roskilde, Denmark

    Tinne Hoff Kjeldsen & Morten Blomhøj

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  1. Tinne Hoff Kjeldsen
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  2. Morten Blomhøj
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Correspondence to Tinne Hoff Kjeldsen.

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Kjeldsen, T.H., Blomhøj, M. Developing Students’ Reflections on the Function and Status of Mathematical Modeling in Different Scientific Practices: History as a Provider of Cases. Sci & Educ 22, 2157–2171 (2013). https://doi.org/10.1007/s11191-012-9555-4

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