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

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.