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Degrees of Freedom in Modeling: Taking Certainty out of Proportion.

In its empirical part this paper establishes a general weak understanding of the process of applying a mathematical model. This is also evident in the way teachers regard the application of alternative sharing in their own problem solving and in relating to children's answers. The theoretical part analyses problems that are considered as applications of proportional reasoning. It suggests that the rationale for applying a proportion model varies and includes, for example, cases with a scientific rationale and others with a social one. In some problems there are no degrees of freedom in applying proportion, but in other cases this model should not be taken as "engraved in stone". This analysis is supported by examples of alternative sharing in Talmudic laws or bankruptcy interpreted by game theoretic models.