This theoretical paper explores the decision-making process involved in modelling and mathematizing situations during problem solving. Specifically, it focuses on the authority behind these choices (i.e., what or who determines the chosen mathematical models). We show that different types of situations involve different sources of authority, thereby creating different degrees of freedom for the problem solver engaged in the modelling process. It also means that mathematics plays different roles in these problems and situations. This epistemological analysis on the meaning of modelling implies that we should reconsider the mathematical status of realistic solutions and raises questions on the validity of some traditional choices of mathematical models and their use in diagnosing children's conceptions. It also suggests constructing modelling tasks by choosing a certain variety of situations that might lead to a better understanding of the roles of mathematics. Recent reviews (Niss, Blum, & Galbraith, 2007; Stillman, Brown, & Galbraith, 2008) indicate that the community of researchers that investigate issues of modelling and applications is involved in many research perspectives. Many of the works deal with the effect of modelling tasks on student learning of mathematical concepts, student developing of modelling skills, teacher attitudes towards modelling tasks and teacher learning from observing students' modelling processes. Within modelling issues, this article touches on several perspectives. From the Epistemological perspective, which is less investigated (Stillman et al., 2008), it aims at analyzing the nature and meaning of choosing mathematical models in a given situation. From the Authenticity and Goals perspective it investigates the roles of mathematics in problem solving, and views the development of this knowledge as a curriculum goal that can have an impact on the choice of modelling tasks. Our goals are also motivated by the need to promote a radical change in teacher beliefs about the roles of everyday knowledge and the roles of mathematics in problem solving and modelling (Bonotto, 2007). For these purposes, we will compare different types of problems in an effort to find what determines the problem solver's modelling choices. We use the term authority in the sense of a source of knowledge or power that either suggests or imposes the choice of mathematical structures. We believe that teachers and students who engage in modelling tasks or any other problem solving activities should be conscious of their reasons for making choices in applying mathematical concepts. We also believe that this awareness can change teacher and student beliefs about the roles of mathematics.
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