How it is Possible to Make Real-World Mathematics More Visible: Some Results from Two Italian Projects
The first project described in this contribution regards the primary school level and is articulated in some teaching experiments aimed at showing how an extensive use of suitable artifacts could prove to be useful instrument in creating a new tension bet
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1 Mathematical Modelling and Problem Posing The first project described in this contribution regards the primary school level and is articulated in some teaching experiments aimed at showing how an extensive use of suitable artifacts could prove to be useful instrument in creating a new tension between school mathematics and real world with its incorporated mathematics. The teaching/learning environment designed in these teaching experiments is characterized by an attempt to establish a new classroom culture also through new socio-mathematical norms, for example norms about what counts as a good or acceptable response, or as a good or acceptable solution procedure, are debated (Bonotto 2005). The focus is on fostering a mindful approach toward realistic mathematical modelling, mathematics applications and also a problem posing attitude, even at the primary school level. The term mathematical modelling is not only used to refer to a process whereby a situation has to be problematized and understood, translated into mathematics, worked out mathematically, translated back into the original (real-world) situation, evaluated and communicated. Besides this type of modelling, which requires that the student has already at his disposal at least some mathematical models and tools to mathematize, there is another kind of modelling, wherein model-eliciting activities are used as a vehicle for the development (rather than the application) of mathematical concepts (Greer et al. 2007). This second type of modeling is called ‘emergent modeling’ in Gravemeijer (2007), and its focus is on long-term learning processes, in which a model develops from an informal, situated model (‘‘a model of’’), into a generalizable mathematical structure (‘‘a model for’’). Although it is very difficult, if not impossible, to make a sharp distinction between the two aspects of mathematical modelling, it is clear that they are
C. Bonotto (&) Department of Mathematics, University of Padua, Padua, Italy e-mail: [email protected]
A. Damlamian et al. (eds.), Educational Interfaces between Mathematics and Industry, New ICMI Study Series 16, DOI: 10.1007/978-3-319-02270-3_9, Springer International Publishing Switzerland 2013
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associated with different phases in the teaching/learning process and with different kinds of instructional activities (Greer et al. 2007). We deem that an early introduction in schools of fundamental ideas about modelling is not only possible but also indeed desirable even at the primary school level. We argue for modelling as a means of recognizing the potential of mathematics as a critical tool to interpret and understand reality, the communities children live in, and society in general. An important aim for compulsory education should be to teach students to interpret critically the reality they live in and understand its codes and messages so as not to be excluded or misled (Bonotto 2007). As regard the problem posing, this process is of central importance in the discipline of mathematics and in the
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