Authentic modelling in stochastics education 1 Authentic modelling in stochastics education – the case of the binomial distribution Rolf Biehler Department of Mathematics/Informatics University of Kassel, Germany Email: biehler@mathematik.uni-kassel.de Giving more room to authentic modelling in the classroom is one of the goals of all the research and development efforts under the heading of application ori- ented mathematics education. From this perspective, I will criticize and analyze pseudo-realistic applications of the binomial distribution that are abundant, even in most recent German textbooks. Alternatives for more authentic modelling with binomial distributions that include real data analyses and modern computer in- tensive statistical methods will be developed. As an intermediate step, paying more attention to “validation awareness“ would be very helpful. 1 Introduction A fundamental object of criticism in the tradition of application oriented mathematics teach- ing is the so called “eingekleidete Aufgabe” (dressed-up mathematical problem). Instead of analysing a real situation and thinking about which mathematical model may be adequate for a better understanding of the real situation students try to find out which model was dressed up and they tend to look at those concepts and models that they have learned recently. Dressed-up mathematical problems can have several effects: Students get a wrong picture and qualification with regard to what is required in authentic modelling situations. Moreover, such problems may cause problems of understanding in the classroom. Whereas the teacher has the seemingly unique model in mind some students may take the real situation more seri- ous, do not recognize the dressed up mathematical model and come up with alternative mod- els that in turn the teacher does not accept. A didactical alternative is giving room to more complete and authentic modelling proc- esses: starting from a real situation, constructing a so-called “reality model” (Realmodell), mathematising the reality model/setting up the mathematical model, working mathematically in the mathematical model, interpreting and validating the model against the real situation with the option of restarting the modelling cycle again (Blum 1995). The real situation should be in the foreground, the initial mathematical model and the final results have to be validated. Nevertheless, a consensual position seems to be that dressed-up problems can also have a di- dactical value if they are used in a didactically advanced sense: the teachers aim at training students’ abilities to translate a reality model into a mathematical model and at outlining the potential range of applications of a concept by means of idealized prototypical situations. For instance, such a prototypical situation for the discrete uniform probability distribution is drawing balls with replacement from a box, where all the balls have the same shape and weight and are well mixed-up so that all have the same chances for being drawn. In the termi- nology of Werner Blum (Blum & vom Hofe 2003), we could identify the “drawing from an urn model” as a “Grundvorstellung” (mental model, basic idea) for the discrete uniform prob- ability distribution. Grundvorstellungen (mental models, basic ideas) function as cognitive Aus: Henn, H.-W., Kaiser, G. (Hrsg.): 2005. Festschrift für Werner Blum, Hildesheim: Franzbecker, S. 19 - 30