Three Approaches for Modelling Situations with Randomness
Three different approaches to the concept of probability dominate the teaching of stochastics: the classical, the frequentistic and the subjectivistic approach. Compared with each other they provide considerably different possibilities to interpret situat
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Abstract Three different approaches to the concept of probability dominate the teaching of stochastics: the classical, the frequentistic and the subjectivistic approach. Compared with each other they provide considerably different possibilities to interpret situations with randomness. With regard to teaching probability, it is useful to clarify interrelations and differences between these three approaches. Thus, students’ probabilistic reasoning in specific random situations could be characterized, classified and finally, understood in more detail. In this chapter, we propose examples that potentially illustrate both, interrelations and differences of the three approaches to probability mentioned above. Thereby, we strictly focus on an educational perspective. At first, we briefly outline a proposal for relevant teachers’ content knowledge concerning the construct of probability. In this short overview, we focus on three approaches to probability, namely the classical, the frequentistic and the subjectivistic approach. Afterwards, we briefly discuss existing research concerning teachers’ knowledge and beliefs about probability approaches. Further, we outline our normative focus on teachers’ potential pedagogical content knowledge concerning the construct of probability. For this, we discuss the construct of probability within a modelling perspective, with regard to a theoretical perspective on the one side and with regard to classroom activities on the other side. We further emphasize considerations about situations which are potentially meaningful with regard to different approaches to probability. Finally, we focus on technological pedagogical content knowledge. Within the perspective of teaching probability, this kind of knowledge is about the question of how technology and, especially simulation, supports students understanding of probabilities.
A. Eichler (B) University of Education Freiburg, Kunzenweg 19, 79117 Freiburg, Germany e-mail: [email protected] M. Vogel University of Education Heidelberg, Keplerstraße 87, 69120 Heidelberg, Germany E.J. Chernoff, B. Sriraman (eds.), Probabilistic Thinking, Advances in Mathematics Education, DOI 10.1007/978-94-007-7155-0_4, © Springer Science+Business Media Dordrecht 2014
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A. Eichler and M. Vogel
Fig. 1 Different kinds of knowledge which teachers should have according to Ball et al. (2008)
1 Introduction The success of any probability curriculum for developing students’ probabilistic reasoning depends greatly on teachers’ understanding of probability as well as much deeper understanding of issues such as students’ misconceptions [. . .]. (Stohl 2005, p. 345)
With these words Stohl (2005) begins her review on teachers’ understanding of probability. The first part of this quote seems to be self-evident. However, probability is known as a difficult mathematical concept yielding “counterintuitive results [. . .] even at very elementary levels” (Batanero and Sanchez 2005, p. 241). Even to grasp the construct of probability itself lasted centuries
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