|Cileda de Queiroz Silva Coutinho (Brazil)||email@example.com|
|Research on teaching and learning probability
has shown the resistance of some naive conceptions about randomness. Based
on some of these results, we designed and carried out a teaching sequence
to guide 14-15 years old students into a first contact with random situations
in a school context. We consider that one of the students' main difficulties
is to model a real situation as a stochastic experiment, in order to replicate
the situation. Our hypothesis was that introducing a pseudo-concrete intermediate
model, such as a diagram in geometry, would help students in this modeling
process. The teaching sequence is based on the introduction of Bernoulli's
urn model to students.
A computer simulation of this model where geometric probabilities are introduced provides an experimental approach. In order to make clear the need of a high number of trials in the experiment to estimate a probability, we compare the results obtained by mechanical simulation of random experiments with those obtained by computer simulation. Students are faced to two kinds of simulation of a random experiment: mechanical and computer based simulations. The mechanical simulation consisted of drawing pearls, with replacement, from a pot. The computer based simulation consisted of "drawing" of pixels from a figure in the computer screen in a computer's mechanism which allows the "drawing" of a pixel ("Urn of Pixels"). This device was designed and implemented using the software Cabri Geometre II in association with Excel.
The use of a geometrical frame to present some random situations allowed the students to compare theoretical results (deduced from geometrical probabilities, such as areas' ratio) with experimental results (frequency's stability). This served to give a meaning to the laplacean and frequentist approaches to probability.
The activities we propose in the second part of our didactical engineering (computational environment) are based on the game of Franc-Carreau, which was proposed by a French mathematician and biologist, Buffon, in the 18th century.
The first results of our research have shown that 14-15 years-old students only need proportional thinking as mathematical background to give a meaning to the link between Bernoulli's urn model and the real situation represented by this model. The concept of relative frequency assumes also a grasp of this modeling process. Therefore, our teaching sequence enable students to recognize the configuration of a Bernoulli's experiment when they are faced to random situations and to give the composition of the Bernoulli's urn which model this situation. As a consequence, students are able to estimate or to calculate some probabilities inside a theoretical model, even in the case they have not the theoretical knowledge on the probability framework, since the work is done in the pseudo-concrete's domain.
Key words : probability, model of Bernoulli's urn, simulation, didactics of mathematics.
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