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t0д╞c ╞4c фф■d■SOME ISSUES IN THE TEACHING-LEARNING OF PROBABILITY
BY MODELLING AND SIMULATION
Jean Claude Girard
IUFM de Lyon, France
jean-claude.girard@lyon.iufm.fr
INTRODUCTION
Since the last reform in France, a probability distribution is defined, at the high school level, as a theoretical frequency distribution modelling a random situation. It uses simulation intensively. This reform obviously intends to link probability and statistics but two main didactic issues can be pointed out.
THE TIME PARAMETER IN THE MODELLING PROCESS
The hypothesis behind the new syllabus is the analogy with geometry and building the Euclidean model of geometry is a long process that extends over more than ten years. The starting point is the observation of concrete objects. These objects are identified on the whole then their properties are progressively checked out with instruments. The corresponding mathematical objects are then defined only by their properties. The road from the first observations to the hypotheticaldeductive reasoning is long.
The curriculum suggests for Probability a modelling process of the same kind and in this case it extends only over two years. It also comes too late when many misconceptions are settled in the pupilsТ brains.
Mental models of randomness are shaped less easily than the mental models of geometric patterns. They need much more time to be built. We must offer the pupils many situations that are likely to shape probabilistic models long before high school. Past researches have shown that junior high school pupils can build equivalent random experiments and model random situations by Bernoulli urns. Nothing is provided about teaching randomness in France, at that age or before. Modelling in this case runs the risk of being fragile but extending over a noticeably long time.
THE AMBIGUITY OF SIMULATION
According to the syllabus УTo simulate a random experiment, is to choose a model of this experiment, then simulate this modelФ and УTo model an experiment whose values are in a space E, is to choose a probability distribution P on E.Ф As a result of this, we must have a model to simulate (in grade 10), i. e. a probability distribution that will be introduced the next year only (grade 11). This is something like a vicious (didactic) circle.
The same syllabus asserts УTo work out and/or check a probabilistic model, the first tool we have is a mathematics theorem called the law of large numbers.Ф The law of large numbers is a theorem internal to the mathematical theory while the frequencies obtained in the experience or by the simulation are facts. So, there is an epistemological ambiguity that leads us not to ask if we are in the model built to describe reality or in the reality itself. . This situation could increase the confusion between frequency and probability instead of solving it.
Furthermore, recent research put forwards that, instead of opening up to other fields, simulation keeps mathematics inside themselves. This is to be linked with the lack of consideration given to experimentation in mathematical work.
CONCLUSION
In conclusion, teaching probability by modelling and simulation is not so easy. It surely needs a longer period than the one planned. The link between statistics and probability as well as the part of simulation in the learning of probability are still to be clarified. This could be the subject of new research leading to the correct curricula throughout all the school years.
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