LEARNING OUTCOMES

The course aims at promoting a reflection on the complex process of mathematical modeling and on the approximate and previsional nature of methods and results. The course addresses technical aspects of modeling, as well as historical/epistemological and didactical ones. Students will be led to reflect on the construction of a mathematical model and on the comparison between deterministic and probabilistic models. Students will also be led to reflect on the required knowledge for a first approach to modeling in school, and on difficulties and potentialities of such a first approach.

AIMS AND LEARNING OUTCOMES

The course will encompass:

• A preliminary questionnaire on students’knowledge in probability, their intuitive image of modeling and their expectations on the course.
• A historical/epistemological overview on the role of mathematical models and the difference between deterministic and probabilistic models, with a special focus to those aspects that are more related to the teaching and learning of modeling at secondary school
• A review (by means of the study of concrete situations, that relevant in history or the contemporary world) of thise basic concepts and methods in probability and statistics that are useful for modeling
• The construction, through examples, of the idea of probabilistic model as a way to represent a complex real phoenomenon. This is also aimed at making prospective teachers reflect on the mathematical representation of reality and the shift from certainty to uncertainty
• The analysis of teaching pathways for a first approach to statistical-probabilistic models in secodnary school
• The study of elementary differential models in the fields of physics, bioogy, demography, epidemiology, …)
• The study of inferential stocastic processes (Markov chains, differential equations of Markovian stocastic processes, …) and the comparison with the deterministic models

The analysis of experimental data in order to evaluate whether the models are suitbale to represent real situations (linear regression, exponential regression, tests…)

PREREQUISITES

Mathematical analysis: one-variable functions, integral calculus, differential equations

Physics:  basic notions of mechanics, thermodynamics and electromagnetism.

Probability: elementary probability, discrete and continuous random variables, law of large numbers and central limit theorem. 

Didactics of mathematics: tools and knowledge of epistemology, history of mathematics, and psychology,  finalized to the analysis of the problems related to the teaching and to the learning of mathematics in the secondary school.

TEACHING METHODS

Lectures, individual work on worksheets, class discussions

SYLLABUS/CONTENT

Mathematical modeling (differential models, stocastic processes) and its teaching and learning in secondary school: from a historical/epistemological reflection on modeling processes to educational design

RECOMMENDED READING/BIBLIOGRAPHY

- Belcastro A., Guala E., Eserciziario di probabilitá e statistica, Dip. di Matematica,Ge

- Costantini D., Monari P., Probabilità e giochi d’azzardo, Franco Muzzio Editore

- Devlin K., La lettera di Pascal, Rizzoli

- Glaymann M., Varga T., La probabilità nella scuola dell’obbligo, Armando Editore 

- Guala E., Dispense di probabilitá e statistica Dipartimento di Matematica, Ge

- Guala E., Modelli deterministici e modelli probabilistici, Dip.di Matematica, Ge

- Hacking I., L’emergenza della probabilità, Il Saggiatore

- Israel G., La visione matematica della realtá, Laterza

- Mood A.M., Graybill F.A., Boes D.C., Introduzione alla statistica, McGrawHill

- Pintacuda N., Insegnare la probabilità, Franco Muzzio Editore

- Vajani L., Saggi sui processi stocastici, Giuffré Editore

- Volterra V., D’Ancona U., Les associations biologiques étudiées au point de vue mathématique, Hermann, Paris