# COMPLEMENTARY MATHEMATICS

## Learning outcomes of the course unit

The student will acquire notions and develop skills related to the History of Mathematics and foundational thematics typical of modern mathematics, eith the general aim of giving him a general overview of basic mathematics, also from an epistemological point of view. In particular the student must be able:

1. To explain the main problems, methods, principles and theorems presented during the lessons

2. Compare different approaches to the same problems in different epochs and with different points of view and tools

3. Organize the knowledge around fundamental themes presented in the course, taking in account the historical collocation and the threads that, in the history, have being enriched time after time from the Ancient era to the last century.

4. To present and discuss in a didactical perspective for middle and high school the crucial themes of basic mathematics and identify epistemological and didactical obstacles.

## Course contents summary

The Babylonian and Egyptian mathematics.

The Greek mathematics: Thales, Pythagoras and his school, the crisis of

incommensurables. Zeno’s paradoxes.

The three famous problems of Greek antiquity: quadrature of the circle, duplication

of the cube, trisection of angle. Hippocrates and the quadrature of lunula.

Euclide: the “Elements”, commons notions, postulates and axioms, theory of

Paralleles, proportions theory, magnitudes, prime numbers, equivalence in plane

and space. Euclide’s work in modern epistemology.

Archimedes: from the measure of circle to the volume of sphere, the method of

exhaustion.

Apollonius: conic sections.

Numerics systems, their properties in a historical perspective and axiomatizations: natural, integer, rational, real.

Non-Euclidean geometries: hystorical and epistemological aspects, Poincaré’s and

Klein’s models.

The Erlangen program and the transformations geometry: congruence, similarity,

affinity, projectivity.

The problem of foundations of Geometry: the Hilbert’s axioms, indipendence,

coherence, completeness. Different notions of completeness. Dialectic relation between intuition and formalization in the evolution of Analysis and in modern axiomatizations. Argumentation and proof in a didactical perspective. The role of History of Mathematics in Mathematics education for supporting the design of teaching-learning activities: epistemological and didactical obstacles (obstacles theory, Brousseau); problematization of the use of History in the didactical practices (actualization or contextualization); relationship between real world and mathematics in the perspective of Euclid and Hilbert, realism and pragmatism and the relevance of the teacher's implicit philosophy of mathematics.

## Recommended readings

F. Speranza, L. Ferrari (2008). Matematiche Complementari. Appunti delle lezioni.

F. Speranza, Scritti di Epistemologia della Matematica, Pitagora, Bologna, 1997.

E. Agazzi, D. Palladino, Le geometrie non euclidee e i fondamenti della geometria

dal punto di vista elementare, La Scuola Editrice, Brescia, 1998.

C.B.Boyer, Storia della Matematica, Mondadori, Milano, 1980.

D'Amore, B. (1999). Elementi di didattica della matematica. Bologna: Pitagora

## Teaching methods

Lectures will be mainly in transmissive style, but with a steady dialogue with students which can be called to the blackboard for discussing problems, or for showing their understanding of and taking part to the course. Also group works to discuss problems and relevant issues will be used as a methodology. Students will be asked to take part to seminar for studying in depth some course topics.

## Assessment methods and criteria

Oral examination.

Assessment will be made by a final oral, in which student must solve mathematical or interpretative problems