# ELEMENTARY MATHEMATICS FROM AN ADVANCED STANDPOINT

OVERVIEW

The lecture course presents the basic theories of mathematics (geometry, arithmetic, calculus, set theory) within the perspective of present day mathematics, with the intent to highlight the technical points which must be known by the lecturer to give a clear presentation of such disciplines to a lay audiience. In particulare basic tools will be offered for the preparation of didactic activities and the discussion with students.

## AIMS AND CONTENT

LEARNING OUTCOMES

The lecture course presents the formal theories for the Euclidean geometry, non-Euclidean geometries, arithmetic, real analysis and set theory, and compares those with the standard informal presentations. From the expert's perspective various didactical questions related to the teaching of geometry, set theory and foundations will be analysed to provide the teacher-to-be with useful advanced tools to organize and develop one's own teaching practice such as, for instance, correct intuitions, relevant themes, significant problems and examples, applications, and interdisciplinary aspects.

AIMS AND LEARNING OUTCOMES

Study of the basic theories from the expert's point of view, to analyze didactic issues and research questions.

PREREQUISITES

None. Standard mathematical practice may be useful.

Teaching methods

Teaching style: In presence

SYLLABUS/CONTENT

Historical setting for the basic mathematical theories: Euclidean geometry, non-Euclidean geometries, arithmetic, real analysis, set theory. Review of the basics of mathematical logic and first order theories related to the basic mathematical theories.

Applications and experiments in didactics: Pythagora's theorem; Euler's line; Ceva's theorem; angles in a triangle; parallel lines; Fibonacci's sequence; Pascal's triangle; Matijasevic-Robinson theorem; algorithms for elementary operations.

RECOMMENDED READING/BIBLIOGRAPHY

Course notes and slides presented during the lectures will be available on aulaweb, complemented by other material. Notes taken at the lectures and the material on aulaweb are enough in preparation for the exam. The books listed below are good references.

Mendelson, E. Introduzione alla logica matematica. Boringhieri

Prodi, Analisi matematica, Boringhieri

Russo, Il primo libro degli elementi di Euclide, Carocci

## TEACHERS AND EXAM BOARD

**Ricevimento:** by appointment

Exam Board

GIUSEPPE ROSOLINI (President)

RUGGERO PAGNAN

FRANCESCA MORSELLI (President Substitute)

## LESSONS

Teaching methods

Teaching style: In presence

LESSONS START

The class will start according to the academic calendar.

## EXAMS

Exam description

The exam consists of a written essay and of an oral examination which can be taken in either order. The written essay is on the arguments of the lecture course and asks for the presentation of particular subjects taught in the course. The oral examination is a presentation and an open discussion of subjects in the syllabus. The final mark determines how the two tests complement each other. The oral examination can be taken in itinere.

Assessment methods

The written essay verifies the actual acquisition of the mathematical knowledge of the basic mathematical theories. The oral examination consists mainly in a presentation of some part of the syllabus and aims at evaluating that the student has acquired an appropriate level of knowledge and analytical skills about the mathematical theories.