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

To focus on some foundational problems relating to the main mathematical areas addressed in upper secondary education and their connection with the cultural and pedagogical choices that a teacher must face in setting up and developing his own teaching activity.

AIMS AND LEARNING OUTCOMES

At the end of the course, students will be able to: understand the logical and axiomatic foundations of elementary mathematics; formalize definitions and proofs in arithmetic, algebra, and geometry; relate elementary contents to higher mathematical structures (sets, numbers, algebraic structures); recognize the principles underlying the construction of number systems; critically analyze epistemological and didactic aspects of elementary mathematics; design and deliver presentations aimed at a non-specialist audience.

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.

Analysis of examples in which the impossibility of solving a problem has led to revolutionary discoveries, exploring how such failures have stimulated the evolution of mathematical thought. Emblematic cases will be treated such as Cantor's theorem, Russell's paradox, the independence of Euclid's fifth postulate, the Entscheidungsproblem and Hilbert's tenth problem, the halting problem and Arrow's impossibility theorem in the theory of social choice.

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.

G. Birkhoff, S. Mac Lane — A Survey of Modern Algebra

E. Landau — Foundations of Analysis

Thomas Jech — The Axiom of Choice

R. Smullyan — Gödel's Incompleteness Theorems

M. Davis — Computability and Unsolvability

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written test and an oral test, which can be taken in any order, even in itinere.

The written test concerns the topics covered during the course and consists of the presentation and written discussion of one or more topics covered in class.

The oral test consists of a critical presentation and discussion of a topic chosen from those in the program, agreed with the teacher.

​The final grade takes into account the coherence between the written paper and the oral presentation, the critical reflection on the foundations of elementary mathematics, the autonomy in reworking the contents.

Si consigliano gli studenti con certificazione di DSA, di disabilità o di altri bisogni educativi speciali di contattare il docente all’inizio del corso per individuare modalità didattiche e d’esame che, nel rispetto degli obiettivi dell’insegnamento, tengano conto delle modalità di apprendimento individuali e forniscano idonei strumenti compensativi.

ASSESSMENT METHODS

The written test evaluates the formal correctness of the mathematical exposition, the ability to connect elementary concepts to broader theoretical structures, the conscious use of mathematical language, the clarity of argumentation.

The oral test evaluates the ability to present clearly and rigorously, the depth of conceptual understanding, the ability to connect the topic to historical, didactic or theoretical contexts.

Students with DSA  (=Specific Learning Disabilities) certification, disability or other special educational needs are advised to contact the teacher at the beginning of the course to establish on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.