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Second Level Degree in Mathematics

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ELEMENTARY MATHEMATICS FROM A SUPERIOR POINT OF VIEW - 6 CFU

Teacher

Luigi Tomasi

Scheduled Period

I Year - 1 Semester | 04/10/2021 - 15/01/2022

Hours: 48 (24 esercitazione, 24 lezione)

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Prerequisites

The mathematical prerequisites are those covered by basic courses of the Bachelor's Degree in Mathematics, in particular Algebra, Geometry, Mathematical Analysis, Probability and Foundations of Mathematics.

Target skills and knowledge

-Developing a critical and in-depth vision of some specific topics of elementary mathematics (Arithmetic and Algebra, Geometry, Relations and Functions, Data and Forecasting) from an epistemological, historical and didactic point of view.
-Knowing broadly the objectives for mathematics set by the current Italian national curriculum for secondary school.
-Knowing the conceptual, epistemological and didactic issues of the teaching and learning of mathematics in secondary school.
-Knowing and knowing how to use technologies (in particular, educational software) in the teaching and learning of Mathematics in secondary school, being aware of their strengths and weaknesses.
-Knowing how to critically explain and connect the various topics covered in the course.

Examination methods

Oral exam plus an in-depth written report on a fundamental topic (assigned by the teacher) covered in the course. The report will be presented in the last part of the course or, alternatively, in the first part of the exam.

Assessment criteria

The student is expected to
-know in a critical way -from a historical, methodological and didactic point of view- the fundamental topics of mathematics curricula for secondary schools
-master the course contents, which are part of the pre-service training of prospective mathematics teachers
-know how to expose and critically connect the notions learned.
The written report submitted by the student will be assessed for one third of the exam grade; the other two thirds of the grade will be assigned for the oral exam.

Course contents

The course will discuss, from an epistemological, historical and didactic point of view, those topics and ideas of basic mathematics that constitute the fundamental themes of the Mathematics curriculum in secondary school:
-Arithmetic and Algebra
-Geometry
-Relations and functions (in particular, Mathematical Analysis)
-Data and Forecasting (in particular, Probability).
In particular, the following topics will be specifically developed (some will be requested in the students' reports):
-Solution of algebraic equations in radicals, with hints on the history of classical algebra
-Straightedge and compass constructions: the classical problems of geometry; points of the plan that can be constructed with straightedge and compass; constructible numbers; constructible regular polygons, with historical notes.
- Elementary number theory topics: Pythagorean triples, prime numbers, fundamental theorem of arithmetic, congruence relation, Euclid's algorithm for the gcd, divisibility criteria; the Euler functions d(n), sigma(n), phi(n); decimal fractions and period of a fraction. The principle of mathematical induction.
- Definitions, theorems and proofs in mathematics teaching (in secondary school); types of proofs; indirect proofs; proofs by contradiction; proofs by induction.
-Axiomatic systems for geometry (Euclid, Hilbert, Choquet,…) and teaching of geometry in secondary school; some approaches to teaching geometry in secondary school. "Paths" for the teaching of geometry in secondary school.
-Sequences and functions; the definition of limit; definition of derivative and definition of definite integral; epistemological, historical and didactic considerations.
-Probability; the definitions of probability; basic theorems of probability.
These elementary mathematics topics will be presented "from a superior point of view", that is, with critical attention to their foundations, their history and their current teaching (in secondary school).
During the course, the use of technological tools for teaching and learning mathematics in secondary school will also be proposed, underlining their methodological value, with frequent examples of use of mathematical software (in particular, the use of the GeoGebra software will be proposed).

Planned learning activities and teaching methods

- Frontal lessons
- Dialogue lessons and guided discussions
- Written reports by students on specific topics (assigned by the teacher)
- Computer exercises with mathematics teaching software (e.g. GeoGebra)

Additional notes about suggested reading

Lecture notes provided by the teacher.
Slides, presentations and recordings of the lessons provided by the teacher.
Books, journals and websites recommended during the course.

Textbooks (and optional supplementary readings)

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