# COMPLEMENTARY MATHEMATICS I

## Learning outcomes of the course unit

Knowledge and understanding. Today mathematics education is a complex phenomenon calling for knowledge from different field, and not only from mathematics. History of mathematics, epistemology and philosophy of mathematics together with specific knowledge of didactical research give important contributions to education and to student’s culture. The course will contribute to the knowledge of relevant historic times of the mathematical thought, by a direct and/or mediate use of original texts. In this way it is possible to follow the evolution of concepts in times and under the influence of various cultural currents. The course will train student to an autonomous comprehension of an ancient text and to recognize the philosophic panorama or the didactic framework influencing the text. The aim is that student turns to able to read the advanced texts and documents produced by research of the field.

Applying knowledge and understanding. Students will be required to find solution of problems from the original texts. Moreover they will become able to choose the suitable frameworks in which to insert the topics in order to devise and to conduct personal argumentation regarding the course subjects.

Making judgement. I will assess the students ability for identifying the historic and contents parameters. The final knowledge will help student on working in team and singly and on individuating the main features of the subjects.

Communication skills. Students will reach a firm knowledge of communication process for mathematics as a scientific subject and as branch of learning.

Learning skills. Students have to acquire flexibility in order to adapt themselves to the need of listeners whom their mathematical communication is addressed.

## Prerequisites

Algebra 1

## Course contents summary

First steps in Mathematics Education

Analogy, generalization, universalization, abstraction, experimental induction. Conceptual fields, dynamics and difficulties of algebraic thinking. Thinking be elements, thinking by structures. The role of substitutions.

History of documents and history of ideas. History of Algebra, from the ancient times to Galois. Algebraic equations of 1st , 2nd, 3rd and 4th degree. Geometrical Algebra in Euclid and in the subsequent epochs.

Origin and organization of Mathematics. Some problems of Philosophy of Mathematics. Concise treatment of cardinals and ordinals. Actual and potential infinity.

Peano’s postulates and recursion. Principles of induction and of minimum. Ordering and between natural numbers. Congruence. Mathematics as the science of the form. An overview of the arithmetization of Analysis.

## Recommended readings

L. Bazzini, R. Iaderosa, Approccio all'Algebra, FRANCO ANGELI, Milano, 2000;

R. Franci, L. Toti Rigatelli, Storia della teoria delle equazioni algebriche, MURSIA,1979;

S.MacLane, Mathematics, Form and Function, Springer, Berlin, 1986

C. Marchini, Appunti di Matematiche Complementari, I Mod. A.A 2008/2009.

## Teaching methods

Knowledge and understanding. Today mathematics education is a complex phenomenon calling for knowledge from different field, and not only from mathematics. History of mathematics, epistemology and philosophy of mathematics together with specific knowledge of didactical research give important contributions to education and to student’s culture. The course will contribute to the knowledge of relevant historic times of the mathematical thought, by a direct and/or mediate use of original texts. In this way it is possible to follow the evolution of concepts in times and under the influence of various cultural currents. The course will train student to an autonomous comprehension of an ancient text and to recognize the philosophic panorama or the didactic framework influencing the text. The aim is that student turns to able to read the advanced texts and documents produced by research of the field.

Applying knowledge and understanding. Students will be required to find solution of problems from the original texts. Moreover they will become able to choose the suitable frameworks in which to insert the topics in order to devise and to conduct personal argumentation regarding the course subjects.

Making judgement. I will assess the students ability for identifying the historic and contents parameters. The final knowledge will help student on working in team and singly and on individuating the main features of the subjects.

Communication skills. Students will reach a firm knowledge of communication process for mathematics as a scientific subject and as branch of learning.

Learning skills. Students have to acquire flexibility in order to adapt themselves to the need of listeners whom their mathematical communication is addressed.