Didactics of Infinitesimal Calculus (first part)
A.Y. 2024/2025
Learning objectives
The main objective of the course is to introduce the theoretical and applicative aspects of the Didactics of Infinitesimal Calculus, with particular attention to the context of secondary school. In the first part of the course the student will revisit the main topics of study of infinitesimal calculus (limits, derivatives, integrals) deepening the historical roots and the motivations behind their development, and will obtain a critical vision of contents and techniques, and their role in modern Mathematical Analysis.
In the second part of the course will be presented an adaptation to the case of infinitesimal calculus of some results of national and international research in mathematics teaching, with reference to the national indications and reference frameworks for the evaluation of skills, to the criteria underlying the design and implementation of mathematical teaching activities for secondary school, to the tools for the analysis of difficulties and teaching strategies oriented to the enhancement of excellence or inclusion in mathematics. Students will develop skills in didactic design and analysis of the criticalities of learning processes in the case of infinitesimal calculus.
In the second part of the course will be presented an adaptation to the case of infinitesimal calculus of some results of national and international research in mathematics teaching, with reference to the national indications and reference frameworks for the evaluation of skills, to the criteria underlying the design and implementation of mathematical teaching activities for secondary school, to the tools for the analysis of difficulties and teaching strategies oriented to the enhancement of excellence or inclusion in mathematics. Students will develop skills in didactic design and analysis of the criticalities of learning processes in the case of infinitesimal calculus.
Expected learning outcomes
Students must be able to present in an exhaustive way their knowledge related to the first and second part of the course and to design teaching activities and evaluation/assessment tests for secondary school students, based on the results and theoretical frameworks of the research in mathematics teaching presented in the course and on the national indications, in relation to the case of infinitesimal calculus. They must also have developed analytical skills, in particular to be able to critically analyze mathematics textbooks and other teaching resources related to the teaching of infinitesimal calculus. At the same time they will have to acquire communication skills, arguing their choices and exposing their knowledge with a good balance between precision in language and discussion of concrete examples.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
First part:
The notions of infinite and infinitesimal
Calculus of lengths, areas, and volumes: the method of exhaustion and the Method of Mechanical Theorems by Archimedes
Development of Calculus: Kepler, Galilei, Cavalieri, Torricelli, Fermat, Pascal, Wallis
Tangent lines and rates of change: the birth of differential calculus
The Leibniz & Newton methods and the birth of Calculus
A rigorous foundation of Calculus, and the development of modern Analysis
Second part: The role of the history of mathematics in teaching
National guidelines for the curriculum
Main learning difficulties of Calculus and mathematical analysis in secondary school
Intuition and rigor in the field of Calculus and mathematical analysis in secondary school Argumentation and proof of Calculus and mathematical analysis in secondary school
Design of teaching activities and evaluation/assessment tests
The notions of infinite and infinitesimal
Calculus of lengths, areas, and volumes: the method of exhaustion and the Method of Mechanical Theorems by Archimedes
Development of Calculus: Kepler, Galilei, Cavalieri, Torricelli, Fermat, Pascal, Wallis
Tangent lines and rates of change: the birth of differential calculus
The Leibniz & Newton methods and the birth of Calculus
A rigorous foundation of Calculus, and the development of modern Analysis
Second part: The role of the history of mathematics in teaching
National guidelines for the curriculum
Main learning difficulties of Calculus and mathematical analysis in secondary school
Intuition and rigor in the field of Calculus and mathematical analysis in secondary school Argumentation and proof of Calculus and mathematical analysis in secondary school
Design of teaching activities and evaluation/assessment tests
Prerequisites for admission
First part: knowledge of the fundamentals of Calculus, as presented in the first classes in Mathematical Analysis: real numbers and their properties; limits of sequences and functions; differentiability; the Riemann integral; antiderivatives and the Fundamental Theorem of Calculus; power series; Taylor series.
Second part:
Theory of didactic situations
Mathematical object and semiotic representations
Concept image/concept definition
Argumentation and proof in the teaching of mathematics in secondary school
Intuition and rigor in the teaching of mathematics in secondary school
Method of the varied research
Role of technologies in the learning of mathematics, with a focus on the use of Geogebra software
Second part:
Theory of didactic situations
Mathematical object and semiotic representations
Concept image/concept definition
Argumentation and proof in the teaching of mathematics in secondary school
Intuition and rigor in the teaching of mathematics in secondary school
Method of the varied research
Role of technologies in the learning of mathematics, with a focus on the use of Geogebra software
Teaching methods
The course will be delivered in the first part with the modality of the frontal lesson, in the second part frontal lessons will alternate with group laboratory activities.
Teaching Resources
E. Giusti - Piccola storia del calcolo infinitesimale dall'antichità al Novecento. ISBN: 8881474565
Carl B. Boyer (1959).The History of the Calculus and its Conceptual Development. Dover Publications.
A. Baccaglini-Frank, P. Di Martino, R. Natalini, G. Rosolini - Didattica della matematica. ISBN: 978-88-6184-550-3
Additional material provided by the teacher
Carl B. Boyer (1959).The History of the Calculus and its Conceptual Development. Dover Publications.
A. Baccaglini-Frank, P. Di Martino, R. Natalini, G. Rosolini - Didattica della matematica. ISBN: 978-88-6184-550-3
Additional material provided by the teacher
Assessment methods and Criteria
First part: Oral exam
Second part: Oral exam and delivery of a project to be discussed during the oral exam.
Both the first and second part of the exam will be evaluated simultaneously in a single oral test, the assessment will be overall and established by mutual agreement by both teachers.
Second part: Oral exam and delivery of a project to be discussed during the oral exam.
Both the first and second part of the exam will be evaluated simultaneously in a single oral test, the assessment will be overall and established by mutual agreement by both teachers.
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 6
Lessons: 42 hours
Professors:
Aspri Andrea, Branchetti Laura
Educational website(s)
Professor(s)
Reception:
By appointment
Online, Microsoft Teams