# TEACHING MATHEMATICS A

## Learning outcomes of the course unit

Knowledge and ability to understand: through the lectures held during the course, the student will acquire the methods and knowledge necessary to analyze teaching proposals, curricula and materials for the mathematics of secondary school (first and second degree), identifying design criteria, strengths and weaknesses. He will learn the main concepts, some empirical methods and research results in mathematics education. He will learn the structure of the National curricula, INVALSI and OECD-PISA frameworks, and other national documents including guidelines for teachers. It will also learn to make an appropriate research of materials and articles to formulate a new teaching proposal and to structure evaluation tests that take into account the results of research and regulatory and institutional constraints, including inclusion and promotion of a deep and meaningful mathematical thought.

In the group works, devoted to deepening some topics and to the preparation of the final project, students learn how to apply the acquired knowledge in a real context of design of teaching units and evaluation tests and questionnaires / interview protocols for the research. In particular, the student will have to apply the acquired knowledge to the analysis of research articles, textbooks, curricula, standardized assessment tests, real student protocols, as well as the design of a complete teaching unit or short survey-based research (questionnaires and interviews).

Autonomy of judgment

The student must be able to evaluate and critically choose a teaching strategy and evaluate the potential impact of planning decisions and implementation of the teaching action on different types of students, analyze data and protocols quantitatively and qualitatively to support decision-making choices and design.

Communication skills

Through the lectures and the comparison with the teacher, the students acquire the specific vocabulary inherent in mathematics education as a scientific discipline. It is expected that, at the end of the course, the student is able to transmit, in oral and written form, the main contents of the course, the key ideas for interpreting classroom situations, learning problems and possible solutions. The student must communicate his / her knowledge with a good balance between precision in the language and exhibition of concrete examples among those analyzed, as well as showing that he has gained his own point of view.

Learning ability

The student who has attended the course will be able to deepen their knowledge in mathematics education through the independent consultation of specialized texts, scientific or popular journals, even outside the topics dealt with strictly in class, in order to deal effectively inclusion in the school world or undertake further training courses.

Knowledge and ability to understand: through the lectures held during the course, the student will acquire the methods and knowledge necessary to analyze teaching proposals, curricula and materials for the mathematics of secondary school first and second degree, and identify the design criteria and strengths and weaknesses. He will learn the main concepts, some methods and empirical results of didactic research in mathematics. He will learn the structure of the national indications, of the INVALSI and OECD-PISA frameworks, of the cultural axes. In particular, the student will have to apply the acquired knowledge to the analysis of research articles, textbooks, curricula, and standardized evaluation tests.

Autonomy of judgment

The student must be able to evaluate and critically choose a teaching strategy and evaluate the potential impact of planning decisions and implementation of the teaching action on different types of students.

Communication skills

Through the lectures and the comparison with the teacher, the student acquires the specific vocabulary inherent in the teaching of mathematics as a scientific discipline. It is expected that, at the end of the course, the student is able to transmit, in oral and written form, the main contents of the course, the key ideas for interpreting classroom situations, learning problems and possible solutions. . The student must communicate his / her knowledge with a good balance between precision in the language and exhibition of concrete examples among those analyzed, as well as showing that he has gained his own point of view.

Learning ability

The student who has attended the course will be able to deepen their knowledge in mathematics education through the independent consultation of specialized texts, scientific or popular journals, even outside the topics dealt with strictly in class, in order to deal effectively inclusion in the school world.

## Prerequisites

There are no compulsory propaedeutical courses

no

## Course contents summary

The course aims to provide students with the general criteria for the design and implementation of mathematics teaching units for secondary school, as well as for formative and summative assessment tests, tools for the analysis of difficulties and teaching strategies oriented towards inclusion in mathematics and contrast of scholastic dispersion. The design and implementation criteria related to teaching practice and assessment are based on national and international research in mathematics education, but with a strong connection with current ministerial indications and international reference frameworks. Therefore, the contents proposed during the course of the lessons concern, on the one hand, key concepts and basic theories elaborated in mathematics education research and their declinations and practical applications in terms of didactic planning and evaluation tools and research, on the other institutional normative references.

Disciplinary and general education: perspectives on research

in education. Current research topics in International

Mathematics education, with particular attention to Secondary

and High School. The role of epistemology and history in

mathematics education research. Classic topics of disciplinary

research and learning difficulties in arithmetic, geometry,

algebra, analysis, probability and statistics. Knowledge,

skills, benchmarks for the development of key competences

for the citizenship (examples from national and international

development and assessment programs). General research topics,

with particular attention to the national research, for a

scientific approach to Mathematics education: Theory of situation

and Didactical Contract, Didactical transposition (Brousseau,

Sarrazy, D'Amore, Chevallard), Obstacles, Misconceptions,

Conceptual Change (Brousseau, Posner, Strike, Hewson & Gertzog), concept image and concept definition (Tall and Vinner), semiotic and

mathematics education (Frege, Peirce, Duval, Arzarello),

theory of figural concepts and mathematical intuition (Fischbein)

embodiment (Lakoff and Nunez), argumentation and proof

(Boero and Morselli), problem solving (Freudenthal,

Schoenfeld, D'Amore), the role of language in the

Learning of mathematics, formative and summative evaluation

(Bolondi), methodologies for teaching mathematics (laboratory,

math discussion, group work, technologies and software),

affect and beliefs (Zan, Di Martino), the role of examples

(Antonini), interdisciplinary approaches to mathematics and

physics.

Teacher, researcher, teacher-researcher: training paths and

possible professions in mathematics, school and research.

## Course contents

CAP1: RESEARCH IN THE DIDACTICS OF MATHEMATICS

Disciplinary teaching and general teaching: perspectives on research in the educational field. Current research topics in Mathematics Education at the international level, with particular attention to first and second level secondary schools: The role of epistemology and history in mathematics education research. Classical themes of disciplinary research and learning difficulties in arithmetic, geometry, algebra, analysis, probability and statistics. Knowledge, competence, reference frameworks for the development of key competences for the citizen (examples from national and international development and competence assessment programs). General research, with particular attention to research conducted in the national field, for a scientific approach to research in mathematics education: theory of situations, didactic contract, didactic transposition (Brousseau, Sarrazy, D'Amore, Chevallard), obstacles, errors , misconceptions (Brousseau, Posner), theory of figural concepts and intuition in mathematics (Fischbein), concept image and concept definition (Tall and Vinner), semiotics and didactics of mathematics (Duval, Mariotti and Bartolini Bussi, D'Amore, Godino and Font), argumentation and demonstration (Boero and Morselli), problem solving (Brousseau, Schoenfeld, D'Amore), the role of language in the learning of mathematics, formative and summative assessment (Bolondi), methodologies for the teaching of mathematics ( laboratory, mathematical discussion, group work, technologies and software), affects and convictions (Zan, Di Martino), interdisciplinarity between mathematics and physics.

CHAP 2: RESEARCH, TRAINING, DIDACTICS OF AULA

Teacher, researcher, teacher-researcher: training courses and possible professions in the field of mathematics education, between school and research. Examples of teaching units on different themes and for different scholastic orders and evaluation tests.

CAP1: RESEARCH IN THE DIDACTICS OF MATHEMATICS

Disciplinary teaching and general teaching: perspectives on research in the educational field. Current research topics in Mathematics Education at the international level, with particular attention to first and second level secondary schools: The role of epistemology and history in mathematics education research. Classical themes of disciplinary research and learning difficulties in arithmetic, geometry, algebra, analysis, probability and statistics. Knowledge, competence, reference frameworks for the development of key competences for the citizen (examples from national and international development and competence assessment programs). General research, with particular attention to research conducted in the national field, for a scientific approach to research in mathematics education: theory of situations, didactic contract, didactic transposition (Brousseau, Sarrazy, D'Amore, Chevallard), obstacles, errors , misconceptions (Brousseau, Posner), theory of figural concepts and intuition in mathematics (Fischbein), concept image and concept definition (Tall and Vinner), semiotics and didactics of mathematics (Duval, Mariotti and Bartolini Bussi, D'Amore, Godino and Font), argumentation and demonstration (Boero and Morselli), problem solving (Brousseau, Schoenfeld, D'Amore), the role of language in the learning of mathematics, formative and summative assessment (Bolondi), methodologies for the teaching of mathematics ( laboratory, mathematical discussion, group work, technologies and software), affects and convictions (Zan, Di Martino), interdisciplinarity between mathematics and physics.

CHAP 2: RESEARCH, TRAINING, DIDACTICS OF AULA

Examples of teaching units on different themes and for different scholastic orders and evaluation tests.

## Recommended readings

The slides projected during the course in PDF format and all the material used during lessons are made available to students and shared on the Elly platform. In addition to the shared material, the student can personally deepen some topics addressed during the course by referring to the following texts:

D’Amore, B. (1999). Elementi di Didattica della matematica. Bologna: Pitagora

Baccaglini-Frank, A., Di Martino, P., Natalini, R., Rosolini, G. (2017). Didattica della matematica. Mondadori.

Further teaching materials in English will be provided to students who will require them.

The slides projected during the course in PDF format and all the material used during lessons and laboratory hours are made available to students and shared on the Elly platform. In addition to the shared material, the student can personally deepen some topics addressed during the course by referring to the following texts:

D’Amore, B. (1999). Elementi di Didattica della matematica. Bologna: Pitagora

Baccaglini-Frank, A., Di Martino, P., Natalini, R., Rosolini, G. (2017). Didattica della matematica. Mondadori.

Further teaching materials in English will be provided to students who will require them.

## Teaching methods

The course has a weight of 9 CFU, which corresponds to 72 hours of lessons. The teaching activities will be conducted by giving lectures in the classroom, alternated with group works . During the lectures the topics of the course are dealt with from a theoretical point of view and with detailed examples. During the group works, students will be required to apply the theory to a case, a textbook, an evaluation test according to the methodological criteria illustrated in the lessons and in the bibliographic and teaching material. In addition to the teaching methods presented so far, in-depth seminars are organized on the topics of the course. The slides and documents used to support the lessons will be uploaded at the beginning of the course on the Elly platform. To download the slides it is necessary to register for the online course. All shared material is considered an integral part of the teaching material. Non-attending students are reminded to check the available teaching materials and the indications provided by the teacher through the Elly platform, the only communication tool used for direct teacher / student contact. On this platform, on a weekly basis, the topics discussed in class are indicated, which will then form the contents index in preparation for the final exam.

The course has a weight of 6 CFU, which corresponds to 48 hours of lessons. The teaching activities will be conducted by giving lectures in the classroom. During the lectures the topics of the course are dealt with from a theoretical point of view and with detailed examples. In addition to the teaching methods presented so far, in-depth seminars are organized on the topics of the course. The slides and documents used to support the lessons will be uploaded at the beginning of the course on the Elly platform; To download the slides it is necessary to register for the online course. All shared material is considered an integral part of the teaching material. Non-attending students are reminded to check the available teaching materials and the indications provided by the teacher through the Elly platform, the only communication tool used for direct teacher / student contact. On this platform, on a weekly basis, the topics discussed in class are indicated, which will then form the contents index in preparation for the final exam.

## Assessment methods and criteria

The course has a weight of 9 CFU, which corresponds to 72 hours of lessons. The teaching activities will be conducted by giving lectures in the classroom, alternated with group works . During the lectures the topics of the course are dealt with from a theoretical point of view and with detailed examples. During the group works, students will be required to apply the theory to a case, a textbook, an evaluation test according to the methodological criteria illustrated in the lessons and in the bibliographic and teaching material. In addition to the teaching methods presented so far, in-depth seminars are organized on the topics of the course. The slides and documents used to support the lessons will be uploaded at the beginning of the course on the Elly platform. To download the slides it is necessary to register for the online course. All shared material is considered an integral part of the teaching material. Non-attending students are reminded to check the available teaching materials and the indications provided by the teacher through the Elly platform, the only communication tool used for direct teacher / student contact. On this platform, on a weekly basis, the topics discussed in class are indicated, which will then form the contents index in preparation for the final exam.

The assessment of learning includes an oral test based on questions related to the contents of the course, aimed at assessing understanding and development of skills indicated in the section of objectives. The test consists of four questions that can focus on research results and teaching theories, institutional references, transversal didactic topics addressed during the course. The vote is calculated by assigning to each question an evaluation from 0 to 30 and making the arithmetic average of the individual evaluations, with final rounding up; the test is passed if it reaches a score of at least 18 points. The praise is assigned in the case of reaching the maximum score on each item to which is added the mastery of the disciplinary lexicon.

## Other informations

There are no compulsory propaedeutical courses