16789 - Teaching of Mathematics

Learning outcomes

At the end of the course, the student: - possesses the main results of international research in mathematics education; - is able to manage concrete classroom situations in the teaching-learning process of mathematics in secondary school; - is able to use, manage, and critically criticize various software tools for teaching; - is able to use this knowledge to develop effective teaching materials to be tested in the classroom.

Course contents

FIRST MODULE

MATHEMATICS TODAY: ELEMENTS OF COMMUNICATION OF MATHEMATICS.
What mathematics is NOT; how the public perception of mathematics is formed; prejudices; the social danger of mathematical illiteracy; the professions of mathematics. Storytelling. We recommend careful reading of the Benvenuti-Natalini article in the bibliography (attached to the slides). Notes on the gender issue.


CURRICULUM AND INSTITUTIONAL ASPECTS
National standard for lower and upper secondary schools: concept of competence, mathematical competence and the European Reference Framework, goals of mathematics teaching, structure and content of the National curriculum for the lower cycle, the Student Profile and the National for High Schools and the Guidelines for Technical and Vocational Institutes. Assessment of mathematical skills in the international and national contexts.


LABORATORY TEACHING
What is meant by laboratory? Elements of laboratory teaching; a new approach?; Before the laboratory: teacher/facilitator training; During the laboratory: the role of rigor; During the laboratory: the role of error; During the laboratory: the role of discussion; After the laboratory: assessment. Educational software. Geogebra Institute and the Geogebra platform. GeoGebra software: characteristics, peculiarities (see also the section on proof-argumentation).


ARGUMENTATION AND PROOF
The function of proof in mathematics and in mathematics teaching. Understanding and convincing. Proof as an object and as a process. Proof as an argumentative form. The social, temporal, and spatial dimensions of proof. The definition of a theorem as a triple. The phases of producing a theorem and student difficulties. Statement and proof as a process and as a product. Student beliefs about proof. Cognitive unity. Geogebra and the introduction to proof.


MATHEMATICAL THINKING AND PROBLEM SOLVING
Mathematical competence and problem solving; the definition of a problem; Gestalt studies on problem solving; problem vs. exercise; problem solving in the classroom; school problem vs. real problem; the narrative dimension; the context-question connection; mathematical modelling as problem solving activity.


HISTORY AND TEACHING OF MATHEMATICS
Pros and cons of using history in the learning-teaching process. The whys and hows of history in teaching. The use of historical sources. The difference between history and legacy.


MODULE TWO


THEORIES OF LEARNING/TEACHING IN MATHEMATICS
Macrotheories of learning: behaviorism, cognitivism, constructivism; consequences of various macrotheories on teaching models.


SITUATION THEORY
The minimal teaching system: Chevallard's triangle; teaching transposition, social context and institutional constraints, the noosphere. Teaching contract: origin and main aspects; the "age of the captain" effect, the need for formal justification and the formal delegation clause, the Topaze effect, situation theory and the structure of an a-didactic situation, the devolution and belief paradox.


THE ROLE OF ERROR
Compromise of correct answers; focus on process and product. The term "misconception"; avoidable and unavoidable misconceptions. Intuitive concepts/procedures; the relationship between intuition and logical reasoning. Models and analogies; Analogies as sources of misconceptions in mathematics. Intuitive models and paradigmatic models. Examples in learning probability and in the multiplicative conceptual field.


SEMIOTICS
Noetics and semiotics: registers of representation and semiotic activities (representing, processing, converting). Duval's paradox. The importance and centrality of semiotics in the teaching and learning process of mathematics and related issues (plurality of representations, one-way register conversion, processing and loss of meaning).


CONCEPT IMAGE AND CONCEPT DEFINITION
Definition of imagery and definition of a concept. Cognitive conflicts and the phenomenon of compartmentalization. Analysis of a research article on concept image and definition related to the concept of function. A brief history of the concept of function and analysis of three possible approaches to introducing the concept of function. The concept of embodiment.


MATHEMATICS AND SPECIAL EDUCATIONAL NEEDS
Special educational needs, disabilities, and specific learning disabilities in mathematics. Characteristics of dyscalculia. The PDP: compensatory tools and remedial measures for mathematics. Mathematics learning in the context of sensory and cognitive disabilities.

Readings/Bibliography

Various study materials will be provided for both modules via the virtual platform.

In addition, the following texts will be used for the second module:

• Principi di base di didattica della matematica, di D'Amore & Sbaragli. Bonomo, 2023
• Metodi e strumenti per l'insegnamento e l'apprendimento della matematica, di Bolondi & FandinoPinilla. Edises, 2012
• Didattica della matematica, di Baccaglini-Frank, Di Martino, Natalini, Rosolini. Mondadori Università, 2017

Teaching methods

Classes are structured as follows: lectures, critical analysis of texts and articles, individual or small-group workshops, cooperative learning and microteaching activities, co-planning, and group discussion.

Assessment methods

The final test consists of a single oral exam in which the student prepares a presentation about one of the course topics. Subsequently, the oral exam consists of questions from the instructors of the two modules. Questions may cover the entire course syllabus.

Teaching tools

An e-learning space is activated on the platform Virtuale

Office hours

See the website of Andrea Maffia

See the website of