31620 - Didactics of Mathematics 2

Academic Year 2023/2024

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Physics (cod. 9245)

    Also valid for Second cycle degree programme (LM) in Mathematics (cod. 5827)

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


REFERENCE FRAMEWORKS AND METHODOLOGIES: National Indications for the first and second cycle secondary school: concept of competence, mathematical competence and European reference framework, purpose of teaching mathematics, structure and contents of the National Indications for the first cycle, the Profile of the student and the National Indications for the Licei and the Guidelines for Technical and Vocational Institutes. Educational software. Geogebra Institute and Geogebra platform. GeoGebra software: features, peculiarities (see also section on demonstration-argumentation).

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

ARGUMENTATION AND DEMONSTRATION Haley-Hoyles research. The function of demonstration in mathematics and mathematics teaching. Understanding and convincing. Demonstration as object and process. Demonstration as an argumentative form. The social, temporal and spatial dimensions of demonstration. The definition of theorem as a triad. The stages of producing a theorem and students' difficulties. Enunciation and demonstration as process and as product. Cognitive unit. Geogebra and the beginnings of demonstration. Basic features of Geogebra and their educational function. Difference between artifact and tool. The theory of semiotic mediation. The microworld. Vision and visualization.

DOING MATHEMATICS TODAY: ELEMENTS OF COMMUNICATING MATHEMATICS. What mathematics is NOT; how the public idea of mathematics is formed; prejudices; social danger of mathematical illiteracy; the professions of the mathematician. Story telling. Careful reading of Benvenuti-Natalini article in bibliography (attached to slides) recommended. Mention of the gender problem.

INTRODUCTION TO LABORATORY TEACHING. What is a laboratory; elements of a laboratory teaching; a new mode?; before the laboratory: the training of the teacher/entertainer; during the laboratory: the role of rigor; during the laboratory: the role of error; during the laboratory: the role of discussion; after the laboratory: evaluation. Careful reading of the Dedò-Di Sieno article in the bibliography (attached to the slides) is recommended.

THEORIES OF LEARNING/TEACHING IN MATHEMATICS. Macro theories of learning: behaviorism, cognitivism, constructivism; consequences of the various macro theories on teaching models; personality theories: emotional intelligence, multiple intelligences, cooperative learning.

THE ROLE OF AFFECTIVE FACTORS IN MATHEMATICS TEACHING. Responding to negative emotions; Beyond the purely cognitive; need for new observational tools; Di Martino study (autobiographical phrases and themes); the central role of the teacher; Di Paola study (on future teachers); correct answer compromise; from reproductive thinking to productive thinking; rethinking the role of time and error.


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

MATHEMATICAL THINKING, COMPUTATIONAL THINKING AND PROBLEM SOLVING Mathematical competence and problem solving; the definition of a problem; Gestalt studies on Problem solving: perception as a structured totality, studies on visual perception, interest in productive thinking, chimpanzee studies, the definition of functional fixity, insight and productive/constraining anxiety; Gestalt studies on Problem solving: From chimpanzee studies to the definition of problem solving stages, how learning works; problem vs. exercise; problem solving in the classroom; school problem vs. real problem; the narrative dimension; the context-question link; directions for problem formulation; rethinking problem solving activity; why do problem solving.

INTRODUCTION TO SCIENCE PUBLISHING - FICTION AND SCHOLASTIC Narrative/nonfiction vs scholastic; what is a book: physical and formal structure; hints on typography: fonts, styles, spaces, layout; editorial standards (only in slides); scholastic publishing: overview of the various jobs open for a mathematician in a publishing house-introduction by the lecturer, insights by Giulia Tosetti (Zanichelli) and Eleonora Pellegrini (Rizzoli) for scholastic publishing, Daniele Gouthier (Scienza Express) for fiction/ non-fiction, Roberto Natalini (CNR) and Francesca Riccioni for comics, Cristina Serra (freelance) for translations.

OBSTACLES, MISCONCEPTIONS AND ERRORS. Behind systematic errors; the term "misconception"; avoidable and unavoidable misconceptions; examples: misconceptions related to the primitive entities of geometry; misconceptions arising from linguistic terms (oblique, diagonal, ... ); misconceptions arising from inconsistencies in textbooks; misconception or error?; errors and difficulties in mathematics; error or nonstandard thinking?

FLIPPED CLASSROOM Flipping teaching: how, when and why. Time/moment reversal; the student at the center of the learning process; a strategy to boost motivation; attitudes of teachers, students and parents. The advantages of the flipped classroom for the math teaching/learning process; what to pay attention to. The use of technologies in the flipped classroom.

CONCEPT IMAGE AND CONCEPT DEFINITION Definition of imagery and concept definition. Cognitive conflicts and the phenomenon of compartmentalization. Analysis of a research paper on concept image and definition related to the concept of function. Brief history of the concept of function and analysis si three possible approaches to the introduction of the concept of function. The concept of embodiment.

HISTORY AND DIDACTICS 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 inheritance.

INTUITION IN MATHEMATICS Intuitive concepts/procedures, relationship between intuition and logical reasoning. Models and analogies; analogies as sources of misconceptions in mathematics. Intuitive models and paradigmatic models. Example of learning about probability. Example of learnings in the multiplicative conceptual field.

MATHEMATICS AND SPECIAL EDUCATIONAL NEEDS Special educational needs, disabilities and specific learning disorders: definition, relevant legislation and diagnosis. Disorder and difficulty. Neurocognitive functioning: the of the triple code, working memory. Characteristics of dyscalculics. The PDP: compensatory tools and dispensatory measures. Mathematics learning in the context of sensory and cognitive disabilities.

The detailed and complete course syllabus will be posted at the end of the lectures on Virtual [https://virtuale.unibo.it/] .


  • Baccaglini Frank, Di Martino, Natalini, Rosolini, Didattica della matematica, Mondadori Università 2018.
  • Bolondi, Fandino Pinilla, Metodi e strumenti per l’insegnamento e l’apprendimento della matematica, EdiSES, 2012.
  • Benvenuti, Natalini, Comunicare la matematica: chi, come, dove, quando e, soprattutto, perché?!, Rivista Umi - Matematica, cultura e società, agosto 2017.
  • Castelnuovo, Pentole, ombre, formiche, Utet 2017.
  • Castelnuovo, Didattica della matematica, Utet 2017.
  • D'Amore, Elementi di didattica della matematica, Pitagora 1999.
  • Dedò, Alla ricerca della geometria perduta 1, Alice e Bob 46 2016.
  • Dedò, Di Sieno, Laboratorio di matematica: una sintesi di contenuti e metodologie, https://arxiv.org/pdf/1211.2159.pdf
  • Di Sieno, Alla ricerca della geometria perduta 2, Alice e Bob 53, 2018.
  • Israel, Millan Gasca, Pensare in matematica, Zanichelli 2015

Teaching methods

Lecture, critical analysis of texts and articles, individual or small group workshop activities, cooperative learning and microteaching, co-design, collective discussion and peer-to-peer evaluation.

Assessment methods



The final examination consists of a project and an oral test.


The topic and how to carry out the project will be clarified in class and posted on the Virtual Platform [https://virtuale.unibo.it/] of teaching.
Oral test

Part I: presentation and discussion from a teaching perspective of the realized project.
Part II: "disciplinary" and "didactic" discussion of concepts or topics covered during the course. This part will assess the student's level of understanding of the concepts and themes covered in the course; the student's ability to analyze such a theme or concept from a didactic point of view, knowing how to recognize its sensitive points for understanding; and the student's ability to place the treatment of such a theme or concept in a broad educational and cultural perspective and within a didactic path.

The final outcome will take into account in equal parts the project (implementation, presentation and discussion) and the "disciplinary" and "didactic" oral discussion of concepts and themes covered during the course.

Registration is required on Almaesami

Teaching tools

All materials will be published in Virtuale [https://virtuale.unibo.it/]

Office hours

See the website of Silvia Benvenuti