Set Theory and Analysis in Mathematics Education


It is intended to suit the students prior scientific knowledge to the didactic needs and requirements of Secondary Education teaching. The recent reforms of the National Curriculum call for a pedagogical practice grounded in scientific rigor, and  ask from future teachers greater math fluency. Accordingly, at the end of this course, the student should have acquired knowledge and capabilities to:

- Safely teach topics relating to Logic, Set Theory and Analysis, more precisely the notion of limit, which constitutes the core content of this theme.

-Distinguish valid from didactic approaches that do not fulfill the objectives listed in the Curriculum.

-Apply to the teaching of other content areas  the transversal themes related to Theory of Sets and Logic, as or example the application of general demonstration techniques or the appropriate use of written in mathematical communication.

General characterization





Responsible teacher

Manuel Messias Rocha de Jesus


Weekly - 4

Total - 60

Teaching language



Basic knowledge of Set Theory and Mathematical Analysis.


1. Bivar A., Grosso C., Oliveira F. e Timóteo M.C. - Caderno de Apoio às Metas Curriculares de Matemática do Secundário – Matemática A, Direção Geral de Educação, 2014.

2. Campos Ferreira, J. - Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982.

3. Figueira, M. - Fundamentos de Análise Infinitesimal, Textos de Matemática, vol. 5, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, 1996.

4. Franco de Oliveira, A. - Lógica e Aritmética, Trajectos Ciência, Gradiva, 1991.

5. Santos Guerreiro, J. - Curso de Análise Matemática, Livraria Escolar Editora, 1989.

6. Sarrico, C. - Análise Matemática, Leituras e Exercícios, Gradiva, 1997.

7. Sebastião e Silva, J. - Guia para a utilização do Compêndio de Matemática (1º volume), Curso Complementar do Ensino Secundário, Edição GEP, Lisboa, 1975.



Teaching method

Lectures + problem-solving sessions (5h00).

Evaluation method

Students must attend, at least, 2/3 of classes.

The students may obtain approval in this course by continuous evaluation.

The continuous evaluation will be  based on:

- The participation of students in class work;

- Individual-written works on programatic topics, their oral presentation and discussion;

- Discussion of some syllabus contents and possible teaching approaches;

-1 test.

Also, they may present themselves to a final exam that will test all the learning goals of this course.

Subject matter

1. Didactic approaches to Bivalent Logic

Logical value of a proposition; Non contradiction and  Law of excluded Middle.

Properties of the operations over propositions.


2. The uses of set theory in the Secondary Education

Conditions and quantified expressions;

Relationship between logical operations over conditions and over sets.

Double inclusion and proof of equivalences by double implication.


3. Proof techniques in the Secondary Education

Proof by contrareciprocal and by Reductio ad Absordum. Choosing of counterexamples in different contexts.


4. Didactic approach of the notion of limit in the 11º grade

The teaching of the notion of limit of a sequence; adequate and inadequate intuitions.

Correct formalization of the notion of convergence: limit of a function in an adherent point to its domain; lateral limits and limits in the infinite; indeterminate situations; asymptote to the graph of a function; Differentiability; applications of derivatives to the study of functions and the modelation of natural phenomena.


Programs where the course is taught: