October 28, 2023

Curriculum trends in secondary education in the past

9h15 | FCT Representative + DM Representative + Organization Representative + GTHMEM Representative

Opening session

09h45 | Circe Mary Silva da Silva 
Differential and integral calculus in the secondary school curriculum in the international context of the twentieth century

The aim of this study is to identify how differential and integral calculus (DIC) was introduced until it was internationally consolidated as one of the rubrics of Secondary School Mathematics. The mapping carried out begins in the early years of the twentieth century and extends for approximately one hundred years. This is a bibliographic research that collects sparse data from journal articles, research reports, and books on the CDI teaching curricula at the international level. Three distinct periods were identified: 1) the first wave of modernization, with the first attempts to introduce CDI in secondary education; 2) the period covering the time between the wars and after World War II, when in some countries CDI was introduced or excluded; 3) the third - the second wave of modernization - which begins with the New Mathematics movement and its repercussions on CDI education. I concluded that its teaching begins gradually in some countries and after a hundred years, it consolidates itself as a mathematical knowledge of secondary school.

10h30 Isabel Oitavem 
The concept of function has a history
This presentation talks about the before and after, the past, and the present of the concept of function.

It is a journey from the 7th century to nowadays. It's a four-dimensional film where you do not need to wear special glasses: let us delve into the timeline and discover the historical dimension of the concept of function.

11h15 Break

11h30 Teresa Monteiro 
The study of the limit of a function in compendia from the 60s, 70s, and 80s of the 20th century
The ideology of Modern Mathematics, in Portugal, began in the ‘60s and extended until the 80s of the 20th century. Sebastião e Silva chaired a committee to review the Mathematics program for the 3rd cycle of liceus, appointed by Minister Galvão Telles, in July of 1963.
Knowing that school culture does not change by decree, we intend to show part of a secondary education path in Mathematics. Therefore, we will carry out a comparative analysis of Portuguese Mathematics Compendiums for the last two years of secondary education, with particular attention to the limit of a function.

Two of the conclusions of this study for the compendia under analysis are: the disappearance of historical references for the study of limits in the 70s and 80s, contrary to what was observed in the 60s; and it was only in the 60s and 70s that the calculation of limits was applied to trigonometric, exponential and logarithmic functions.

12h15 Lunch

14h00 José Manuel Matos 
Beginnings of professional knowledge of secondary school teachers

Most research on teacher's professional knowledge is focused on the present. Centred on school mathematics, in this session I will discuss the ways in which this knowledge emerged in the early decades of the 20th century, analysing the contributions of the new disciplines created in the Escolas Normais Superiores.

14h45 Mária Almeida 

Reflections on the infinitesimal analysis at liceus in Portugal (1948 – 1958)
The mathematics contents we teach today are sometimes questioned not only by teachers and students but also by other sectors of society. The contents that appear or are removed from the programs show us aspects related to the mathematical knowledge desirable in each epoch. In 1936, infinitesimal analysis was suppressed from the Mathematics programs of the Portuguese liceus’ 3rd cycle, however, it appeared in the programs promulgated in 1948. This communication aims to analyse the reactions to this curricular change in liceus education, using legislation, pedagogical and scientific magazines, teaching books and works produced in the context of liceus teachers’ training.

15h15 Break

15h30 António Domingos 
Learning Theories. Contributions to understanding concepts in their cognitive and social dimensions

In this communication, we intend to revisit some of the learning theories that had their heyday in the second half of the last century, but which still continue to make important contributions today. The aim is to list the main characteristics of Reification Theory, APOS Theory, and David Tall's work on the conceptual approach to concepts. Activity theory is an extension of the previous theories, giving them a social aspect, which allows us to complete a more up-to-date and global vision of learning processes.

16h15 Rui Candeias (Moderator)
Discussion of the day's presentations
November 11, 2023

Curriculum trends in secondary education in the present

09h45 António Domingos & Cristina Cruchinho
Curricular trends in secondary education today
With the approval of the Essential Mathematical Learning for Secondary Education (Aprendizagens Essenciais de Matemática para o Ensino Secundário) on 13 January 2023, we now have a new curriculum proposal that aims to provide students with mathematics that will enable them to finish compulsory education with mathematical and social skills that will facilitate their integration into today's society, while at the same time providing them with a solid preparation that will enable them to go on to study in the most diverse areas of knowledge.

In this conference, we intend to present the main innovative ideas that are present in the documents, along with the key ideas and the main methodological options that underlie their implementation.

10h30 Alexandra Sofia Rodrigues
Mathematics in Professional Education
In 2023, International Mathematics Day celebrates the theme “Mathematics for everyone!”. This is particularly true when we refer to vocational education and the training of specialized intermediate-level technicians. Professional courses in Portugal, since their creation in 1989, provide for the existence of synergies between the school, the profile of the students, the social and cultural framework, and the profession. Should the mathematics to be taught/learned be the same for a future photography technician or for a future industrial maintenance technician? Portugal has close to 400 different professional qualifications. Is it possible for mathematics to respond to such a large number of qualifications?

At this conference, we will analyse the Portuguese professional education curriculum in light of the importance of adapting it to different areas of training.

11h15 Break 

11h30 Antonio M. Oller Marcén

Proportionality: a topic in between two worlds
Since its historical origin, there has been a certain duality between the arithmetical and geometrical treatment of proportionality. Euclid's Elements, for example, devote different books to it (Book V for arithmetic and Book VI for geometry). In current education, proportionality continues to be a link between these two areas: arithmetic and geometry. In secondary education in Spain, arithmetic proportionality is studied prior to geometric proportionality. In many cases, ideas, concepts, or procedures introduced in the arithmetic treatment are reintroduced when dealing with proportionality from a geometric point of view. Consequently, it is of interest to investigate whether the conceptual and procedural apparatus associated with arithmetic proportionality is used "consistently" in the geometric context. In this talk, we will present some examples of inconsistencies present in Spanish textbooks and reflect on possible implications for the teaching of this important mathematical knowledge.

12h15 Lunch

14h00 Amarildo Melchiades da Silva
Financial Education at School: tension between practice and research

In this conference we will discuss the insertion of Financial Education in Brazilian schools, in particular in High Schools, based on the official document of the Brazilian government called the National Common Curriculum Base (BNCC) and the results of research on the subject developed in the Postgraduate Programme in Mathematics Education at the Federal University of Juiz de Fora (UFJF) by researchers from the Research, Dissemination and Studies in Mathematics Education Centre (NIDEEM) suggesting a tension between the proposition coming from the official document directed at teachers and the proposition coming from research.

14h45 Marianna Bosch
Curricular changes and transparencies in the knowledge to be taught

Where does the mathematics that is proposed in the curriculum as knowledge to teach come from? How is it decided what should be taught - and studied - in each year of secondary education and how is it materialized in concrete activities carried out by teachers and students in the classrooms? Investigating these questions will lead us to consider the processes of didactic transposition through which different people and institutions intervene in the construction of "knowledge for teaching." It will also allow us to identify aspects or elements of this knowledge that are not very visible despite having an important role in the development of the mathematical activities of teachers and students.

15h30 Helena Rocha (Moderator)
Discussion of the day's presentations