Curriculum trends in secondary education in the past

Opening session

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.

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

Two of the conclusions of this study for the compendia under analysis are: the disappearance of historical references for the study of limits in the 70s and 80s, contrary to what was observed in the 60s; and it was only in the 60s and 70s that the calculation of limits was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

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 20^th century, analysing the contributions of the new disciplines created in the Escolas Normais Superiores.

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

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.

Curriculum trends in secondary education in the present

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.

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 

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

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.

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.