The different representations of functions are assumed as central on the development of the concept of function. Being widely recognized the complexity of this concept, the different representations allow the student to understand in a representation what could not be understood in another representation. And the integration of technology into the teaching and learning process provides an easy and quick way to access different representations. This study intends to analyse the understanding of upper secondary students about the information transmitted by each of the representations of functions usually available on technology. Specifically, it intends to understand which transitions between representations are more easily understood by the students and which ones are more difficult to perform. It also intended to identify some aspects that may contribute to this. This study adopts a quantitative methodology in which the answers given by a class to a test focused on the transition from one representation to another are analysed; and a qualitative methodology based on interviews to three of the students in the class, as a way of seeking comprehension about their answers. The results achieved suggest a greater ease of understanding associated to the graphical representation and a greater difficulty associated to the tabular representation. The reasons for this seem to be related to the specific characteristics of each representation, but fundamentally with aspects related to the experiences lived by the students on the mathematics classes, being the integration of technology an influence not to neglect.

Abstract An experimental work on reinforced concrete flat slab specimens to test the efficiency of postinstalled bolts, as punching shear reinforcement in resisting vertical and cyclic horizontal loads, was conducted and is presented in this paper. The test protocol consisted in increasing horizontal drifts combined with constant vertical load until failure. Two different detailing solutions for the shear reinforcement were considered, one using a radial distribution around the column and another using a cross distribution, being the results compared with a previously tested reference specimen. The dimensions of the specimens were 4.25 x 1.85 x 0.15 m3. The test setup used for these tests was developed by the research team and simulates the boundary conditions with already recognized good results. Postinstalled steel bolts were proven to be an efficient solution for strengthening of existing structures, improving the structural behavior, and the punching resistance.

Abstract An experimental work on reinforced concrete flat slab specimens to test the efficiency of postinstalled bolts, as punching shear reinforcement in resisting vertical and cyclic horizontal loads, was conducted and is presented in this paper. The test protocol consisted in increasing horizontal drifts combined with constant vertical load until failure. Two different detailing solutions for the shear reinforcement were considered, one using a radial distribution around the column and another using a cross distribution, being the results compared with a previously tested reference specimen. The dimensions of the specimens were 4.25 x 1.85 x 0.15 m3. The test setup used for these tests was developed by the research team and simulates the boundary conditions with already recognized good results. Postinstalled steel bolts were proven to be an efficient solution for strengthening of existing structures, improving the structural behavior, and the punching resistance.

The main goal of this work is to characterize how the knowledge of pre-service teachers about the characteristics of the tasks and the role of technology evolves. Based on a case study carried out around a pair of pre-service teachers, the main conclusions point to the contribution of the reflection around a set of six tasks on Functions selected by the pre-service teachers. Central to this reflection was an analyze of the role technology can play in tasks, the comments made by the colleagues to their tasks and some experiences on modeling and open-ended tasks. These elements provided the development of a greater awareness regarding aspects such as the level of structuring of the task and its degree of challenge. And this was determinant for an appropriation of the different characteristics of the tasks and to the development of the pre-service teachers’ knowledge.

n/a

This chapter provides an overview of the alpha paleobiodiversity of Angola based on the available fossil record that is limited to the sedimentary rocks, ranging in age from Precambrian to the present. The geological period with the highest paleobiodiversity in the Angolan fossil record is the Cretaceous, with more than 80{%} of the total known fossil taxa, especially marine molluscs, including ammonites as a majority among them. The vertebrates represent about 15{%} of the known fauna and about one tenth of them are species firstly described based on specimens from Angola.

In 1900, Hilbert gave a lecture at the International Congress of Mathematicians in Paris, for which he prepared 23 problems that mathematicians should solve during the twentieth century. It was found that there was a note on a 24th Problem focusing on the problem of simplicity of proofs. One of the lines of research that was generated from this problem was the identification of proofs. In this article, we present a possible method for exploring the identification of proofs based on the membership problem original from the theory of polynomial rings. To show this, we start by giving a complete worked-out example of a membership problem, that is, the problem of checking if a given polynomial belongs to an ideal generated by finitely many polynomials. This problem can be solved by considering Gröbner bases and the corresponding reductions. Each reduction is a simplification of the polynomial and it corresponds to a rewriting step. In proving that a polynomial is a member of an ideal, a rewriting process is used, and many different such processes can be considered. To better illustrate this, we consider a graph where each rewriting step corresponds to an edge, and thus a path corresponds to a rewriting process. In this paper, we consider the identification of paths, within the context of the membership problem, to propose a criterion of identification of proofs.
This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert’s 24th problem’.

The potential of using different representations is widely recognized, but not much is known about how teachers use them nor about the impact of the technology on such use. The goal of this study is to characterize the teachers’ representational fluency when teaching functions at high school level, discussing, at the same time, the impact in the use of representations resulting from the use of technology. Adopting a qualitative approach, I analyze one teacher’s practice. The results suggest that algebraic and graphical representations are seen as more important, that tabular representation is assumed as irrelevant and that the access to technology impacts the learning, the representations used and how they are used.

This study focusses on the criteria used by pre-service teachers of Mathematics to choose interdisciplinary tasks. The pre-service teachers’ knowledge is assumed as the basis of the actions taken and used as the origin of the choices and approaches observed. The study adopts a qualitative and interpretative methodology and the data were collected using class observation and interviews. The analysis is guided by the Application and Pedagogical Content Knowledge, a model inspired on TPACK (from Mishra and Koehler) and MKT (from Ball and colleagues). The conclusions point to an appreciation of the mathematical part of the tasks and to a devaluation of the remaining components. This suggests difficulty in articulating and integrating different domains of knowledge and points to a fragmented view of the potential of using mathematical applications.

Proof plays a central role in developing, establishing, and communicating mathematical knowledge. Nevertheless it is not such a central element in school mathematics. This article discusses some issues involving mathematical proof in school, intending to characterize the understanding of mathematical proof in school, its function and the meaning and relevance attributed to the notion of simple proof. The main conclusions suggest that the idea of addressing mathematical proof at all levels of school is a recent idea that is not yet fully implemented in schools. It requires an adaptation of the understanding of proof to the age of the students, reducing the level of formality, and allowing the students to experience the different functions of proof and not only the function of verification. Among the different functions of proof, the function of explanation deserves special attention due to the illumination and empowerment that it can bring to the students and their learning. The way this function of proof relates to the notion of simple proof (and the related aesthetic issues) seems relevant enough to make it, in the future, a focus of attention for the teachers who address mathematical proof in the classroom.

The current curricular guidelines for mathematics education in Portugal emphasize the relevance of working with different representations of functions to promote understanding. Given this relevance, we seek understanding about the notion of function held by 37 basic education pre-service teachers in their first year of a master’s course. Data were collected through a task focusing on identifying functions in situations based on different representations. The content analysis technique was then adopted in the search for an understanding of the justifications given by the participants. The results achieved suggest it is easier for the pre-service teachers to identify examples that are not functions than examples that are functions. There is also a tendency for greater accuracy in the identification of examples expressed by tables than by algebraic expressions. The justifications presented show a notion of function as a relation between values of two non-empty sets, but without guaranteeing that this relation is single-valued.