Mathematics has a strong presence in the school curriculum, often justified by its usefulness in social life, in the world of work and by its connections with other sciences. This interdisciplinary connection, in particular when it requires constructing and refining mathematical models and discussing their applications to solve problems of other sciences, can assist students to understand why mathematics is so important in school. In the development of interdisciplinary activities, the characteristics of the tasks emerge as an important aspect. The emphasis is on the use of technological materials and the way they can support the development of concepts, provide different representations and support deeper understandings, and offer a multifaceted support to collect data and simulate experiences. Based on these assumptions, the aim of this chapter is to present, analyse and discuss tasks that promote interdisciplinary technological approaches from a mathematical point of view. In this chapter we assume interdisciplinarity as a complex construct, and in order to clarify its meaning we will discuss several types of conceptions, from multidisciplinary, to interdisciplinary, and to transdisciplinary. We will then address related concepts, such as modelling and STEM, highlighting similarities and differences between them, to reach an understanding of interdisciplinarity. In the process of the interdiciplinary approach, digital technologies arise as a central element. Based on a set of tasks on mathematics and on different sciences, we discuss what can change on an interdisciplinary approach to the teaching and learning of mathematical content and on the articulation between subjects.

This paper studies the influence of different longitudinal top reinforcement detailing on the behaviour and punching capacity of flat slabs. Experimental and numerical studies were carried out. The experimental campaign consisted in testing three reinforced concrete slabs 2.20 m wide with a thickness equal to 0.15 m. The numerical study was conducted using the non-linear finite element software ATENA 3D. Using numerical models, which were calibrated using the experimental tests, a parametric study was carried out to include other design solutions beyond the tested ones. The parameters that were changed in this parametric study were the steel reinforcement ratio, the concrete strength and the detailing of the top steel reinforcement, namely using a solution with uniform distribution and other with a higher concentration of reinforcement near the column. Finally, the results were compared with predictions obtained using some of the existing codes (Eurocode 2 and Model Code 2010). From this study it can be concluded that concentrating the top flexural reinforcement, keeping the same total amount of top longitudinal reinforcement, results in a stiffer response, an increment in the punching resistance, and a decrease in the maximum vertical displacement.

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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.

Nowadays, adhesively bonded structures have received exhaustive attention mainly because, contrary to mechanical joints, they are able to avoid stress concentration. When a material is externally bonded to another structural member to improve the strength or stiffness of the latter, the adhesive joint is supposed to perform well for a long time, independently of the type of loading the bonded joint will be subjected to. However, studies dedicated to this topic are scarce when it comes to the influence of thermal action. The influence of temperature variations on bonded joints is not yet well understood, so more studies are needed to improve the current level of knowledge. The present study aims to develop an analytical solution capable of simulating the interfacial bond behaviour between two structural materials subjected to thermal loading. The complete debonding processes of such adhesively bonded joints are estimated based on a bi-linear bond–slip relationship. The proposed analytical model is validated by the numerical simulation of several examples, where some parameters previously identified as potentially affecting the bond behaviour are investigated. A commercial software based on the Finite Element Method (FEM) is used to support those examples in which either the analytical or the numerical simulations agreed very well. © 2019 Elsevier Ltd

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