This study aims to understand the functional thinking of 10th-grade students while studying functions. Specifically, we intend to answer the following research questions: what are the functional thinking processes used by 10th-grade students when studying functions? What difficulties do students present while learning functions? In view of the nature of this research objective, we adopted a qualitative and interpretative approach. In order to answer these questions, data were collected from the written records produced by the students while solving the proposed tasks, from records of the oral interactions during discussions and from a questionnaire. The results show that functional thinking processes were implicit in the resolution of the tasks proposed to the students. The students expressed an understanding of how the variables were related, presenting evidence of their functional thinking while working on the new concepts represented by the functions addressed in the proposed tasks. Some students expressed difficulties in interpreting the different types of representations associated with the functions, in retaining the necessary information from a graphical representation that would help them to draw conclusions and establish correspondences, in explaining functional relationships, and in interpreting the information provided by algebraic expressions. These difficulties can reduce the recognition of the relationships between variables and their behavior in the different representations, becoming an obstacle to learning for some students.

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Technology is recognized for its potential to implement exploration tasks. The ease and speed with which it becomes possible to observe many cases of a situation, allows the development of conjectures and brings conviction about their veracity. Mathematical proof, assumed as the essence of Mathematics, tends to appear to the students as something dispensable. Based on KTMT – Knowledge for Teaching Mathematics with Technology model, this study intends to understand the impact of the teachers’ knowledge on mathematical proof in a context of technology integration. The study adopts a qualitative and interpretative methodology, based on case study, analyzing the practice of one teacher. The conclusions emphasize the relevance of the teacher’s MTK – Mathematics and Technology Knowledge, and TLTK – Teaching and Learning and Technology Knowledge. The teacher's MTK guides her decisions, leading her to focus on helping students understand the meaning of conjecture and proof, valuing, at the same time, the relevance of algebraic manipulations. However, the teacher’s TLTK guides her practice, where the knowledge about the students is determinant. The study provides evidence about the difficulty of articulating proof and technology, but it also clarifies the relevance of this articulation and of how the teacher’s KTMT can impact the teacher’s decisions.

Abstract We present the first detailed braincase anatomical description and neuroanatomical study of Portugalosuchus azenhae, from the Cenomanian (Late Cretaceous) of Portugal. This eusuchian crocodylomorph was originally described as a putative Crocodylia and one of the oldest representatives of this clade; however, its phylogenetic position remains controversial. Based on new data obtained from high resolution Computed Tomography images (by micro-CT scan), this study aims to improve the original description of this taxon and also update the scarce neuroanatomical knowledge of Eusuchia and Crocodylia from this time interval, a key period to understand the origin and evolution of these clades. The resulting three-dimensional models from the CT data allowed a detailed description of its well-preserved neurocranium and internal cavities. Therefore, it was possible to reconstruct the cavities of the olfactory region, nasopharyngeal ducts, brain, nerves, carotid arteries, blood vessels, paratympanic sinus system and inner ear, which allowed to estimate some neurosensorial capabilities. By comparison with other crocodylomorphs, these analyses showed that Portugalosuchus, back in the Cenomanian, already displayed an olfactive acuity, sight, hearing and cognitive skills within the range of that observed in other basal eusuchians and crocodylians, including extant species. In addition, and in order to test its disputed phylogenetic position, these new anatomical data, which helped to correct and complete some of the original observations, were included in one of the most recent morphology-based phylogenies. The position of Portugalosuchus differs slightly from the original publication since it is now located as a “thoracosaurid” within Gavialoidea, but still as a crocodylian. Despite all this, to better contrast these results, additional phylogenetic analyses including this new morphological character coding together with DNA data should be performed.

An incomplete, yet remarkably-sized dentated rostrum and associated partial cervical vertebrae of a pterosaur (ML 2554) were recently discovered from the Late Jurassic (Late Kimmeridgian-Early Tithonian) Lourinhã Formation of Praia do Caniçal, of central west Portugal. This specimen exhibits features such as a spatulated anterior expansion of the rostrum, robust comb-like dentition, and pronounced rims of the tooth alveoli, indicating gnathosaurine affinities. Based on its further unique tooth and dentary morphology, a new genus and species, \textit{Lusognathus almadrava} gen. et spec. nov., is proposed, making this the first named pterosaur species found within Portugal. The presence of this taxon adds yet another element to the fluvio-deltaic lagoonal environment that has been suggested as representative of the Lourinhã Formation in the Late Jurassic, further contributing to the diversity and distribution of gnathosaurines worldwide.

In this paper we consider the monoid DPC_n of all partial isometries of a n-cycle graph C_n. We show that DPC_n is the submonoid of the monoid of all oriented partial permutations on a n-chain whose elements are precisely all restrictions of a dihedral group of order 2n. Our main aim is to exhibit a presentation of DPC_n. We also describe Green's relations of DPC_n and calculate its cardinality and rank.

In this paper we consider the monoid DPSn of all partial isometries of a star graph Sn with n vertices. Our main objectives are to determine the rank and to exhibit a presentation of DPSn. We also describe Green’s relations of DPSn and calculate its cardinal.

In this paper we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank of the monoid wEndP_n of all weak endomorphisms of a directed path with n vertices, which is a submonoid of the widely studied monoid O_n of all order-preserving transformations of a n-chain. Also, we describe the regular elements of wEndP_n and calculate its size and number of idempotents.

Following the new description of an oriented full transformation on a finite chain given recently by Higgins and Vernitsk,
in this short note we present a refinement of this description which is extendable to partial transformations and to injective partial transformations.

In this paper we consider the submonoids OPDI_n, MDI_n and ODI_n of the dihedral inverse monoid DI_n of all orientation-preserving, monotone and order-preserving transformations, respectively. Our goal is to exhibit presentations for each of these three monoids.

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O sítio de Silveirinha é uma das localidades de mamíferos mais conhecidas da Paleontologia do Cenozoico de Portugal e da Europa em geral. Graças à sua rica e diversificada associação de mamíferos, com mais de 30 taxa, foi posicionado no Eocénico inferior (início do Ypresiano, MP7, ca. 55,8 M.a.), sendo o local mais antigo da Europa desta Época, devido à presença única de taxa típicos do Paleocénico superior, juntamente com outras espécies já características do Eocénico inferior. Este estudo irá rever o material de pequenos mamíferos deste sítio, conservado na coleção clássica da Universidade Nova de Lisboa, a fim de fazer uma actualização taxonómica à luz das publicações mais recentes.

The emergence of the platform economy is the most recent model of outsourcing practices and the flexibilization of the organisation of production supported by the development of information and communication technologies. The spread of this business model has led to the expansion of different precarious forms of employment, the most common being the bogus self-employed. The business model based on digital platforms also has the effect of limiting the capacity of trade unions to represent and defend workers, preventing or limiting, for example, resource to collective bargaining. This article compares the trade union revitalisation strategies deployed in Spain and Portugal in the rider sector by traditional trade unions and associations of delivery workers and precarious workers. The article shows how, in both countries, unions have combined organising and mobilisation strategies with actions aimed at influencing political and regulatory expectations. However, there is evidence of a greater experimentation of practices in the Spanish case, as well as a greater impact and influence of their strategies in political terms. The article also reveals important differences between the two countries regarding the relationship between traditional trade unions and delivery workers' associations.