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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.