Os números imaginários: (um estudo sobre) a sua “realidade”

Dissertação de mestrado em Matemática, área de especialização em Ensino.Um pouco por todo o mundo, os programas do Ensino Secundário em Matemática estão condicionados por respostas para inúmeras dúvidas sobre “O que ensinar em Matemática?” “Como ensinar Matemática?”, “A Matemática é, ou deveria ser, útil? E em que sentido?”. Na Introdução e no capítulo I da presente monografia referimos o conceito de número como um dos conceitos fundamentais em Matemática que percorre, enquanto processo de ensino, uma vasta gama de níveis desde os 1º, 2º e 3º ciclos do Ensino Básico e ainda do Ensino Secundário até ao Ensino Superior. É neste percurso educativo que surge, de forma natural, o tema dos Números Complexos: usualmente, como uma questão de generalização do conceito de número, para além dos números reais. Constatamos, em particular, que a presença dos Números Complexos, nesta fase de formação escolar, é particularmente significativa porquanto pode, por si só e muito melhor do que a maioria dos outros capítulos, remeter-nos para o conjunto de questões evocadas no parágrafo anterior. No capítulo II analisamos os programas oficiais de Matemática ao longo dos últimos 50 anos e pudemos constatar que o tema dos Números Complexos tem ora sido retido no Ensino Secundário, ora tem sido adiado para o Ensino Superior. Analisamos também questões sobre Números Complexos tal qual foram surgindo em exames nacionais. No capítulo III, estudamos a História dos Números Complexos e constatamos que são razões de utilidade que estiveram na sua génese, embora fossem razões de natureza teórica (filosóficas, de existência, de coerência lógica com a restante Matemática) que levaram inúmeros autores a interrogarem-se sobre a natureza dos Números Complexos e a construírem, pouco a pouco, uma interpretação geométrica sem a qual a Teoria das Funções Analíticas não teria sido alcançada depois de 1825. Reconhecemos também a importância pedagógica da tomada de consciência dos erros e das discussões que procederam o período fecundo em que os matemáticos inventaram a Teoria das Funções Analíticas. No capítulo IV estabelecemos uma grelha de avaliação de manuais escolares de forma a estudar, em termos de existência e coerência lógica, o ensino dos Números Complexos nas escolas portuguesas. No capítulo V, abordamos a problemática do recurso à história da Matemática como instrumento facilitador da aprendizagem dos Números Complexos e sugerimos algumas propostas de actividades deste ensino. Com esta monografia de Mestrado pretendeu-se clarificar, através de um estudo histórico e didáctico sisemático, as variadas implicações para o ensino actual dos Números Complexos de toda a riqueza e a fecundidade do conhecimento acumulado por séculos de História. Concluímos finalmente que, porventura menosprezada pelos professores de Matemática (porventura por razões de condicionalismos práticos de agenda lectiva), a presença dos Números Complexos, nos programas nacionais do Ensino Secundário da Matemática, está plenamente justificada. Os Números Complexos podem efectivamente afirmar-se como um “instrumento” didáctico indispensável na concretização de objectivos gerais e específicos fundamentais da aprendizagem da Matemática como é o caso de: t - organizar e relacionar conhecimentos prévios dos alunos, envolvendo-os na descoberta; - ter em linha de conta tanto as capacidades de raciocínio abstracto como as da intuição; - responder a questões fundamentais como “O que é um número?”, “Para que servem os números?”, “Quem inventou os números?” ou “Como foram inventados os números?”All over the world, the teaching of Mathematics for Secondary Schools is conditioned by questions on “What Mathematics is due to be taught?”, “How to teach Mathematics?” or “Should Mathematics be useful? In what sense?” In this research study we refer, in the Introduction and Chapter I, that the concept of number is one of the most important in Mathematics that runs from the basic levels of Mathematics teaching to University instruction. It is during this path that we naturally find the Complex Numbers: usually as a generalization process beyond real numbers. We agreed, in particular, that Complex Numbers may be particularly important in relation to the questions that were posed in the last paragraph. Chapter II deals with the national curricula for Secondary Schools on the teaching of Complex Numbers through the past 50 years and we also analyzed questions on exams related to Complex Numbers. In Chapter III we studied the History of Complex Numbers and we came to acknowledge the reasons, both utilitarian and philosophical, related to its evolution. We point out to the doubts and the difficulties felt by the authors who built, bit by bit, the concept of Complex Number without whom the Theory of Analytical Functions would never be reached in 1825. We also recognized the pedagogical importance of acknowledging errors and faults committed by well known mathematicians. Chapter IV offers an evaluation grid for schooltexts, both for existence and logical coherence, on teaching Complex Numbers in Portuguese schools. In Chapter V we approach the use of History of Mathematics on teaching Mathematics by suggesting some school activities for teaching Complex Numbers. The aim of the conducted research on the actual teaching of Complex Numbers was the clarification, through a didactical-historical systematic study, of the various implications on the richness of accumulated knowledge by centuries of History. We finally concluded that the theme of Complex Numbers is perfectly justified in the National Curricula for Secondary Schools because: - it refers both to abstract thinking and intuitive capacities; - it organizes and it relates previous knowledge to actual mathematical knowledge, by involving pupils on discovery activities; - it allows both teachers and pupils to deal with fundamental questions on “What is a number?”, “What are numbers good for?”, “Who invented numbers?” or “How were numbers invented?

NEOTROPICAL ALIEN MAMMALS: a data set of occurrence and abundance of alien mammals in the Neotropics

Biological invasion is one of the main threats to native biodiversity. For a species to become invasive, it must be voluntarily or involuntarily introduced by humans into a nonnative habitat. Mammals were among first taxa to be introduced worldwide for game, meat, and labor, yet the number of species introduced in the Neotropics remains unknown. In this data set, we make available occurrence and abundance data on mammal species that (1) transposed a geographical barrier and (2) were voluntarily or involuntarily introduced by humans into the Neotropics. Our data set is composed of 73,738 historical and current georeferenced records on alien mammal species of which around 96% correspond to occurrence data on 77 species belonging to eight orders and 26 families. Data cover 26 continental countries in the Neotropics, ranging from Mexico and its frontier regions (southern Florida and coastal-central Florida in the southeast United States) to Argentina, Paraguay, Chile, and Uruguay, and the 13 countries of Caribbean islands. Our data set also includes neotropical species (e.g., Callithrix sp., Myocastor coypus, Nasua nasua) considered alien in particular areas of Neotropics. The most numerous species in terms of records are from Bos sp. (n = 37,782), Sus scrofa (n = 6,730), and Canis familiaris (n = 10,084); 17 species were represented by only one record (e.g., Syncerus caffer, Cervus timorensis, Cervus unicolor, Canis latrans). Primates have the highest number of species in the data set (n = 20 species), partly because of uncertainties regarding taxonomic identification of the genera Callithrix, which includes the species Callithrix aurita, Callithrix flaviceps, Callithrix geoffroyi, Callithrix jacchus, Callithrix kuhlii, Callithrix penicillata, and their hybrids. This unique data set will be a valuable source of information on invasion risk assessments, biodiversity redistribution and conservation-related research. There are no copyright restrictions. Please cite this data paper when using the data in publications. We also request that researchers and teachers inform us on how they are using the data

NEOTROPICAL CARNIVORES: a data set on carnivore distribution in the Neotropics

Mammalian carnivores are considered a key group in maintaining ecological health and can indicate potential ecological integrity in landscapes where they occur. Carnivores also hold high conservation value and their habitat requirements can guide management and conservation plans. The order Carnivora has 84 species from 8 families in the Neotropical region: Canidae; Felidae; Mephitidae; Mustelidae; Otariidae; Phocidae; Procyonidae; and Ursidae. Herein, we include published and unpublished data on native terrestrial Neotropical carnivores (Canidae; Felidae; Mephitidae; Mustelidae; Procyonidae; and Ursidae). NEOTROPICAL CARNIVORES is a publicly available data set that includes 99,605 data entries from 35,511 unique georeferenced coordinates. Detection/non-detection and quantitative data were obtained from 1818 to 2018 by researchers, governmental agencies, non-governmental organizations, and private consultants. Data were collected using several methods including camera trapping, museum collections, roadkill, line transect, and opportunistic records. Literature (peer-reviewed and grey literature) from Portuguese, Spanish and English were incorporated in this compilation. Most of the data set consists of detection data entries (n = 79,343; 79.7%) but also includes non-detection data (n = 20,262; 20.3%). Of those, 43.3% also include count data (n = 43,151). The information available in NEOTROPICAL CARNIVORES will contribute to macroecological, ecological, and conservation questions in multiple spatio-temporal perspectives. As carnivores play key roles in trophic interactions, a better understanding of their distribution and habitat requirements are essential to establish conservation management plans and safeguard the future ecological health of Neotropical ecosystems. Our data paper, combined with other large-scale data sets, has great potential to clarify species distribution and related ecological processes within the Neotropics. There are no copyright restrictions and no restriction for using data from this data paper, as long as the data paper is cited as the source of the information used. We also request that users inform us of how they intend to use the data

Characterisation of microbial attack on archaeological bone

As part of an EU funded project to investigate the factors influencing bone preservation in the archaeological record, more than 250 bones from 41 archaeological sites in five countries spanning four climatic regions were studied for diagenetic alteration. Sites were selected to cover a range of environmental conditions and archaeological contexts. Microscopic and physical (mercury intrusion porosimetry) analyses of these bones revealed that the majority (68%) had suffered microbial attack. Furthermore, significant differences were found between animal and human bone in both the state of preservation and the type of microbial attack present. These differences in preservation might result from differences in early taphonomy of the bones. © 2003 Elsevier Science Ltd. All rights reserved