: Resolving problems involving proof and demonstration: a difficulty for primary and secondary school teachers

http://www.icmes2009.pucpr.br/index.phpInternational audienceIn the last decade, the importance attributed to proofs and demonstrations in Mathematics has led to a large variety of research in this area. We consider, generally, the demonstration as a validation procedure that characterizes Mathematics and distinguishes it from experimental sciences. We are part of a research group that develops a project together with teachers from the public and private school systems in the city of São Paulo and that, in its initial phase, discussed deductive reasoning in the process of teaching and learning Mathematics in the Middle School grades. This project provides continuing education for teachers concerned with their pedagogical practices and who would like to increment them, rethinking knowledge or even coming into contact with mathematics topics for the first time. In this work, we initially raise some theoretical questions that are related to demonstrations and that seek to better understand logical reasoning. We next present a case study, in a qualitative approach, in which we observe a teacher's difficulties in identifying the hypothesis and thesis of a mathematical affirmation in the area of geometry, especially when it does not present the famous expression "if and only if." We also discuss the issue of whether or not a proposition corresponds to a reciprocal theorem. Comprehension of the information given in the problem statement of a mathematics proposition and recognition of crucial elements such as hypothesis and thesis are fundamental to the process of constructing an acceptable demonstration.Na última década, a importância atribuída a provas e demonstrações em Matemática levou a uma enorme variedade de pesquisas nessa área. Consideramos, usualmente, a demonstração como um procedimento de validação que caracteriza a Matemática e a distingue das ciências experimentais. Fazemos parte de um grupo de pesquisa que desenvolve um projeto junto a professores da rede pública e particular da cidade de São Paulo e que discutiu, na sua fase inicial, o raciocínio dedutivo no processo de ensino e aprendizagem da Matemática nas séries finais do Ensino Fundamental. Esse projeto realiza uma formação continuada para professores preocupados com a sua prática pedagógica e que gostariam de incrementá-la, repensando saberes ou até mesmo entrando em contato com conteúdos matemáticos pela primeira vez. Nesse trabalho, inicialmente, levantamos algumas questões teóricas relacionadas à demonstração e que procuram compreender melhor o raciocínio lógico. Em seguida, apresentamos um estudo de caso, na abordagem qualitativa, em que observamos as dificuldades de um professor do ensino básico em identificar a hipótese e a tese em uma afirmação matemática da área de geometria, especialmente, quando essa não apresenta a famosa expressão "se e somen uma proposição corresponder ou não a um teorema recíproco. A compreensão da informação dada no enunciado de uma proposição matemática e o reconhecimento de elementos cruciais como hipótese e tese são fundamentais para o processo de construção de uma demonstração aceitável

As Operações com Números Racionais e seus Significados a partir da Concepção Parte-todo

We present a reflection regarding operations with fractional numbers focusing on the part-all conception through some activities that can contribute to teaching practices at the elementary school level. A brief characterization of the all-part conception is provided along with descriptions of activities involving the four basic operations based on the use of flat figures, although other conceptions may emerge during the solution. We limit our considerations to this conception because it is the most frequent, in textbooks as well as teacher practice, although it is not used to justify operational rules of fractional numbers. Key-words: Fractions. Fractional Numbers. Operations.Neste artigo, apresentamos uma reflexão a respeito das operações com números fracionários focalizando a concepção parte-todo por meio de algumas atividades que possam contribuir para a prática docente na escola básica. O estudo explicita uma breve caracterização da concepção parte-todo e atividades envolvendo as quatro operações fundamentais tratadas em figuras planas, embora outras concepções possam ser mobilizadas durante a solução. Restringimos nossas considerações a essa concepção visto ser a mais freqüente, tanto em livros didáticos, quanto na prática do professor, embora não seja utilizada para justificar as regras operatórias dos números fracionários. Palavras-chave: Frações. Números Fracionários. Operações

Covid-19 Dynamic Monitoring and Real-Time Spatio-Temporal Forecasting

Background: Periodically, humanity is often faced with new and emerging viruses that can be a significant global threat. It has already been over a century post—the Spanish Flu pandemic, and we are witnessing a new type of coronavirus, the SARS-CoV-2, which is responsible for Covid-19. It emerged from the city of Wuhan (China) in December 2019, and within a few months, the virus propagated itself globally now resulting more than 50 million cases with over 1 million deaths. The high infection rates coupled with dynamic population movement demands for tools, especially within a Brazilian context, that will support health managers to develop policies for controlling and combating the new virus. / Methods: In this work, we propose a tool for real-time spatio-temporal analysis using a machine learning approach. The COVID-SGIS system brings together routinely collected health data on Covid-19 distributed across public health systems in Brazil, as well as taking to under consideration the geographic and time-dependent features of Covid-19 so as to make spatio-temporal predictions. The data are sub-divided by federative unit and municipality. In our case study, we made spatio-temporal predictions of the distribution of cases and deaths in Brazil and in each federative unit. Four regression methods were investigated: linear regression, support vector machines (polynomial kernels and RBF), multilayer perceptrons, and random forests. We use the percentage RMSE and the correlation coefficient as quality metrics. / Results: For qualitative evaluation, we made spatio-temporal predictions for the period from 25 to 27 May 2020. Considering qualitatively and quantitatively the case of the State of Pernambuco and Brazil as a whole, linear regression presented the best prediction results (thematic maps with good data distribution, correlation coefficient >0.99 and RMSE (%) <4% for Pernambuco and around 5% for Brazil) with low training time: [0.00; 0.04 ms], CI 95%. / Conclusion: Spatio-temporal analysis provided a broader assessment of those in the regions where the accumulated confirmed cases of Covid-19 were concentrated. It was possible to differentiate in the thematic maps the regions with the highest concentration of cases from the regions with low concentration and regions in the transition range. This approach is fundamental to support health managers and epidemiologists to elaborate policies and plans to control the Covid-19 pandemics

High efficiency blue organic light-emitting diodes with below-bandgap electroluminescence

Blue organic light-emitting diodes require high triplet interlayer materials, which induce large energetic barriers at the interfaces resulting in high device voltages and reduced efficiencies. Here, we alleviate this issue by designing a low triplet energy hole transporting interlayer with high mobility, combined with an interface exciplex that confines excitons at the emissive layer/electron transporting material interface. As a result, blue thermally activated delay fluorescent organic light emitting diodes with a below-bandgap turn-on voltage of 2.5 V and an external quantum efficiency of 41.2% were successfully fabricated. These devices also showed suppressed efficiency roll-off maintaining an EQE of 34.8% at 1000 cd m-2. Our approach paves the way for further progress through exploring alternative device engineering approaches instead of only focusing on the demanding synthesis of organic compounds with complex structures

In Vitro Interactions between Bacteria, Osteoblast-Like Cells and Macrophages in the Pathogenesis of Biomaterial-Associated Infections

Biomaterial-associated infections constitute a major clinical problem that is difficult to treat and often necessitates implant replacement. Pathogens can be introduced on an implant surface during surgery and compete with host cells attempting to integrate the implant. The fate of a biomaterial implant depends on the outcome of this race for the surface. Here we studied the competition between different bacterial strains and human U2OS osteoblast-like cells (ATCC HTB-94) for a poly(methylmethacrylate) surface in the absence or presence of macrophages in vitro using a peri-operative contamination model. Bacteria were seeded on the surface at a shear rate of 11 1/s prior to adhesion of U2OS cells and macrophages. Next, bacteria, U2OS cells and macrophages were allowed to grow simultaneously under low shear conditions (0.14 1/s). The outcome of the competition between bacteria and U2OS cells for the surface critically depended on bacterial virulence. In absence of macrophages, highly virulent Staphylococcus aureus or Pseudomonas aeruginosa stimulated U2OS cell death within 18 h of simultaneous growth on a surface. Moreover, these strains also caused cell death despite phagocytosis of adhering bacteria in presence of murine macrophages. Thus U2OS cells are bound to loose the race for a biomaterial surface against S. aureus or P. aeruginosa, even in presence of macrophages. In contrast, low-virulent Staphylococcus epidermidis did not cause U2OS cell death even after 48 h, regardless of the absence or presence of macrophages. Clinically, S. aureus and P. aeruginosa are known to yield acute and severe biomaterial-associated infections in contrast to S. epidermidis, mostly known to cause more low-grade infection. Thus it can be concluded that the model described possesses features concurring with clinical observations and therewith has potential for further studies on the simultaneous competition for an implant surface between tissue cells and pathogenic bacteria in presence of immune system components

Cosmology of the selfaccelerating third order Galileon

In this paper we start from the original formulation of the galileon model with the original choice for couplings to gravity. Within this framework we find that there is still a subset of possible Lagrangians that give selfaccelerating solutions with stable spherically symmetric solutions. This is a certain constrained subset of the third order galileon which has not been explored before. We develop and explore the background cosmological evolution of this model drawing intuition from other even more restricted galileon models. The numerical results confirm the presence of selfacceleration, but also reveals a possible instability with respect to galileon perturbations.Comment: 30 pages, 24 figure

Maternal Undernutrition Significantly Impacts Ovarian Follicle Number and Increases Ovarian Oxidative Stress in Adult Rat Offspring

BACKGROUND: We have shown recently that maternal undernutrition (UN) advanced female pubertal onset in a manner that is dependent upon the timing of UN. The long-term consequence of this accelerated puberty on ovarian function is unknown. Recent findings suggest that oxidative stress may be one mechanism whereby early life events impact on later physiological functioning. Therefore, using an established rodent model of maternal UN at critical windows of development, we examined maternal UN-induced changes in offspring ovarian function and determined whether these changes were underpinned by ovarian oxidative stress. METHODOLOGY/PRINCIPAL FINDINGS: Our study is the first to show that maternal UN significantly reduced primordial and secondary follicle number in offspring in a manner that was dependent upon the timing of maternal UN. Specifically, a reduction in these early stage follicles was observed in offspring born to mothers undernourished throughout both pregnancy and lactation. Additionally, antral follicle number was reduced in offspring born to all mothers that were UN regardless of whether the period of UN was restricted to pregnancy or lactation or both. These reductions were associated with decreased mRNA levels of genes critical for follicle maturation and ovulation. Increased ovarian protein carbonyls were observed in offspring born to mothers UN during pregnancy and/or lactation and this was associated with peroxiredoxin 3 hyperoxidation and reduced mRNA levels; suggesting compromised antioxidant defence. This was not observed in offspring of mothers UN during lactation alone. CONCLUSIONS: We propose that maternal UN, particularly at a time-point that includes pregnancy, results in reduced offspring ovarian follicle numbers and mRNA levels of regulatory genes and may be mediated by increased ovarian oxidative stress coupled with a decreased ability to repair the resultant oxidative damage. Together these data are suggestive of maternal UN potentially contributing to premature ovarian ageing in offspring