Theorem Justification and Acquisition in Dynamic Geometry: A Case of Proof by Contradiction

Theorem acquisition and deductive proof have always been core elements in the study and teaching of Euclidean geometry. The introduction of dynamic geometry environments,DGE (e.g., Cabri-Géomètre, Geometer's Sketchpad), into classrooms in the past decade has posed a challenge to this praxis. Student scan experiment through different dragging modalities on geometrical objects that they construct, and consequently infer properties(generalities, theorems) about the geometrical artefacts. Because of the inductive nature of the DGE, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical and epistemological concern. In this paper, we will describe and study a ‘Cabri proof by contradiction’ of a theorem on cyclic quadrilaterals given by a pair of 16 year-old students in a Hong Kong secondary school. We will discuss how their construction motivates a visual-cognitive scheme on `seeing' proof in DGE, and how this scheme could fit into the theoretical construct of cognitive unity of theorems proposed by Boero, Garuti and Mariotti(1996). The issue of a cognitive duality and its relation to visualization will be raised and discussed. Finally, we propose a possible perspective to bridge the experimental-theoretical gap in DGE by introducing the idea of a dynamic template as a visualizer to geometrical theorem justification and acquisition.postprin

Properties of tangential and cyclic polygons: an application of circulant matrices

In this paper, the properties of tangential and cyclic polygons proposed by Lopez-Real are proved rigorously using the theory of circulant matrices. In particular, the concepts of slippable tangential polygons and conformable cyclic polygons are defined. It is shown that an n-sided tangential (or cyclic) polygon Pn with n even is slippable (or conformable) and the sum of a set of non-adjacent sides (or interior angles) of Pn satisfies certain equalities. On the other hand, for a tangential (or cyclic) polygon Pn with n odd, it is rigid and the sum of a set of non-adjacent sides (or interior angles) of Pn satisfies certain inequalities. These inequalities give a definite answer to the question raised by Lopez-Real concerning the alternating sum of interior angles of a cyclic polygon.postprin

Deletion of iRhom2 protects against diet-induced obesity by increasing thermogenesis

Objective: Obesity is the result of positive energy balance. It can be caused by excessive energy consumption but also by decreased energy dissipation, which occurs under several conditions including when the development or activation of brown adipose tissue (BAT) is impaired. Here we evaluated whether iRhom2, the essential cofactor for the Tumour Necrosis Factor (TNF) sheddase ADAM17/TACE, plays a role in the pathophysiology of metabolic syndrome.Methods: We challenged WT versus iRhom2 KO mice to positive energy balance by chronic exposure to a high fat diet and then compared their metabolic phenotypes. We also carried out ex vivo assays with primary and immortalized mouse brown adipocytes to establish the autonomy of the effect of loss of iRhom2 on thermogenesis and respiration.Results: Deletion of iRhom2 protected mice from weight gain, dyslipidemia, adipose tissue inflammation, and hepatic steatosis and improved insulin sensitivity when challenged by a high fat diet. Crucially, the loss of iRhom2 promotes thermogenesis via BAT activation and beige adipocyte recruitment, enabling iRhom2 KO mice to dissipate excess energy more efficiently than WT animals. This effect on enhanced thermogenesis is cell-autonomous in brown adipocytes as iRhom2 KOs exhibit elevated UCP1 levels and increased mitochondrial proton leak.Conclusion: Our data suggest that iRhom2 is a negative regulator of thermogenesis and plays a role in the control of adipose tissue homeostasis during metabolic disease. (C) 2019 The Authors. Published by Elsevier GmbH

International Comparative Studies In Mathematics: Lessons For Improving students’ Learning

Comparing is one of the most basic intellectual activities. We consciously make comparisons to understand where we stand, both in relation to others as well as to our own past experiences. There has been a long history of international comparative studies in education (Alexander 2000). Especially in the past several decades, many international comparative studies of mathematics have been conducted, either to examine differences in mathematical proficiency and dispositions among students from various countries or to understand the possible influence on the observed differences of various factors such as curriculum, teacher preparation, quality of classroom instruction, and parental involvement. Some of these studies are large-scale, and others are small-scale in-depth analyses from cognitive or social perspectives. These international comparative studies in mathematics are valuable because they provide a large body of knowledge showing how students do mathematics in the context of the world’s varied educational institutions. In addition, they examine the cultural and educational factors that influence the learning of mathematics.published_or_final_versio

Mortality and pulmonary complications in patients undergoing surgery with perioperative SARS-CoV-2 infection: an international cohort study

Background: The impact of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) on postoperative recovery needs to be understood to inform clinical decision making during and after the COVID-19 pandemic. This study reports 30-day mortality and pulmonary complication rates in patients with perioperative SARS-CoV-2 infection. Methods: This international, multicentre, cohort study at 235 hospitals in 24 countries included all patients undergoing surgery who had SARS-CoV-2 infection confirmed within 7 days before or 30 days after surgery. The primary outcome measure was 30-day postoperative mortality and was assessed in all enrolled patients. The main secondary outcome measure was pulmonary complications, defined as pneumonia, acute respiratory distress syndrome, or unexpected postoperative ventilation. Findings: This analysis includes 1128 patients who had surgery between Jan 1 and March 31, 2020, of whom 835 (74·0%) had emergency surgery and 280 (24·8%) had elective surgery. SARS-CoV-2 infection was confirmed preoperatively in 294 (26·1%) patients. 30-day mortality was 23·8% (268 of 1128). Pulmonary complications occurred in 577 (51·2%) of 1128 patients; 30-day mortality in these patients was 38·0% (219 of 577), accounting for 81·7% (219 of 268) of all deaths. In adjusted analyses, 30-day mortality was associated with male sex (odds ratio 1·75 [95% CI 1·28–2·40], p\textless0·0001), age 70 years or older versus younger than 70 years (2·30 [1·65–3·22], p\textless0·0001), American Society of Anesthesiologists grades 3–5 versus grades 1–2 (2·35 [1·57–3·53], p\textless0·0001), malignant versus benign or obstetric diagnosis (1·55 [1·01–2·39], p=0·046), emergency versus elective surgery (1·67 [1·06–2·63], p=0·026), and major versus minor surgery (1·52 [1·01–2·31], p=0·047). Interpretation: Postoperative pulmonary complications occur in half of patients with perioperative SARS-CoV-2 infection and are associated with high mortality. Thresholds for surgery during the COVID-19 pandemic should be higher than during normal practice, particularly in men aged 70 years and older. Consideration should be given for postponing non-urgent procedures and promoting non-operative treatment to delay or avoid the need for surgery. Funding: National Institute for Health Research (NIHR), Association of Coloproctology of Great Britain and Ireland, Bowel and Cancer Research, Bowel Disease Research Foundation, Association of Upper Gastrointestinal Surgeons, British Association of Surgical Oncology, British Gynaecological Cancer Society, European Society of Coloproctology, NIHR Academy, Sarcoma UK, Vascular Society for Great Britain and Ireland, and Yorkshire Cancer Research