Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis

The tangent line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the tangent line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the tangent line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis

Computer simulation of syringomyelia in dogs

Syringomyelia is a pathological condition in which fluid-filled cavities (syringes) form and expand in the spinal cord. Syringomyelia is often linked with obstruction of the craniocervical junction and a Chiari malformation, which is similar in both humans and animals. Some brachycephalic toy breed dogs such as Cavalier King Charles Spaniels (CKCS) are particularly predisposed. The exact mechanism of the formation of syringomyelia is undetermined and consequently with the lack of clinical explanation, engineers and mathematicians have resorted to computer models to identify possible physical mechanisms that can lead to syringes. We developed a computer model of the spinal cavity of a CKCS suffering from a large syrinx. The model was excited at the cranial end to simulate the movement of the cerebrospinal fluid (CSF) and the spinal cord due to the shift of blood volume in the cranium related to the cardiac cycle. To simulate the normal condition, the movement was prescribed to the CSF. To simulate the pathological condition, the movement of CSF was blocked

Probabilidade de acontecimentos envolvendo aspetos lógicos

In this article, we study the Probability's knowledge of prospective primary school teachers, when there are logical aspects involved in the formulation of events. Forty-six students participated in the study, from a University in Northern Portugal, who had to resolve a three-item task in the context of a formal evaluation: the first one, on the union probability (connective or), the second one, on the joint probability (connective and), and the third one, on the conditional probability, in which the conditioned event is, in turn, a joint probability (connective and). In terms of the main results of the study, it was found that the students showed a very limited performance, frequently changing the connectives and and or, the inclusive disjunction with the exclusive disjunction and in the disjunction considered as incompatible non-disjoint events.Neste artigo estuda-se o conhecimento de Probabilidades de futuros professores dos primeiros anos escolares, quando na formulação dos acontecimentos estão envolvidos aspetos lógicos. Participaram no estudo 46 alunos da Licenciatura em Educação Básica, de uma Universidade do Norte de Portugal, os quais resolveram uma tarefa, com três itens, em contexto de avaliação formal: o primeiro sobre a probabilidade da reunião (conetivo ), o segundo sobre a probabilidade conjunta (conetivo ) e o terceiro sobre a probabilidade condicionada, em que o acontecimento condicionado é, por sua vez, uma probabilidade conjunta (conetivo ). Em termos dos principais resultados do estudo, verificou-se que os alunos revelaram um desempenho muito limitado, trocando, frequentemente, os conetivos e , a disjunção inclusiva com a disjunção exclusiva e na disjunção consideraram como sendo incompatíveis acontecimentos não disjuntos.Este trabalho contou com o apoio de Fundos Nacionais através da FCT – Fundação para a Ciência e a Tecnologia no âmbito do projeto PEst-OE/CED/UI1661/2014, do CIEd-UM e do projeto UID/Multi/04016/2016

Development of intuitive rules: Evaluating the application of the dual-system framework to understanding children's intuitive reasoning

This is an author-created version of this article. The original source of publication is Psychon Bull Rev. 2006 Dec;13(6):935-53 The final publication is available at www.springerlink.com Published version: http://dx.doi.org/10.3758/BF0321390

Connie Myers v. Albertsons, Inc. : Brief of Appellee

Appeal of the Judgment of Michael Glasmann Based upon a Jury Verdict Second Judicial District Court Weber County, State of Uta

Research on Teaching and Learning Probability

This book summarizes the vast amount of research related to teaching and learning probability that has been conducted for more than 50 years in a variety of disciplines. It begins with a synthesis of the most important probability interpretations throughout history: intuitive, classical, frequentist, subjective, logical propensity and axiomatic views. It discusses their possible applications, philosophical problems, as well as their potential and the level of interest they enjoy at different educational levels. Next, the book describes the main features of probabilistic thinking and reasoning, including the contrast to classical logic, probability language features, the role of intuitions, as well as paradoxes and the relevance of modeling. It presents an analysis of the differences between conditioning and causation, the variability expression in data as a sum of random and causal variations, as well as those of probabilistic versus statistical thinking. This is followed by an analysis of probability’s role and main presence in school curricula and an outline of the central expectations in recent curricular guidelines at the primary, secondary and high school level in several countries. This book classifies and discusses in detail the three different research periods on students’ and people’s intuitions and difficulties concerning probability: early research focused on cognitive development, a period of heuristics and biases programs, and the current period marked by a multitude of foci, approaches and theoretical frameworks

Intuitive and Informal Knowledge in Preschoolers’ Development of Probabilistic Thinking

Preschoolers develop a wide range of mathematical informal knowledge and intuitive thinking before they enter formal, goal-oriented education. In their everyday activities young children get engaged with situations that enhance them to develop skills, concepts, strategies, representations, attitudes, constructs and operations concerning a wide range of mathematical notions. Recently there is scientific interest in linking children’s informal and formal knowledge in order to provide them with opportunities to avoid biases aiming at formulating, perceiving, reflecting on and exercising probabilistic notions. The current study investigates preschoolers’ (N=90) intuitive understanding of the likelihood of events in a probabilistic task with spinners. Participants, at the age of 4 to 6, are tested on their predictions of the most probable outcome prior to and after an instructive session of reasoning. The probabilistic task, based on constructivist principles, includes methodological alterations concerning the sample space and the themes of the stimuli. Educational implications are further discussed under the general point of view that in order to link informal to formal mathematical learning in preschool classroom, the subject content and the cognitive capacity of children are important to match

Simultaneous genetic analysis of longitudinal means and covariance structure in the simplex model using twin data

A longitudinal model based on the simplex model is presented to analyze simultaneously means and covariance structure using univariate longitudinal twin data. The objective of the model is to decompose the mean trend into components which can be attributed to those genetic and environmental factors which give rise to phenotypic individual differences and a component of unknown constitution which does not involve individual differences. Illustrations are given using simulated data and repeatedly measured weight obtained in a sample of 82 female twin pairs on sbc occasions. KEY WORDS: repeated measures; genetic and environmental covariance structure; mean trend; longitudinal twin data; genetic simplex mode; LISREL