Analysis of applications and of conceptions for an application-oriented mathematics instruction

In connection with the (revived) demand for considering applications in the teaching of mathematics, various schemata or lists of criteria have been developed since the end of the sixties, which set up requirements about closeness to the real world or about the type of mathematics being used, and which have made it possible to analyze the available applications in their light. After having stated the problem (in section 1), we present (in section 2) a sketch of some of the best known of these and of some earlier schemata, although we are not aiming for a complete picture. Then (in section 3) we distinguish among different dimensions.in the analysis of applications. With this as a basis, we develop (in section 4) our own suggestion for categorizing types of applications and conceptions for an application-oriented mathematics instruction. Then (in section 5) we illustrate our schemata by some examples of performed evaluations. Finally (in section 6), we present some preliminary first results of the analysis of teaching conceptions.

ONTIC: A Knowledge Representation System for Mathematics

Ontic is an interactive system for developing and verifying mathematics. Ontic's verification mechanism is capable of automatically finding and applying information from a library containing hundreds of mathematical facts. Starting with only the axioms of Zermelo-Fraenkel set theory, the Ontic system has been used to build a data base of definitions and lemmas leading to a proof of the Stone representation theorem for Boolean lattices. The Ontic system has been used to explore issues in knowledge representation, automated deduction, and the automatic use of large data bases.

Translation Invariance in Object Recognition, and Its Relation to Other Visual Transformations

Human object recognition is generally considered to tolerate changes of the stimulus position in the visual field. A number of recent studies, however, have cast doubt on the completeness of translation invariance. In a new series of experiments we tried to investigate whether positional specificity of short-term memory is a general property of visual perception. We tested same/different discrimination of computer graphics models that were displayed at the same or at different locations of the visual field, and found complete translation invariance, regardless of the similarity of the animals and irrespective of direction and size of the displacement (Exp. 1 and 2). Decisions were strongly biased towards same decisions if stimuli appeared at a constant location, while after translation subjects displayed a tendency towards different decisions. Even if the spatial order of animal limbs was randomized ("scrambled animals"), no deteriorating effect of shifts in the field of view could be detected (Exp. 3). However, if the influence of single features was reduced (Exp. 4 and 5) small but significant effects of translation could be obtained. Under conditions that do not reveal an influence of translation, rotation in depth strongly interferes with recognition (Exp. 6). Changes of stimulus size did not reduce performance (Exp. 7). Tolerance to these object transformations seems to rely on different brain mechanisms, with translation and scale invariance being achieved in principle, while rotation invariance is not.

A comparison of the alr and ilr transformations for kernel density estimation of compositional data

In a seminal paper, Aitchison and Lauder (1985) introduced classical kernel density estimation techniques in the context of compositional data analysis. Indeed, they gave two options for the choice of the kernel to be used in the kernel estimator. One of these kernels is based on the use the alr transformation on the simplex SD jointly with the normal distribution on RD-1. However, these authors themselves recognized that this method has some deficiencies. A method for overcoming these dificulties based on recent developments for compositional data analysis and multivariate kernel estimation theory, combining the ilr transformation with the use of the normal density with a full bandwidth matrix, was recently proposed in Martín-Fernández, Chacón and Mateu- Figueras (2006). Here we present an extensive simulation study that compares both methods in practice, thus exploring the finite-sample behaviour of both estimators

Inputs for the incorporation of the UNESCO guidelines on ICT competency standards for teachers : the training of teachers of mathematics in Central America

Resumen tomado de la publicaci??n

Introduction to Network Mathematics

Introduction to Network Mathematics provides college students with basic graph theory to better understand the Internet

School of Mathematics Computing Guide

Guide for computing in the School of Mathematics. Intended for new staff and PG students. Originally written by Anton Prowse from a number of earlier documents.

WAIS seminar:Mathematics for Web Science Part3-1

ABSTRACT In the first two seminars we looked at the evolution of Ontologies from the current OWL level towards more powerful/expressive models and the corresponding hierarchy of Logics that underpin every stage of this evolution. We examined this in the more general context of the general evolution of the Web as a mathematical (directed and weighed) graph and the archetypical “living network” In the third seminar we will analyze further some of the startling properties that the Web has as a graph/network and which it shares with an array of “real-life” networks as well as some key elements of the mathematics (probability, statistics and graph theory) that underpin all this. No mathematical prerequisites are assumed or required. We will outline some directions that current (2005-now) research is taking and conclude with some illustrations/examples from ongoing research and applications that show great promise.

WAIS seminar:Mathematics for Web Science Part 3-2

ABSTRACT In the first two seminars we looked at the evolution of Ontologies from the current OWL level towards more powerful/expressive models and the corresponding hierarchy of Logics that underpin every stage of this evolution. We examined this in the more general context of the general evolution of the Web as a mathematical (directed and weighed) graph and the archetypical “living network” In the third seminar we will analyze further some of the startling properties that the Web has as a graph/network and which it shares with an array of “real-life” networks as well as some key elements of the mathematics (probability, statistics and graph theory) that underpin all this. No mathematical prerequisites are assumed or required. We will outline some directions that current (2005-now) research is taking and conclude with some illustrations/examples from ongoing research and applications that show great promise.

WAIS seminar: Mathematics for Web Science: Part 3-3

ABSTRACT In the first two seminars we looked at the evolution of Ontologies from the current OWL level towards more powerful/expressive models and the corresponding hierarchy of Logics that underpin every stage of this evolution. We examined this in the more general context of the general evolution of the Web as a mathematical (directed and weighed) graph and the archetypical “living network” In the third seminar we will analyze further some of the startling properties that the Web has as a graph/network and which it shares with an array of “real-life” networks as well as some key elements of the mathematics (probability, statistics and graph theory) that underpin all this. No mathematical prerequisites are assumed or required. We will outline some directions that current (2005-now) research is taking and conclude with some illustrations/examples from ongoing research and applications that show great promise.

Intervention strategies on teachers focused on creating meaningful educational environments for mathematical abilities development

This article describes an intervention process undertaken in a training program for preschool and first grade teachers from public schools in Cali, Colombia. The objective of this process is to provide a space for teachers to reflect on pedagogical practices which allow them to generate educational processes that foster children’s understanding of mathematical knowledge in the classroom. A set of support strategies was presented for helping teachers in the design, analysis and implementation of learning environments as meaningful educational spaces. Furthermore, participants engaged in an analysis of their own intervention modalities to identify which modalities facilitate the development of mathematical abilities in children. In order to ascertain the transformations in the teachers’ learning environments, the mathematical competences and cognitive processes underlying the activities proposed in the classroom, as well as teacher intervention modalities and the types of student participation in classroom activities were examined both before and after the intervention process. Transformations in the teachers’ conceptions about the children’s abilities and their own practices in teaching mathematics in the classroom were evidenced.

Option pricing in market models driven by telegraph processes with jumps

Esta tesis está dividida en dos partes: en la primera parte se presentan y estudian los procesos telegráficos, los procesos de Poisson con compensador telegráfico y los procesos telegráficos con saltos. El estudio presentado en esta primera parte incluye el cálculo de las distribuciones de cada proceso, las medias y varianzas, así como las funciones generadoras de momentos entre otras propiedades. Utilizando estas propiedades en la segunda parte se estudian los modelos de valoración de opciones basados en procesos telegráficos con saltos. En esta parte se da una descripción de cómo calcular las medidas neutrales al riesgo, se encuentra la condición de no arbitraje en este tipo de modelos y por último se calcula el precio de las opciones Europeas de compra y venta.

Las finanzas en la frontera del conocimiento

En los últimos treinta años las finanzas se convirtieron en una disciplina que ha sido objeto de estudio y transformaciones continuas. Su evolución y desarrollo ha sido sorprendente acercándose cada vez más a la frontera del conocimiento. A pesar de que sus primeros estudios fueron solamente prácticos y conceptuales, a partir de las década de los cincuenta surgió la teoría en las finanzas debido al reconocimiento de la incertidumbre y el riesgo tanto en los negocios como en la economía. En la búsqueda de una fundamentación teórica que permitiera predecir los comportamientos en el futuro de los asuntos financieros, se acudió a los conceptos desarrollados por la teoría económica. La complejidad y el surgimiento de nuevos cuestionamientos hicieron que la teoría financiera se sirviera tanto de la economía como de otras disciplinas como las matemáticas, estadística y econometría. En la actualidad es indispensable tener una completa formación en estas disciplinas para incursionar en el estudio de las finanzas.