Associativitet : Grundskoleelever urskiljer den associativa lagen

De grundläggande aritmetiska räknelagarna är centrala för elevers utveckling inom algebra. Det är därför viktigt att elever ges möjlighet att urskilja och utveckla förståelse för dessa. Genom denna kvalitativa intervjustudie undersöktes hur 16 elever i årskurs 2-5, utifrån en i förväg designad instruktionssekvens, resonerar om, generaliserar och använder den associativa egenskapen för addition. Studien visar att många elever faktiskt urskiljer den associativa egenskapen för addition genom arbetet med instruktionssekvensen, men att endast ett fåtal tillämpar denna egenskap vid beräkning. Studien visar även att flera elever efter att ha urskilt associativitet kan göra generaliseringar av egenskapen relaterad till addition eller subtraktion. Slutsatsen av studien är att en instruktionssekvens som erbjuder systematisk variation och upprepning av uttryck med samma struktur, möjliggör för elever att urskilja och beskriva associativitet. Urskiljandet möjliggörs även av att elever uppmanas att betrakta uttrycken från flera perspektiv och beskriva dem utifrån frågor om likheter och skillnader.

Tals del-helhetsrelationer : Elevers sätt att urskilja del-helhetsrelationer i öppna utsagor.

På vilket sätt kan vi hjälpa alla elever att bli förtrogna med matematikens uttrycksformer? Ett sätt är att bygga en stadig aritmetisk grund för eleverna där de befäster talens innehåll. Det är vad den här uppsatsen handlar om. Uppsatsen beskriver vad som skiljer användandet av del-helhetsrelationer från andra sätt att lösa öppna utsagor på. Uppsatsen beskriver även vilka kritiska aspekter om öppna utsagor som kan förekomma hos elever i årskurs 1 och 2. Uppsat-sen är skriven ur en fenomenografisk ansats med variationsteoretiska inslag eftersom de två teorierna är nära besläktade. Studien genomfördes genom filmade intervjuer med 11 elever som valdes ut genom en munt-lig och en skriftlig diagnos samt ett skriftligt arbetsblad. Resultatet visar att elever som använ-der automatiserade del-helhetsrelationer har en fördel när de löser öppna utsagor jämfört med elever som använder andra lösningsmetoder. Skillnaderna syns tydligt när det gäller lösandet av öppna subtraktionsutsagor där helheten saknas. En väg till den abstrakta förståelsen för tals del-helhetsrelationer går via fingertalen. Min slutsats är att eleverna redan tidigt i skolan måste få undervisning om fingertalen samt talens del-helhetsrelationer för att undvika att de utvecklar matematiksvårigheter.

The parallel solution of early-exercise Asian options with stochastic volatility

This paper describes an parallel semi-Lagrangian finite difference approach to the pricing of early exercise Asian Options on assets with a stochastic volatility. A multigrid procedure is described for the fast iterative solution of the discrete linear complementarity problems that result. The accuracy and performance of this approach is improved considerably by a strike-price related analytic transformation of asset prices. Asian options are contingent claims with payoffs that depend on the average price of an asset over some time interval. The payoff may depend on this average and a fixed strike price (Fixed Strike Asians) or it may depend on the average and the asset price (Floating Strike Asians). The option may also permit early exercise (American contract) or confine the holder to a fixed exercise date (European contract). The Fixed Strike Asian with early exercise is considered here where continuous arithmetic averaging has been used. Pricing such an option where the asset price has a stochastic volatility leads to the requirement to solve a tri-variate partial differential inequation in the three state variables of asset price, average price and volatility (or equivalently, variance). The similarity transformations [6] used with Floating Strike Asian options to reduce the dimensionality of the problem are not applicable to Fixed Strikes and so the numerical solution of a tri-variate problem is necessary. The computational challenge is to provide accurate solutions sufficiently quickly to support realtime trading activities at a reasonable cost in terms of hardware requirements.

The stochastic geometric machine model

This paper introduces the stochastic version of the Geometric Machine Model for the modelling of sequential, alternative, parallel (synchronous) and nondeterministic computations with stochastic numbers stored in a (possibly infinite) shared memory. The programming language L(D! 1), induced by the Coherence Space of Processes D! 1, can be applied to sequential and parallel products in order to provide recursive definitions for such processes, together with a domain-theoretic semantics of the Stochastic Arithmetic. We analyze both the spacial (ordinal) recursion, related to spacial modelling of the stochastic memory, and the temporal (structural) recursion, given by the inclusion relation modelling partial objects in the ordered structure of process construction.

'Day number': a promoter routine of flexibility and conceptual understanding

This paper is part of the Project “Adaptive thinking and flexible computation: Critical issues”. In this paper we discuss different perspectives of flexibility and adaptive thinking in literature. We also discuss the idea of proceptual thinking and how this idea is important in our perspective of adaptive thinking. The paper analyses a situation developed with a first grade classroom and its teacher named the day number. It is a daily activity at the beginning of the school day. It consists on to look for the date number and to think about different ways of writing it using the four arithmetic operations. The analyzed activity was developed on March 19, so the challenge was to write 19 in several ways. The data show the pupils’ enthusiasm and their efforts to find different ways of writing the number. Some used large numbers and division, which they were just starting to learn. The pupils presented symbolic expressions of 19, decomposing and recomposing it in a flexible manner.

Situações-problema no ensino de matemática financeira

O presente trabalho descreve uma proposta de atividade educacional direcionada para professores de Matemática, envolvendo situações-problema no ensino de Matemática Financeira para ser aplicado com alunos do Ensino Médio. Tais atividades tem como objetivo fornecer um contexto real, no qual o estudante esteja inserido. O trabalho se divide em quatro partes: a introdução de uma situaçãoproblema envolvendo juros simples, o conhecimento matemático, a resolução da situação-problema e a proposta de atividade educacional. Diferenciando-se do que usualmente é encontrado nos livros didáticos, a proposta aqui apresentada propõe estudar conteúdos matemáticos de forma articulada, envolvendo o conceito de porcentagem vinculado com funções lineares e juros simples com função afim e progressão aritmética. Dessa forma, é apresentada uma sequência de aulas envolvendo situações-problema através de atividades, adequadas para os alunos.

The Countdown Problem

We systematically develop a functional program that solves the countdown problem, a numbers game in which the aim is to construct arithmetic expressions satisfying certain constraints. Starting from a formal specification of the problem, we present a simple but inefficient program that solves the problem, and prove that this program is correct. We then use program fusion to calculate an equivalent but more efficient program, which is then further improved by exploiting arithmetic properties.

Fold and Unfold for Program Semantics

In this paper we explain how recursion operators can be used to structure and reason about program semantics within a functional language. In particular, we show how the recursion operator fold can be used to structure denotational semantics, how the dual recursion operator unfold can be used to structure operational semantics, and how algebraic properties of these operators can be used to reason about program semantics. The techniques are explained with the aid of two main examples, the first concerning arithmetic expressions, and the second concerning Milner's concurrent language CCS. The aim of the paper is to give functional programmers new insights into recursion operators, program semantics, and the relationships between them.

Design and implementation of a floating point unit for rigel, a massively parallel accelerator

Scientific applications rely heavily on floating point data types. Floating point operations are complex and require complicated hardware that is both area and power intensive. The emergence of massively parallel architectures like Rigel creates new challenges and poses new questions with respect to floating point support. The massively parallel aspect of Rigel places great emphasis on area efficient, low power designs. At the same time, Rigel is a general purpose accelerator and must provide high performance for a wide class of applications. This thesis presents an analysis of various floating point unit (FPU) components with respect to Rigel, and attempts to present a candidate design of an FPU that balances performance, area, and power and is suitable for massively parallel architectures like Rigel.

Aplicação da metodologia numérica “fast bounds crack” para uma estimativa eficiente da evolução do tamanho de trinca

A significant part of the life of a mechanical component occurs, the crack propagation stage in fatigue. Currently, it is had several mathematical models to describe the crack growth behavior. These models are classified into two categories in terms of stress range amplitude: constant and variable. In general, these propagation models are formulated as an initial value problem, and from this, the evolution curve of the crack is obtained by applying a numerical method. This dissertation presented the application of the methodology "Fast Bounds Crack" for the establishment of upper and lower bounds functions for model evolution of crack size. The performance of this methodology was evaluated by the relative deviation and computational times, in relation to approximate numerical solutions obtained by the Runge-Kutta method of 4th explicit order (RK4). Has been reached a maximum relative deviation of 5.92% and the computational time was, for examples solved, 130,000 times more higher than achieved by the method RK4. Was performed yet an Engineering application in order to obtain an approximate numerical solution, from the arithmetic mean of the upper and lower bounds obtained in the methodology applied in this work, when you don’t know the law of evolution. The maximum relative error found in this application was 2.08% which proves the efficiency of the methodology "Fast Bounds Crack".

Viel bringt viel? Effekte lernzeitverlängernder Maßnahmen auf die Lese- und Mathematikleistungen von Grundschülerinnen und Grundschülern mit ungünstiger Lernausgangslage

Der vorliegende Beitrag untersucht die Effekte von lernzeitverlängernden Maßnahmen für Grundschülerinnen und Grundschüler in Mecklenburg-Vorpommern mit einer ungünstigen Lernausgangslage zum Zeitpunkt der Einschulung. Dazu wurde über die gesamte Grundschulzeit hinweg die Leistungsentwicklung von 67 Kindern in den Bereichen Lesen und Rechnen erfasst. Bei 19 Kindern wurde eine Lernzeitverlängerung durch Diagnoseförderklassen (DFK) realisiert, bei 18 durch eine Klassenwiederholung (KW) und 30 Kinder lernten in regulären Grundschulklassen (GSK) ohne eine Lernzeitverlängerung. Die Auswertungen der Daten mittels Hierarchisch-linearer Modelle (HLM) weisen auf gleiche Entwicklungsverläufe der drei Untersuchungsgruppen in den Bereichen Lesen und Mathematik hin. Zum Ende der Klasse 4 erreichten die drei Gruppen ähnliche Leistungsniveaus. In allen drei Settings fiel auf, dass die Entwicklung mathematischer Kompetenzen über die Schulzeit hinweg verzögert erfolgte. Ungünstige Lernausgangslagen im Bereich Mathematik konnten den Analysen zufolge durch keine der untersuchten Beschulungsformen ausreichend kompensiert werden. (DIPF/Orig.)

Motivação para a prática da modalidade de atletismo por parte dos atletas de diferentes regiões de Portugal

Dissertação apresentado à Escola Superior de Educação do Instituto Politécnico de Castelo Branco para cumprimento dos requisitos necessários à obtenção do grau de Mestre em Atividade Física.

Beiträge zum Mathematikunterricht 2015. Vorträge auf der 49. Tagung für Didaktik der Mathematik vom 09.02.2015 bis 13.02.2015 in Basel [Band 2]

Die Jahrestagung der Gesellschaft für Didaktik der Mathematik fand im Jahr 2015 zum dritten Mal in der Schweiz statt. [...] Mit rund 300 Vorträgen, 16 moderierten Sektionen, 15 Arbeitskreistreffen und 21 Posterpräsentationen eröffnete sich ein breites Spektrum an Themen und unterschiedlichen Zugangsweisen zur Erforschung von Fragen rund um das Lernen und Lehren von Mathematik. (DIPF/Orig.)

Beiträge zum Mathematikunterricht 2015. Vorträge auf der 49. Tagung für Didaktik der Mathematik vom 09.02.2015 bis 13.02.2015 in Basel [Band 1]

Die Jahrestagung der Gesellschaft für Didaktik der Mathematik fand im Jahr 2015 zum dritten Mal in der Schweiz statt. [...] Mit rund 300 Vorträgen, 16 moderierten Sektionen, 15 Arbeitskreistreffen und 21 Posterpräsentationen eröffnete sich ein breites Spektrum an Themen und unterschiedlichen Zugangsweisen zur Erforschung von Fragen rund um das Lernen und Lehren von Mathematik. (DIPF/Orig.)

Preliminary assessment of the numerical competencies of children from a public preschool and kindergarten in Cuenca

Existe abundante evidencia de que los niños pequeños son capaces de desarrollar un conocimiento matemático y que las destrezas aritméticas de estos niños son predictores de su desempeño académico futuro. También existe un acuerdo común de que la calidad de la educación matemática inicial tiene una importante influencia en el aprendizaje posterior de los niños. En Ecuador hay escasos estudios sobre las competencias matemáticas tempranas de los niños y sobre su enseñanza. Por ello, se inició un estudio para (1) evaluar las competencias numéricas de los niños de pre-escolar y kindergarten (primero de básica) que asisten a una escuela pública de Cuenca, con el objetivo de analizar críticamente su pensamiento y razonamiento numérico; y (2) examinar las prácticas y creencias de los profesores con relación a la enseñanza de la matemática y a las competencias matemáticas de los niños. La aplicación del Test de Conocimiento Numérico (Griffin, 2005) demostró que la mayoría de los niños participantes no habían desarrollado habilidades numéricas básicas. Adicionalmente, los profesores expresaron una fuerte creencia de que los niños pequeños no son capaces de tener un pensamiento matemático. Como consecuencia, las actividades matemáticas que realizan los niños y profesores son desarrolladas de manera insuficiente. Las implicaciones científicas y prácticas de estos resultados son discutidas.