It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using our examples students were encouraged to think independently and came to improve their understanding of the subject.

Several years of my experiences in the use of real-time experiments are now upgraded in order to enhance also the teaching of mathematics. The motion sensor device enables us to get real time x(t) and v(t) graphs of a moving object or person. We can productively use these graphs to introduce differentiation on visual level as well as to show the integration procedure. The students are fully involved in the teaching as they are invited to walk in front of the sensor. This approach motivates them by the realistic aspects of mathematical structures. The method could help to fulfill the credo of teaching: comprehension before computation. The steps of such an approach are explained and discussed in further detail below.

Certainly one of the most powerful and important modeling languages of our time is the Calculus. But research consistently shows that students do not understand how the variables in calculus-based mathematical models relate to aspects of the systems that those models are supposed to represent. Because of this, students never access the true power of calculus: its suitability to model a wide variety of real-world systems across domains. In this paper, we describe the motivation and theoretical foundations for the DeltaTick and HotLink Replay applications, an effort to address these difficulties by a) enabling students to model a wide variety of systems in the world that change over time by defining the behaviors of that system, and b) making explicit how a system\'s behavior relates to the mathematical trends that behavior creates. These applications employ the visualization and codification of behavior rules within the NetLogo agent-based modeling environment (Wilensky, 1999), rather than mathematical symbols, as their primary building blocks. As such, they provide an alternative to traditional mathematical techniques for exploring and solving advanced modeling problems, as well as exploring the major underlying concepts of calculus.

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1378473 / Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2010

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1378473 / Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2010

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / A presente dissertação tem o objetivo de propor a reinclusão de elementos de Cálculo no ensino médio, pois no passado o Cálculo fazia parte do currículo e foi abolido após uma reforma no ensino da matemática. O trabalho apresenta os resultados de um levantamento estatístico sobre os índices de reprovação na disciplina Cálculo Diferencial e Integral I nos períodos mais recentes da Universidade do Estado do Rio de Janeiro (UERJ) e, também, uma pesquisa quantitativa com quarenta professores de matemática com o objetivo de analisar a viabilidade do projeto e os problemas a serem enfrentados. Por fim, a dissertação conta com uma série de atividades comentadas sobre o tema de limites, que é o foco do trabalho. Tais atividades podem ser incluídas já no 1 ano do ensino médio, paralelamente ao conteúdo de funções, e visam proporcionar aos estudantes o contato com elementos de Cálculo em uma linguagem acessível, e orientar o professor nesta tarefa / This dissertation has the objective of proposing the reinclusion of Calculus elements in high school, because in the past Calculus was part of the curriculum and it was abolished after a reform in mathematics teaching. This paper presents the results of a statistical return about the rates of fails in the subject Differential and integral Calculus I in recent terms at Universidade do Estado do Rio de Janeiro (UERJ) and also a quantitative research with forty mathematics teachers, which has the objective of analyzing the viability of the project and the problems to be faced. Finally, the dissertation has a series of discussed activities about the theme of limits, which is the focus of this paper. These activities can be included in the first year of high school, at the same time as functions content. They aim to offer students a contact with Calculus elements in an accessible language and also to orientate the teacher in this task.

Représentation et raisonnement qualitatifs fonctionnent avec des relations non-numériques entre les objets d'un univers. Les formalismes généraux développés dans ce domaine sont basés sur différents types d'algèbres de relations, comme les algèbres de Tarski. Tous ces formalismes, qui sont appelés des calculs qualitatifs, partagent l'hypothèse implicite que l'univers est homogène, c'est-à-dire qu'il se compose d'objets de même nature. Toutefois, les objets de différents types peuvent aussi entretenir des relations. L'état de l'art du raisonnement qualitatif ne permet pas de combiner les calculs qualitatifs pour les différents types d'objets en un seul calcul.De nombreuses applications discriminent entre différents types d'objets. Par exemple, certains modèles spatiaux discriminent entre les régions, les lignes et les points, et différentes relations sont utilisées pour chaque type d'objets. Dans l'alignement d'ontologies, les calculs qualitatifs sont utiles pour exprimer des alignements entre un seul type d'entités, telles que des concepts ou des individus. Cependant, les relations entre les individus et les concepts, qui imposent des contraintes supplémentaires, ne sont pas exploitées.Cette thèse introduit la modularité dans les calculs qualitatifs et fournit une méthodologie pour la modélisation de calculs qualitatifs des univers hétérogènes. Notre contribution principale est un cadre basé sur une classe spéciale de schémas de partition que nous appelons modulaires. Pour un calcul qualitatif engendré par un schéma de partition modulaire, nous définissons une structure qui associe chaque symbole de relation avec un domaine et codomain abstrait à partir d'un treillis booléen de sortes. Un module d'un tel calcul qualitatif est un sous-calcul limité à une sorte donnée, qui est obtenu par une opération appelée relativisation à une sorte. D'un intérêt pratique plus grand est l'opération inverse, qui permet de combiner plusieurs calculs qualitatifs en un seul calcul. Nous définissons une opération appelée combinaison modulo liaison, qui combine deux ou plusieurs calculs qualitatifs sur différents univers, en fonction de quelques relations de liaison entre ces univers. Le cadre est suffisamment général pour soutenir la plupart des calculs spatio-temporels qualitatifs connus. / Qualitative representation and reasoning operate with non-numerical relations holding between objects of some universe. The general formalisms developed in this field are based on various kinds of algebras of relations, such as Tarskian relation algebras. All these formalisms, which are called qualitative calculi, share an implicit assumption that the universe is homogeneous, i.e., consists of objects of the same kind. However, objects of different kinds may also entertain relations. The state of the art of qualitative reasoning does not offer a combination operation of qualitative calculi for different kinds of objects into a single calculus.Many applications discriminate between different kinds of objects. For example, some spatial models discriminate between regions, lines and points, and different relations are used for each kind of objects. In ontology matching, qualitative calculi were shown useful for expressing alignments between only one kind of entities, such as concepts or individuals. However, relations between individuals and concepts, which impose additional constraints, are not exploited.This dissertation introduces modularity in qualitative calculi and provides a methodology for modeling qualitative calculi with heterogeneous universes. Our central contribution is a framework based on a special class of partition schemes which we call modular. For a qualitative calculus generated by a modular partition scheme, we define a structure that associates each relation symbol with an abstract domain and codomain from a Boolean lattice of sorts. A module of such a qualitative calculus is a sub-calculus restricted to a given sort, which is obtained through an operation called relativization to a sort. Of a greater practical interest is the opposite operation, which allows for combining several qualitative calculi into a single calculus. We define an operation called combination modulo glue, which combines two or more qualitative calculi over different universes, provided some glue relations between these universes. The framework is general enough to support most known qualitative spatio-temporal calculi.

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abstract: A sample of 127 high school Advanced Placement (AP) Calculus students from two schools was utilized to study the effects of an engineering design-based problem solving strategy on student performance with AP style Related Rate questions and changes in conceptions, beliefs, and influences. The research design followed a treatment-control multiple post-assessment model with three periods of data collection. Four high school calculus classes were selected for the study, with one class designated as the treatment and three as the controls. Measures for this study include a skills assessment, Related Rate word problem assessments, and a motivation problem solving survey. Data analysis utilized a mixed methods approach. Quantitative analysis consisted of descriptive and inferential methods utilizing nonparametric statistics for performance comparisons and structural equation modeling to determine the underlying structure of the problem solving motivation survey. Statistical results indicate that time on task was a major factor in enhanced performance between measurement time points 1 and 2. In the experimental classroom, the engineering design process as a problem solving strategy emerged as an important factor in demonstrating sustained achievement across the measurement time series when solving volumetric rates of change as compared to traditional problem solving strategies. In the control classrooms, where traditional problem solving strategies were emphasized, a greater percentage of students than in the experimental classroom demonstrated enhanced achievement from point 1 to 2, but showed decrease in achievement from point 2 to 3 in the measurement time series. Results from the problem solving motivation survey demonstrated that neither time on task nor instruction strategy produced any effect on student beliefs about and perceptions of problem solving. Qualitative error analysis showed that type of instruction had little effect on the type and number of errors committed, with the exception of procedural errors from performing a derivative and errors decoding the problem statement. Results demonstrated that students who engaged in the engineering design-based committed a larger number of decoding errors specific to Pythagorean type Related Rate problems; while students who engaged in routine problem solving did not sustain their ability to correctly differentiate a volume equation over time. As a whole, students committed a larger number of misused data errors than other types of errors. Where, misused data errors are the discrepancy between the data as given in a problem and how the student used the data in problem solving. / Dissertation/Thesis / Ph.D. Mathematics Education 2012

Submitted by Raniere Barreto (raniere.barros@ufvjm.edu.br) on 2018-04-11T18:25:08Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) bruce_franca_guimaraes.pdf: 1772676 bytes, checksum: 70105fe5762f394af52a5b015acc42e3 (MD5) / Approved for entry into archive by Rodrigo Martins Cruz (rodrigo.cruz@ufvjm.edu.br) on 2018-04-20T13:42:12Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) bruce_franca_guimaraes.pdf: 1772676 bytes, checksum: 70105fe5762f394af52a5b015acc42e3 (MD5) / Made available in DSpace on 2018-04-20T13:42:12Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) bruce_franca_guimaraes.pdf: 1772676 bytes, checksum: 70105fe5762f394af52a5b015acc42e3 (MD5) Previous issue date: 2017 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior (CAPES) / O foco do presente trabalho ? apresentar um estudo sobre fun??es reais de v?rias vari?veis reais, bem como, estudar as principais t?cnicas envolvendo esse tipo de fun??o. Conv?m destacar os conceitos de limites, derivadas parciais, m?ximos e m?nimos multiplicadores de Lagrange. Na parte final da disserta??o, apresentamos uma aplica??o na Engenharia. / Disserta??o (Mestrado Profissional) ? Programa de P?s-Gradua??o Matem?tica, Universidade Federal dos Vales do Jequitinhonha e Mucuri, 2017. / The focus of the present work is to present a study on the real functions of several real variables, as well as to study the main techniques involving this type of function. It is worth mentioning the concepts of limits, partial derivatives, maximum and minimum and Lagrange multipliers. In the final part of the dissertation, we present an application in Engineering.