This master thesis is focused on fractions as a problematic area of mathematics education. The goal of the thesis is to find out which mistakes students make and which wrong strategies they adopt in dealing with fractions. Another aim is to study how the historical ways of dealing with fractions could help the students nowadays. The thesis consists of a theoretical part introduces the matematical thinking of the Ancient Mesopotomia and Egypt periods, as well as present day teaching of fractions, using analysis of the textbooks and a brief glimpse into Czech as well as international researches. The practical part is based on an investigation in three different seventh grade classes in two primary schools and one high school class of the prima grade. In total 73 students participated in the research which cotains a diagnostic test, a series of individual interviews and an educational experiment with historical tasks. The result of this thesis is the finding that most of the mistaken strategies are not connected with only one particular school, because they appear in all of the schools and the Ancient Egyptian methods could be helpful in teaching fractions today as well. For this reason I recommend to spend more time in the educational proces with unit fractions (fractions of the form 1 n ), which are...

Made available in DSpace on 2018-06-27T19:13:42Z (GMT). No. of bitstreams: 2 maria do carmo trevisab.pdf: 3684071 bytes, checksum: 54548083b79cf21d41221c0e5611f14c (MD5) maria do carmo trevisab.pdf.jpg: 3636 bytes, checksum: 9ae74f3f5f367c61e1b1c1873174d665 (MD5) Previous issue date: 2006-11-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The reason for this work emerged from the importance of the Systems of Numeracy in the basic subjects of digital eletronics, that are responsible for the operation of the microcomputers. The idea of conceiving and developing a multimedia application about this matter comes from the experience as a teacher of the Técnico em Informática course of Escola Estadual de Ensino Médio Profª. Maria Rocha. The objective of the operation is to reinforce the learning of the students from the technical courses in computer science that is related to this matter and utilize the mathematical operations at different bases. It was included at the application the History of Mathematics because it is believed that it is a significant contribution for the understanding of the matter. The systems of Numeracy is also used at primary education and the application can be used by students through an appropriate methodology because the relation between an educational software and the learning are specially related to the methodology of the teaching used by the teacher. / A motivação para esse trabalho surgiu da importância do conteúdo de Sistemas de Numeração nas disciplinas básicas de eletrônica digital, responsáveis pelo funcionamento dos microcomputadores. A idéia de conceber e desenvolver uma aplicação multimídia sobre esse conteúdo surgiu a partir da experiência como docente no curso profissionalizante Técnico em Informática da Escola Estadual de Ensino Médio Profª Maria Rocha. O objetivo da aplicação é potencializar a aprendizagem deste conteúdo e trabalhar as operações matemáticas nas diversas bases pelos alunos dos cursos técnicos em Informática. Foi incluída na aplicação a História da Matemática, porque se acredita que isso contribui de forma significativa para a real compreensão do conteúdo. Sistemas de Numeração também é trabalhado na educação básica e a aplicação pode ser aproveitada por esses alunos com uma metodologia apropriada, pois, as relações entre um software educacional e a aprendizagem estão, em especial, diretamente condicionadas à metodologia de ensino utilizada pelo professor.

Två vanliga räkneregler som elever lär sig om i matematikundervisningen är prioriteringsregeln och vänster-till-höger-principen. Tidigare forskning har dock visat att elever också använder påhittade regler som vanligtvis inte brukar användas inom matematiken. Syftet med den här studien är att undersöka dessa ”egenskapade” regler. Syftet uppnås genom att studera vad det är för mindre kända räkneregler som eleverna tillämpar samt om hur konsekventa eleverna är i sin användning av en typ av räkneregel.  I studien gjorde 55 elever i årskurs 5 ett arbetsblad bestående av fem numeriska uttryck. Av de 55 eleverna använde 16 av dem någon form av regel som gick ut på att tal i de numeriska uttrycken parades ihop. 13 av de här 16 eleverna blev intervjuade om hur de hade tänkt när de löste uppgifterna. Data för studien utgörs därför av elevernas arbetsblad såväl som transkriberingarna från intervjuerna. Studien visar tre olika slags ”regler” som eleverna använder, förutom de vanliga räknereglerna vänster-till-höger-principen och prioriteringsregeln. De tre räknereglerna bygger alla på att tal paras ihop på ett eller annat sätt. Trots att nästan ingen av de 13 eleverna hade fått undervisning om de vanliga räknereglerna, så använder eleverna egna regler som följer logiska strukturer. Dessutom visar studien att de flesta eleverna inte är speciellt konsekventa när det kommer till valet av regel. Många av eleverna väljer att använda olika slags räkneregler för att beräkna uttryck som är uppbyggda på nästan samma sätt. / Two common rules of arithmetic that students learn about in education are the order of operations and the counting from left to right. However, previous research has shown that students also use made-up rules which are not usually used in mathematics. The aim of this study is to investigate the rules of arithmetic created by the students themselves. The aim is achieved by examine what kind of less-known rules of arithmetic that students apply and also how consistent students are in their use of a type of rule.   In the study, 55 students did a worksheet consisting of five tasks. In total, 16 of the 55 students used some kind of rule where numbers in the numerical expressions were paired in some way. Furthermore, 13 of the 16 students were interviewed about their way of thinking when solving the tasks. The data therefore consists of the students’ worksheets and transcriptions from the interviews.  The study shows that, in addition to the usual conventions left-to-right and order of operations, students use three different kinds of rules of arithmetic. The three rules of arithmetic are based on the principle that numbers are paired in one way or another. Despite that almost none of the 13 students had been taught the conventional rules of arithmetic, most students use own rules that follow logical structures. In addition, the study shows that most students are not particularly consistent when it comes to choosing strategy. Many students choose to use different kind of rules of arithmetic when they are calculating expressions that are structured in almost the same way.

Made available in DSpace on 2016-04-29T13:17:55Z (GMT). No. of bitstreams: 1 Patricia Nogueira Azevedo.pdf: 3065901 bytes, checksum: 01828ef21534ce60a06f95f4854b6812 (MD5) Previous issue date: 2015-03-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Plan contingencies of reinforcement in the school context involves a number of variables that must be analyzed and taken into account. Therefore, invest in the analysis of contingencies that produce effective teaching that can assist teachers in the exercise of its function becomes essential. This work aimed to investigate the effects of contingencies of positive reinforcement and punishment / negative reinforcement on the answer of children in arithmetic activities. The participants were eight students of 1st and 2nd years of elementary school. To perform the experiment was developed a computer application, with which participants perform arithmetic accounts (addition or subtraction) of two digits, no loan. The participants were divided randomly into four groups of two participants each, which underwent four experimental conditions, two baselines - addition or subtraction operations without reinforcement - and two experimental conditions - Positive Reinforcement with addition operation or Positive Reinforcement with subtraction operation and Negative Reinforcement with addition operation or Negative Reinforcement with subtraction operation. The number of accounts held and the amount of hits and misses of each participant in each session were assessed. The results showed that both contingency positive reinforcement and negative reinforcement can produce changes in the answer of the participants, that is, children learn when subjected to two types of contingencies tested in this study, it is not possible to claim that one produces better results than the other in terms of student learning. There was also an increase in the number of hits as much as the addition of the subtraction operations independently of the contingency effect. Identifies the need for more work to research the relationship between teaching conditions and their products, which can help in the development of new teaching procedures that favor learning / Planejar contingências de reforçamento no contexto escolar envolve uma série de variáveis que devem ser analisadas e levadas em consideração. Para tanto, investir na análise de contingências que produzam ensino efetivo que possa auxiliar professores no exercício de sua função torna-se imprescindível. Este trabalho teve o objetivo de investigar os efeitos de contingências de reforçamento positivo e de punição/reforçamento negativo sobre o responder de crianças em atividades de aritmética. Participaram da pesquisa oito alunos de 1º e 2º anos do ensino fundamental. Para realização do experimento foi desenvolvido um aplicativo de computador, com o qual os participantes realizam contas aritméticas (operações de soma ou subtração) de dois dígitos, sem empréstimo. Os participantes foram divididos, de forma aleatória, em quatro grupos de dois participantes cada, que passaram por quatro condições experimentais, sendo duas linhas de base - Operações de Soma ou Subtração sem reforçamento - e duas condições experimentais - Reforçamento Positivo com Operação de Soma ou Reforçamento Positivo com Operação de Subtração e Reforçamento Negativo com Operação de Soma ou Reforçamento Negativo com Operação de Subtração. Foram avaliadas a quantidade de contas realizadas e a quantidade de acertos e erros de cada participante a cada sessão. Os resultados demonstraram que tanto contingências de reforçamento positivo quanto negativo podem produzir alterações no responder dos participantes, ou seja, as crianças aprendem quando submetidas aos dois tipos de contingências testadas neste estudo, não sendo possível a afirmação de que uma delas produza melhores resultados do que a outra em termos da aprendizagem dos alunos. Verificou-se, também, um aumento no número de acertos tanto nas operações de soma quanto de subtração, independentemente da contingência em efeito. Identifica-se a necessidade de mais trabalhos que pesquisem a relação entre as contingências de ensino e seus produtos, o que pode ajudar no desenvolvimento de novos procedimentos de ensino que favoreçam o aprendizado

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2017-01-18T10:05:54Z No. of bitstreams: 2 Dissertação - Henrique Bernardes da Silva - 2016.pdf: 11781455 bytes, checksum: 3b2abe26f1a59c53402fa294d12ee6da (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-01-18T10:06:51Z (GMT) No. of bitstreams: 2 Dissertação - Henrique Bernardes da Silva - 2016.pdf: 11781455 bytes, checksum: 3b2abe26f1a59c53402fa294d12ee6da (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-01-18T10:06:51Z (GMT). No. of bitstreams: 2 Dissertação - Henrique Bernardes da Silva - 2016.pdf: 11781455 bytes, checksum: 3b2abe26f1a59c53402fa294d12ee6da (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-12-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The meaning given to the definitions and arithmetical properties necessary to the process of constructing the numerical sets are related to the understanding of this process by the teacher and to the situations proposed to the student. In this sense, prioritizing the sets of natural and integers, this text proposes a construction of the numerical sets and highlights, considering the dissemination of technological resources that can be used in the classroom, how the meanings of numbers and operations are present in situations with use software. The text presented in this paper is divided into three parts. The first one is dedicated to the construction of the numerical sets based on their arithmetic characteristics. In a second moment some softwares with potential for the teaching of Mathematics are presented and finally a proposal of use of software to carry out activities directed to the signification. The objective of this work is, therefore, to offer teachers theoretical subsidies for mathematical reasoning, constructing a simplified theoretical basis of arithmetic addressed in basic education and, besides, to present suggestions of software for the work of mathematical significance highlighting their potentialities and a didactic situation involving one of them. / O significado dado às definições e propriedades aritméticas necessárias ao processo de construção dos conjuntos numéricos estão relacionados à compreensão deste processo, pelo professor, e às situações propostas ao aluno. Neste sentido, priorizando os conjuntos dos números naturais e inteiros, este texto propõe uma construção dos conjuntos numéricos e destaca, considerando disseminação dos recursos tecnológicos que podem ser utilizados em sala de aula, como os significados dos números e operações estão presentes em situações com uso de aplicativos. O texto apresentado neste trabalho está dividido em três partes. A primeira delas é dedicada a construção dos conjuntos numérico com base nas suas características aritméticas. Em um segundo momento são apresentados alguns aplicativos com potencial para o ensino de Matemática e por fim uma proposta de utilização de software para realização de atividades voltadas à significação. O objetivo deste trabalho é, portanto, oferecer aos professores subsídios teóricos para a fundamentação matemática, construindo uma base teórica simplificada da aritmética abordada na educação básica e, além disto, apresentar sugestões de aplicativos para o trabalho de significação matemática destacando suas potencialidades e uma situação didática envolvendo um deles.

This thesis focuses on errors and difficulties that students face when solving problems with algebraic expressions in mathematics at secondary school. Its aim was to describe the factors that affect pupils' achievement while dealing with algebraic expressions, classify them on the basis of a classification of pupils' errors used in mathematics and identify the biggest pupils' difficulties. The thesis consists of theoretical and experimental parts. The theoretical part focuses on the factors that influence the success of pupils in their learning process. I present their summary based on information gained from literature and I complete them with my own teaching experience of mathematics at secondary school. Next I deal with the concept of error, error classification and one of the most important phases of learning process, which is a description of teacher's work with pupil's error (again on the basis of information gained from the literature). The theoretical part ends with definitions of basic concepts from specialized literature on algebraic expressions at the end of the theoretical part. In the experimental part I deal with my own experiment during teaching of mathematics at secondary school, which is based on individual written work of students in the first year of their study and on the...

The aim of this thesis is to deal with teaching of negative numbers in mathematics in lower primary school. The thesis brings an overview of exercises related to negative numbers, which are available in textbooks dedicated to lower and upper primary school. Main part of the thesis is devoted to three different experiments - pupils experiment, teaching experiment and parlour game. The pupils experiment analysis the successes and troubles of pupils attending 4th and 5th class when filling out the worksheets focused on various models of negative numbers. Furthermore, the teaching experiment based on a questionnaire survey describes comments of lower and upper primary school teachers in regards to implementation and usefulness of negative numbers. Parlour game experiment demonstrates the ability of pupils to grasp mathematical phenomenon by playing a game. Key words: ­ additive operations with negative numbers ­ experiment ­ models of negative number: thermometer, floor, surface ­ environment Stepping ­ environment Stairs ­ word problems ­ board game ­ negative numbers

Les travaux développés dans ce manuscrit de thèse concernent la proposition et la mise au point d’une nouvelle architecture optique basée sur la modulation de cohérence (MC) d’une source à spectre large dédiée aux calculs arithmétiques et au traitement des signaux. La modulation de cohérence de lumière est une technique particulière de codage optique qui autorise, entre autres, le multiplexage des signaux à travers une seule porteuse. Les signaux à traiter sont codés en MC par le biais de paires modulateur de lumière/lame de phase placées en série et éclairées par un seul faisceau de lumière polarisée. Cette technique est basée sur l’introduction d’un retard optique supérieur à la longueur de cohérence de la source utilisée. La validation expérimentale de l’approche proposée pour la réalisation d’opérations arithmétiques, telles que la somme et la soustraction, a été effectuée par le biais de signaux temporels. Différentes formes et fréquences des signaux ont été testées et ont parfaitement validé l’approche. L’impact du cross-talk des signaux et de la divergence du faisceau gaussien sur la qualité des opérations effectuées a été étudié. Ces effets se traduisent par un bruit de modulation d’intensité affectant le résultat des opérations effectuées. Dans ce travail, nous avons proposé des solutions permettant de minimiser son impact. L’originalité de la technique proposée est qu’elle permet la réalisation d’opérations multiples entre plusieurs signaux. Des tests ont été réalisés sur des images à deux et plusieurs niveaux de gris. Les résultats obtenus ont été évalués par les figures de mérite incluant le rapport signal sur bruit (SNR, signal to noise ratio), le rapport signal/bruit crête à crête (PSNR, peak signal to noise ratio) et l’erreur quadratique moyenne (MSE, mean squared error). Enfin, nous avons appliqué la technique à la cryptographie par contenu. Nous avons démontré la performance et la robustesse de la technique. Comme perspectives de ce travail, nous envisageons exploiter d’avantage la technique dans le domaine de la cryptographie (ie: utilisation d’une phase aléatoire pour le codage des images). De plus, une extension de l’étude à la compression des images sera utile. Une autre perspective de ce travail de recherche est l’étude de l’impact de l’incohérence spatiale sur le codage et le décodage des signaux / In this thesis, we study and developed the use of coherence multiplexing (also called path-difference, by analogy with WDM optical communications) to achieve simultaneous coding and decoding of analogue signals. The coherence modulation of light consists in encoding a signal on a light beam as an optical path-difference larger than its coherence length. This opens the way to the use of broadband sources in systems that thought to be restricted to quasi-monochromatic light. The different signals to be processed are encoded by using an Electro-optic Modulator and a birefringent plate placed between two polarizers. First, we have shown how the coherence multiplexing process can be exploited to achieve parallel real-time all optical signal addition and subtraction. Then, we have studied the impact of the crosstalk, due to the imperfection of the opto-geometrical parameters of the elements in the architecture, in the quality of the obtained results. The second part of the work consists of the validation of the technique to image signals. Thus, we have tested both image with binary and several gray levels. Also, we have confirmed that the method can be used for simple and multiplex encoding module. After that, we have evaluated the performance of the processor as a function of the continuous optical path-difference ratio in terms of Signal to Noise Ratio (SNR), Mean Square Error (MSE) and Peak to peak Signal to Noise Ratio (PSNR). Finally, we have tested the coherence multiplexing method to the encryption method based on merging together multiple-images. Therefore, we have evaluated the performance and the robustness of the method

This master’s thesis deals with the issue of digital to analog conversion and possibility of its realization in digital circuits. Goal of this project is to design sigma-delta digital to analog converter with resolution of 14 bits and frequency band (0 ÷ 20) kHz. Main functional blocks: interpolator and modulator sigma-delta will be realized like digital structures. Reconstruction filter will be realized like an analog structure. For design a check of parameters of designed converter programs MATLAB and Simulink are used. Designed digital structures will be described by VHDL language.

In this doctoral dissertation, we studied and solved several questions regarding positional and abstract numeration systems. Each particular problem is the focus of a chapter. The first problem concerns the study of the preservation of recognizability under multiplication by a constant in abstract numeration systems built on polynomial regular languages. We obtained several results generalizing those from P. Lecomte and M. Rigo. The second problem we considered is a decidability problem, which was already studied, most notably, by J. Honkala and A. Muchnik. For our part, we studied this problem for two new cases: the linear positional numeration systems and the abstract numeration systems. Next, we focused on the extension to the multidimensional setting of a result of A. Maes and M.~Rigo regarding S-automatic infinite words. We obtained a characterization of multidimensional S-automatic words in terms of multidimensional (non-necessarily uniform) morphisms. This result can be viewed as the analogous of O. Salon's extension of a theorem of A. Cobham. Finally, generalizing results of P. Lecomte and M. Rigo, we proposed a formalism to represent real numbers in the general framework of abstract numeration systems built on languages that are not necessarily regular. This formalism encompasses in particular the rational base numeration systems, which have been recently introduced by S. Akiyama, Ch. Frougny, and J. Sakarovitch. Finally, we ended with a list of open questions in the continuation of this work./Dans cette dissertation, nous étudions et résolvons plusieurs questions autour des systèmes de numération abstraits. Chaque problème étudié fait l'objet d'un chapitre. Le premier concerne l'étude de la conservation de la reconnaissabilité par la multiplication par une constante dans des systèmes de numération abstraits construits sur des langages réguliers polynomiaux. Nous avons obtenus plusieurs résultats intéressants généralisant ceux de P. Lecomte et M. Rigo. Le deuxième problème auquel je me suis intéressée est un problème de décidabilité déjà étudié notamment par J. Honkala et A. Muchnik et ici décliné en deux nouvelles versions : les systèmes de numération de position linéaires et les systèmes de numération abstraits. Ensuite, nous nous penchons sur l'extension au cas multidimensionnel d'un résultat d'A. Maes et de M. Rigo à propos des mots infinis S-automatiques. Nous avons obtenu une caractérisation des mots S-automatiques multidimensionnels en termes de morphismes multidimensionnels (non nécessairement uniformes). Ce résultat peut être vu comme un analogue de l'extension obtenue par O. Salon d'un théorème de A. Cobham. Finalement, nous proposons un formalisme de la représentation des nombres réels dans le cadre général des systèmes de numération abstraits basés sur des langages qui ne sont pas nécessairement réguliers. Ce formalisme englobe notamment le cas des numérations en bases rationnelles introduits récemment par S. Akiyama, Ch. Frougny et J. Sakarovitch. Nous terminons par une liste de questions ouvertes dans la continuité de ce travail.