The objective of this work is to provide a support material for teachers to teach the numbering system using problem solving, games and applications. What pushed us to choose this theme was the fact that we observed in most textbooks a general approach and few attractions to the extent that, on its own, motivate students to learn such content because they often finish teaching fundamental without knowing what non-decimal systems are and how the mathematical operations are done in those systems, as well as the relationships of these systems with the decimal system. Thinking about this, we decided to explore this theme using, in addition to the textbook, also the playful one because we believe that working with the methodology of didactic games and problem solving will obtain a more meaningful learning. We hope to contribute to the teaching and learning processes of this content to the students of the sixth year. / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O objetivo desse trabalho é fornecer um material de apoio aos professores para ensinar o sistema de numeração utilizando resolução de problemas, jogos e aplicações. O que nos impeliu à escolha desse tema foi o fato de termos observado, na maioria dos livros didáticos, uma abordagem geral e de poucos atrativos a ponto de por si só motivar os alunos a aprenderem tal conteúdo, pois muitas vezes estes terminam o ensino fundamental sem saber o que são os sistemas não decimais e como se fazem as operações matemáticas nesses sistemas, bem como as relações desses sistemas com o sistema decimal. Pensando nisso, resolvemos explorar este tema utilizando, além do livro didático, também o lúdico por acreditarmos que trabalhando com a metodologia dos jogos didáticos e resolução de problemas obteremos uma aprendizagem mais significativa. Esperamos com isso, contribuir nos processos de ensino e de aprendizagem desse conteúdo aos alunos dos sextos anos.

<p>U disertaciji je predstavljen postupak GSSKN (generalizovani postupak sumiranja<br />struja i korekcija napona) integrisan u DMS. Integracija postupka GSSKN u DMS se<br />sastoji od sledećeg: 1) postupak GSSKN se naslanja na jedinstvenu bazu podataka DMS i<br />2) rezultati dobijeni proračunom tokova snaga korišćenjem postupka GSSKN koriste<br />se u velikom broju ostalih funkcija DMS. Standardni postupak BFS (čišćenje<br />unaprjed/unazad), koji je razvijen za proračun tokva snaga uravnoteženih<br />distributivnih mreža u simetričnim režimima, u ovoj disertaciji je generalizovan na<br />uravnotežene mreže u nesimetričnim režimima, neuravnotežene distributivne mreže<br />(u nesimetričnim režimima), sa obuhvaćenim miksovanim mrežama, a da su pri tome u<br />potpunosti zapaženi robustnost, efikasnost i brzina u odnosu na ostale postupke.<br />Odnosno, u ovoj disertaciji je predložen GSSKN za proračun bilo kako složenih<br />slaboupetljanih distributivnih mreža s distributivnim generatorima, koji ima sve<br />osobine i prednosti u odnosu na ostale postupke, kao što su ih imali standardni<br />postupci BFS za proračun tokova snaga uravnoteženih slaboupetljanih<br />distributivnih mreža sa distributivnim generatorima u simetričnim režimima.<br />Dakle, osnovni cilj ove disertacije glasi: pokazati da standardne postupke BFS ne<br />samo da ne treba napuštati prilikom proračuna tokova snaga neuravnoteženih<br />slaboupetljanih aktivnih mreža (uključene miksovane), već da je to napuštanje<br />kontraproduktivno, bar sa aspekta postupaka koji su do sada objavljeni u literaturi.</p> / <p>The dissertation presents GBFS (General Backward/Forward Sweeping) procedure<br />which is integrated in DMS. The integration of the GBFS procedure in DMS<br />consists of the following: 1) the GBFS procedure is based on the unique DMS<br />database and 2) the results obtained from load flow calculation by application of<br />GBFS procedure are used in many other DMS applications. The standard BFS<br />procedure (Backward/Forward Sweeping), developed for the load flow calculation<br />of balanced distribution networks in symmetrical states, in this dissertation is<br />generalized to balanced networks in asymmetrical states, unbalanced distribution<br />networks (in asymmetrical states), including mixed networks. In addition, the<br />robustness, efficiency and speed comparing to other procedures, have been saved.<br />In other words, this dissertation proposes the GBFS procedure for calculation of<br />any composite weakly meshed distribution networks with distribution generators. It<br />has all the properties and advantages as compared to other procedures, like<br />standard BFS procedures applied for load flow calculation of balanced weakly<br />meshed distribution networks with distribution generators in symmetrical states.<br />Therefore, the basic objective of this dissertation is as follows: to show that the<br />standard BFS procedures should not be abandoned when performing the load flow<br />calculation of unbalanced weakly meshed active networks (mixed included), and<br />that, on the contrary, abandoning them would be counterproductive, at least from<br />the aspect of procedures published so far.</p>

Made available in DSpace on 2014-12-17T15:04:46Z (GMT). No. of bitstreams: 1 AnildaPSG.pdf: 471746 bytes, checksum: ca6e9ed04b0deccfe6bb0e02877d63fe (MD5) Previous issue date: 2006-05-19 / The central question of the present study is to identify the epistemological knowledge that the teachers-trainees possess regarding the characteristics (properties) of the decimal numbering system; its purpose is to offer a contribution to the pedagogic practice of the teachers who work within the Basic Literacy Cycle, in terms of what concerns both the acquisition of contents and the development of the knowledge that helps them in the elaboration of adequate strategies to working with the Decimal Numbering System in the classroom. The study is based on the constructivist sociointeractionist approach to teaching Mathematics and it constitutes, in itself, a methodological intervention with the teachers-trainees engaged in the Professional Qualification Program in Basic Education of the Federal University of Rio Grande do Norte. The foundations of the study were found in investigations of researchers who had carried out studies on the construction of numerical writing, showing, for instance, that the construction process of ideas and procedures involved in groupings and changes to base 10 take a lot longer to be accomplished than one can imagine. A set of activities was then elaborated which could not only contribute to the acquisition of contents but that could also make the teachers-trainees reflect upon their teaching practices in the classroom so that in this way they will be able to elaborate more consistent didactic approaches, taking into consideration the previous knowledge of the students and also some obstacles that often appear along the way. Even when teachers have access to the most appropriate dicactic resources, the lack of knowledge of the content and of the real meaning of that content make the Decimal Numbering System, a subject of fundamental importance, be taught most times in a mechanical way. The analisys of the discussions and behaviours of the teachers-trainees during the activities reavealed that they made them reflect upon their current practices in the classroom and that, as a whole, the aims of each of the activities carried out with the teachers-trainers were reached / O presente estudo tem como quest?o central identificar o conhecimento epistemol?gico que os professores-alunos possuem a respeito das caracter?sticas (propriedades) do sistema de numera??o decimal, e tem como finalidade oferecer uma contribui??o para a pr?tica pedag?gica dos professores que atuam nas turmas de Ciclo B?sico de Alfabetiza??o, tanto no que se refere ? aquisi??o de conte?dos quanto ao aprimoramento de conhecimentos que os auxiliem na elabora??o de estrat?gias adequadas para o trabalho com o Sistema de Numera??o Decimal em sala de aula. O estudo est? baseado na proposta construtivista sociointeracionista para o ensino da Matem?tica e se constitui numa interven??o metodol?gica, com os professores-alunos ligados ao Programa de Qualifica??o Profissional para a Educa??o B?sica da Universidade Federal do Rio Grande do Norte. Buscaram-se fontes de sustenta??o em investiga??es de pesquisadores que realizaram estudos sobre a constru??o das escritas num?ricas, mostrando, por exemplo, que o processo de constru??o das id?ias e procedimentos envolvidos nos agrupamentos e trocas na base 10 leva muito mais tempo para ser realizado do que se pode imaginar. Foi elaborado um conjunto de atividades as quais pudessem n?o s? contribuir para a aquisi??o de conte?dos, mas fizessem os professores-alunos refletir sobre suas pr?ticas em sala de aula, para que, assim, eles pudessem elaborar propostas did?ticas mais consistentes, levando em conta os conhecimentos pr?vios dos alunos e alguns obst?culos que se interp?em nessa trajet?ria. Mesmo o professor dispondo dos mais apropriados recursos did?ticos, a falta de dom?nio do conte?do, do real significado desse conte?do, faz com que o Sistema de Numera??o Decimal, tema de fundamental import?ncia, seja ensinado, na maioria das vezes, de forma mec?nica. A an?lise das discuss?es e o comportamento dos professores-alunos durante a realiza??o das atividades revelaram que estas provocaram reflex?es sobre as pr?ticas de sala de aula como tamb?m que, de modo geral, foram atingidos os objetivos propostos em cada atividade realizada com os professores-alunos

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Esta pesquisa analisa os saberes docentes e os conhecimentos discentes do 3 ano do Ensino Fundamental sobre o sistema de numeraÃÃo decimal â SND. A HistÃria do SND, de acordo com Ifrah (2005) e Eves (2011), permite conhecer o desenvolvimento das suas caracterÃsticas â as bases, os algarismos, a criaÃÃo do zero â o que favorece uma EducaÃÃo MatemÃtica problematizadora. Os objetivos dessa pesquisa, que à um estudo de caso, sÃo: i) identificar os conhecimentos de estudantes do 3 ano do Ensino Fundamental na escrita de nÃmeros, com 2 e 3 ordens, e os saberes docentes mobilizados na interpretaÃÃo de tais registros; ii) conhecer registros de representaÃÃo de estudantes do 3 ano do Ensino Fundamental na escrita de nÃmeros, com 2 e 3 ordens; e iii) investigar como a professora analisa as escritas discentes de nÃmeros, com 2 e 3 ordens, em diferentes registros de representaÃÃo. Participaram da pesquisa 24 estudantes do 3 ano do Ensino Fundamental e uma professora de uma escola pÃblica do municÃpio de Maranguape, regiÃo Metropolitana de Fortaleza. Os saberes discentes foram avaliados nos seguintes aspectos: ComparaÃÃo de numerais com quantidade diferente de algarismos; ComparaÃÃo de numerais com a mesma quantidade de algarismos; Do numeral verbal falado para o numeral arÃbico (escrever); Do numeral verbal falado para o numeral arÃbico (escolher uma opÃÃo); Do numeral arÃbico para numeral verbal escrito (por extenso); Do numeral escrito (por extenso) para o numeral arÃbico. As questÃes foram organizadas em itens que continham numerais com 2, 3 e 4 algarismos. ApÃs a aplicaÃÃo do teste, realizou-se com a professora regente uma entrevista estruturada dividida em 3 momentos: o primeiro, relacionado aos seus saberes do conhecimento, pedagÃgicos e existenciais sobre o SND; o segundo, com perguntas com o objetivo de compreender como esta analisa as produÃÃes dos seus estudantes; e o terceiro, abordando as reflexÃes da professora sobre a pesquisa realizada. Os resultados com os estudantes revelaram a necessidade do trabalho com as diversas representaÃÃes do SND e o fato que mais da metade dos estudantes jà possui algum conceito sobre a quarta ordem do SND, mesmo sem esse conteÃdo constar do currÃculo referente ao seu ano e nÃo ter sido estudado, ratificando outros estudos os quais afirmam que os estudantes estÃo na escola com aprendizagens que esta nÃo os proporcionou. O currÃculo, portanto, precisa ser revisto, pois o engessamento de alguns conteÃdos a determinado momento restringe a aprendizagem dos estudantes. Os resultados com a professora evidenciam uma prÃtica que tem no livro didÃtico seu principal recurso metodolÃgico e desconhecimento das caracterÃsticas do SND. Ratifica-se, dessa forma, a necessidade de uma formaÃÃo docente dessa etapa da escolarizaÃÃo que englobe todos os saberes do conhecimento. Espero que este estudo contribua para novas pesquisas, favorecendo o desenvolvimento de uma EducaÃÃo MatemÃtica que as crianÃas merecem para uma vida mais plena.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / O Sistema de NumeraÃÃo Decimal â SND à um conhecimento matemÃtico essencial no inÃcio da EducaÃÃo BÃsica, o qual requer do estudante a habilidade de ler e escrever numerais utilizando variados tipos de registros â pictÃrico, com objetos, com algarismos, com letras â conforme explica a Teoria dos Registros de RepresentaÃÃo SemiÃtica. Hà de se destacar, ainda, a importÃncia da oralidade â escuta e fala â em todos os processos de aprendizagem do sujeito. Esta pesquisa, de natureza qualitativa, realizada em uma escola pÃblica do municÃpio de Caucaia, objetivou: i) identificar a relaÃÃo entre a diversidade de registros numÃricos de crianÃas do 2 ano do Ensino Fundamental; ii) conhecer os registros de representaÃÃo de estudantes do 2 ano do Ensino Fundamental na escrita de nÃmeros com 2, 3 e 4 ordens; e iii) interpretar os conhecimentos discentes com registros numÃricos de 2, 3 e 4 ordens. Foram aplicados dois instrumentos: o Instrumento 1 tinha numerais com 2 e 3 ordens e o Instrumento 2 tinha numerais com 4 ordens. Participaram da pesquisa 14 estudantes do 2 ano do Ensino Fundamental, que tiveram seus conhecimentos analisados mediante diferentes tipologias, elencadas por Barguil (2015), e à luz da teoria da TranscodificaÃÃo NumÃrica (OROZCO, 2005), que classifica os erros como sintÃtico ou lÃxico. As questÃes versavam sobre a comparaÃÃo de numerais e o registro numÃrico, com algarismos e letras, observando a relaÃÃo desse com a oralidade. A anÃlise dos dados foi organizada em duas etapas: i) desempenho dos estudantes em cada questÃo; e ii) desempenho global de cada estudante em todas as questÃes. Os resultados revelaram uma expressiva quantidade de erros sintÃticos, que se referenciam na expressÃo verbal, bem como indicam a oralidade como o principal ponto de partida para a construÃÃo correta destes registros, sinalizando a necessidade de ampliar o uso de diversidades de registros para a aprendizagem do SND. / The Decimal Numbering System â DNS is an essential mathematical knowledge at the beginning of Basic Education, which student requires the ability to read and write numerals using various types of records â pictorial, with objects, with digits, with letters â as explained by theory of Semiotics Representation Registers. It must be noted also the importance of oral communication â listening and speaking â in all subject learning processes. This research, qualitative, held in a public school in the municipality of Caucaia, aimed to: i) identify the relationship between the diversity of numerical records of children in the 2nd year of elementary school; ii) know the students representative records of the 2nd year of elementary school writing numbers with 2, 3 and 4 orders; and iii) interpret the students with knowledge of numerical records 2, 3 and 4 orders. Two instruments were applied: the Instrument 1 had numerals 2 and 3 orders and the Instrument 2 had numerals with four orders. The participants were 14 students of 2nd year of elementary school, who had their knowledge analyzed by different types, listed by Barguil (2015), and in the light of the theory of Numerical Transcoding (OROZCO, 2005), which classifies errors as syntactic or lexicon. The questions were about the comparison of numerals and the record numbers, with digits and letters, noting the relationship of this with orality. Data analysis was organized in two stages: i) performance of students in each issue; and ii) the overall performance of each student in all matters. The results revealed a significant amount of syntactical errors, which are referred to in verbal expression, as well as indicate the orality as the main starting point for the correct construction of these records, which indicate the need to expand the use of records of diversity for learning DNS.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2017-10-10T13:16:18Z No. of bitstreams: 2 Dissertação - Rosélia José da Silva Carvalho - 2017.pdf: 5608804 bytes, checksum: d5bbc8da96b6580c4599b9f243eb64be (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-10-10T13:16:40Z (GMT) No. of bitstreams: 2 Dissertação - Rosélia José da Silva Carvalho - 2017.pdf: 5608804 bytes, checksum: d5bbc8da96b6580c4599b9f243eb64be (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-10-10T13:16:40Z (GMT). No. of bitstreams: 2 Dissertação - Rosélia José da Silva Carvalho - 2017.pdf: 5608804 bytes, checksum: d5bbc8da96b6580c4599b9f243eb64be (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-09-06 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / The present research discusses the relationships between the mathematics teaching’s way of organization for the early years of elementary school through the Learning Trigger Situations (“LTSs”) and the appropriation of the Decimal Number System’s (DNS) conceptual nexus by participating students of Mathematics Club (CluMat). The objective is “to investigate the clues of appropriation of DNS’ conceptual nexus by students in the 4th year of Elementary School of a Municipal Public School in Goiânia/GO (Brazil), by developing “LTSs” in CluMat." The assumptions of Historical-Cultural Theory, Activity Theory, Developmental Teaching and Teaching Guiding Activity are adopted as theoretical basis, believing they guide the education process for the students’ humanization and development, and, in this context, may help to understand the teaching’s organization processes as a way of overcoming the "encapsulation of school learning". In order to understand the study’s object, a didactic experiment was organized in which Learning Trigger Situations (“LTSs”) of the SND module in CluMat were implemented in a public school in Goiânia, with twelve students. The oral and written reports of participating students in the didactic experiment subsidized the search for answers to the research question: "What are the indications that the mathematics teaching's organization through “LTSs” contributed to the appropriation of DNS’ conceptual nexus by participating students in CluMat "? To unveil the phenomenon, the analysis was performed based on the concept of ‘units’ proposed by Vigotski (2001). In this context, three units were used: 1) the collective actions and reflections in the learning space; 2) the playfulness as a characteristic in the organization of learning trigger situations and 3) the evidence of SND’s knowledge appropriation through “LTSs”. The analysis’ movement made it possible to understand that the elements 'sharing and playfulness', present in the “LTSs” organization mode of SND module, mobilized students to the study, discussion, reflection and to doubts and knowledge sharing and to the collective synthesis of ideas. This process also encouraged students to develop, in cooperation with each other, strategies, operations and actions in the search for problem solving, which contributed to a qualitative change in their actions. In general, it’s possible to infer that “LTSs” organization and dynamization contributed to CluMat students achieving a new level of development. / A presente pesquisa discute as relações entre o modo de organização do ensino de matemática para os anos iniciais do Ensino Fundamental por meio de Situações Desencadeadoras de Aprendizagem (SDAs) e a apropriação dos nexos conceituais do Sistema de Numeração Decimal (SND) por estudantes participantes do Clube de Matemática (CluMat). O objetivo é “investigar os indícios de apropriação de nexos conceituais do SND por estudantes do 4º ano do Ensino Fundamental de uma escola pública municipal de Goiânia, com o desenvolvimento de SDAs no CluMat”. Adotam-se como base teórica os pressupostos da Teoria Histórico-Cultural, da Teoria da Atividade, do Ensino Desenvolvimental e da Atividade Orientadora de Ensino, por acreditar-se que são orientadores de um processo de educação para a humanização e o desenvolvimento dos sujeitos e, nesse contexto, podem auxiliar na compreensão dos processos de organização de ensino como meio de superação do “encapsulamento da aprendizagem escolar”. Para apreender o objeto de estudo, organizou-se um experimento didático no qual foram implementadas as SDAs do módulo SND no CluMat em uma escola pública municipal de Goiânia, com doze estudantes. Os relatos orais e escritos dos estudantes participantes do experimento didático subsidiaram a busca por respostas à pergunta de pesquisa, qual seja: “Quais os indícios de que a organização do ensino de matemática por meio de SDAs contribuíram para a apropriação de nexos conceituais do SND por estudantes participantes do CluMat”? Para desvelar o fenômeno, realizou-se a análise com base no conceito de unidades proposto por Vigotski (2001). Deste modo, fez-se uso de três unidades, a saber: 1) as ações e reflexões coletivas no espaço de aprendizagem; 2) a ludicidade como característica na organização das situações desencadeadoras de aprendizagem e 3) os indícios de apropriação do conhecimento acerca do SND por meio de SDAs. O movimento de análise permitiu compreender que os elementos ‘compartilhamento e ludicidade’, presentes no modo de organização das SDAs do módulo SND, mobilizaram os estudantes ao estudo, à discussão, à reflexão, ao compartilhamento de dúvidas e conhecimentos e à síntese coletiva das ideias. Esse processo ainda incentivou os estudantes a desenvolverem, em cooperação com seus pares, estratégias, operações e ações na busca pela resolução dos problemas, o que contribuiu para uma mudança qualitativa em suas ações. De modo geral, inferimos que a forma de organização e a dinamização das SDAs contribuíram para que os estudantes participantes do CluMat atingissem um novo nível de desenvolvimento.

The thesis deals with the numbering plan, PBX, tariffs, draft disable tariff system and its possible extensions. Within numbering plans work deals with shapes of national and international figures, describes different types of these numbers, description of the structure and definition of fields, including the length of these fields. Within countries and regions of the Czech Republic are given individual prefix for both fixed and mobile networks. There are mentioned special shapes that are different in length compared to standard types. All these types of shapes are listed according to ITU-T. Within the exchanges are given generation switches, PBXs use within the telephone network and a description of the selected functions. Furthermore, there are kinds of pricing and options of valuation call. The work contains a description of automated acconting system, according to the principle of work of the program and a description of the individual steps and functions that thanks to the cooperation process individually each calls, convert numbers to international form, assigns the call tariff and appreciate the call according to the tariff. Finally, there are described possible extensions of this automated acconting system.

Made available in DSpace on 2016-04-27T16:57:53Z (GMT). No. of bitstreams: 1 dissertacao_marinete_fontoura_garcia.pdf: 592464 bytes, checksum: c714eb2a28776b300b99c76808bc6186 (MD5) Previous issue date: 2006-09-21 / This work treats of the formation in service of teachers of Infantile Education and yours know in relation to the process of numeric construction. Organized starting from reflections and observations produced through discussions in a group of studies with professionals of a same school unit, did it stop in the following subject: How are the teachers that participate in a cooperative group, going building new you know and new forms of to understand and to work with the numeric construction, developing a posture of investigating teacher?. The qualitative approach of research was used, initially with the application of individual interviews, following by the formation of a group of studies, that checked to the work a cooperation nature between the participant teachers and the researcher; this felt through studies, choice, application and reflection of teaching activities Leaning in researchers' investigations that studied the numeric construction, it tries to evidence, through the reports presented in the interviews and in the discussions of the cooperative group (involving teachers of Infantile Education that they teach in the municipal net of a municipal district of Great São Paulo), as they feels the pedagogic practice in relation to the numeric construction and in that measured that practice can change through the teachers' participation in a cooperative group. In this group, the teachers discussed texts, they investigated their own students, they promoted discussions and they analyzed the results of the investigations to the light of the theoretical referential. Results of researches and studies already accomplished, about the numeric construction, they are little published and, through the interviews, it was verified that the teachers, doesn't tend access to such studies, they don't consider the children's hypotheses concerning the numbers. Like this, the work has as purpose to contribute with the teachers' pedagogic practice, presenting and developing relative subjects to the axis Mathematics in Infantile Education / Este trabalho trata da formação em serviço de Professoras de Educação Infantil e seus saberes em relação ao processo de construção numérica. Organizado a partir de reflexões e observações produzidas por meio de discussões em um grupo de estudos com profissionais de uma mesma unidade escolar, deteve-se na seguinte questão: Como as professoras, que participam de um grupo cooperativo, vão construindo novos saberes e novas formas de compreender e lidar com a construção numérica, desenvolvendo uma postura de professora investigadora? . Foi utilizada a abordagem qualitativa de pesquisa, inicialmente com a aplicação de entrevistas individuais, seguida da formação de um grupo de estudos, que conferiu ao trabalho uma natureza de cooperação entre as professoras participantes e a pesquisadora; isto se deu por meio de estudos, escolha, aplicação e reflexão de atividades de ensino Apoiado em investigações de pesquisadores que estudaram a construção numérica, procura evidenciar, por meio dos relatos apresentados nas entrevistas e nas discussões do grupo cooperativo (envolvendo professoras de Educação Infantil que lecionam na rede municipal de um município da Grande São Paulo), como se dá a prática pedagógica em relação à construção numérica e em que medida essa prática pode mudar por meio da participação das professoras em um grupo cooperativo. Neste grupo, as professoras discutiram textos, investigaram seus próprios alunos, promoveram discussões e analisaram os resultados das investigações à luz dos referenciais teóricos. Resultados de pesquisas e estudos já realizados, sobre a construção numérica, são pouco divulgados e, por meio das entrevistas, verificou-se que as professoras, não tendo acesso a tais estudos, não consideram as hipóteses das crianças acerca dos números. Assim, o trabalho tem como finalidade contribuir com a prática pedagógica das professoras, apresentando e desenvolvendo questões relativas ao eixo Matemática em Educação Infantil

Η διαμόρφωση με πολύπλεξη συχνότητας ορθογωνίων φερουσών (Orthogonal Frequency Division Multiplexing - OFDM) έχει εδραιωθεί ως μία από τις επικρατέστερες μεθόδους διαμόρφωσης για την υψηλού ρυθμού μετάδοση πληροφορίας μέσω ασύρματων μέσων. Σε ένα σύστημα OFDM, ένα από τα βασικότερα και υπολογιστικά πολυπλοκότερα τμήματα είναι ο υπολογισμούς του Ταχύ Μετασχηματισμού Fourier. Αντικείμενο της εργασίας αυτής είναι η μελέτη της χρήσης εναλλακτικής αριθμητικής για την υλοποίηση κυκλωμάτων FFT. Τυπικά, τέτοιου είδους κυκλώματα υλοποιούνται χρησιμοποιώντας κάποια γραμμική αναπαράσταση σταθερής υποδιαστολής. Στη βιβλιογραφία έχουν προταθεί υλοποιήσεις του FFT με χρήση του Λογαριθμικού Συστήματος Αρίθμησης (Logarithmic Numbering System – LNS) και έχουν αναφερθεί κέρδη για συγκεκριμένους παράγοντες όπως το σφάλμα κβαντισμού, η επιφάνεια ολοκλήρωσης και η κατανάλωση ισχύος. Η αποδοτικότητα αυτών των λύσεων ερευνάται για τη συγκεκριμένη περίπτωση της εφαρμογής του FFT σε OFDM modems. Εστιάζοντας στην περίπτωση του FFT 64 σημείων για OFDM modem για ασύρματα δίκτυα 802.11a, μία από τις πλέον επιτυχημένες αρχιτεκτονικές που έχουν προταθεί για την υλοποίηση του, στηρίζεται στη λογική του FFT γραμμής – στήλης και παρουσιάζει έναν τρόπο πραγματοποίησης του υπολογισμού χωρίς κανένα ψηφιακό πολλαπλασιαστή. Με το βασικό πλεονέκτημα της λογαριθμικής αναπαράστασης να είναι η απλοποίηση των κυκλωμάτων πολλαπλασιασμού (με ταυτόχρονη όμως αύξηση του κόστους για την πραγματοποίηση προσθέσεων), δείχνεται ότι τελικά η υλοποίηση ενός FFT αμιγώς σε LNS δεν είναι προτιμητέα. Αν και η αρχιτεκτονική του FFT γραμμής – στήλης μπορεί να προσφέρει υψηλή απόδοση με χαμηλό κόστος υλοποίησης, παρουσιάζει μια σειρά από αδυναμίες, που σχετίζονται κυρίως με τη χρήση ειδικών κυκλωμάτων για την εκτέλεση των πολλαπλασιασμών με τις σταθερές που εμφανίζονται στον FFT (twiddle factors). Για την αντιμετώπιση αυτών των περιορισμών προτείνεται η εισαγωγή του LNS σε κάποια τμήματα του κυκλώματος του FFT, οδηγώντας έτσι στη δημιουργία ενός συστήματος μικτής αναπαράστασης. Σε τέτοιου είδους υβριδικά συστήματα τίθενται δύο βασικά ζητήματα. Το πρώτο αφορά τον ορισμό της ισοδυναμίας μεταξύ των διαφορετικών αναπαραστάσεων και το δεύτερο τον αποδοτικό τρόπο υλοποίησης των κυκλωμάτων μετατροπής από το ένα αριθμητικό σύστημα στο άλλο. Τυπικά, τα κριτήρια ισοδυναμίας που επιλέγονται είναι αυστηρά μαθηματικά ορισμένα, όπως για παράδειγμα ο Λόγος Σήματος προς Θόρυβο (Signal-to-Noise Ratio - SNR) ή το Μέσο Σχετικό Σφάλμα Αναπαράστασης (Average Relative Representation Error – ARRE). Στη συγκεκριμένη εργασία ακολουθείται μια λιγότερο δεσμευτική προσέγγιση, ορίζοντας την ισοδυναμία δύο αναπαραστάσεων με βάση την τελική απόδοση του συστήματος OFDM όσον αφορά το ρυθμό λαθών στο δέκτη (Bit Error Rate - BER). Με βάση αυτή τη λογική, αποδεικνύεται ότι μπορούν να χρησιμοποιηθούν αναπαραστάσεις πολύ μικρού μεγέθους λέξης και οι προσεγγίσεις που χρειάζεται να γίνουν κατά τις μετατροπές μεταξύ των δύο συστημάτων δεν είναι ανάγκη να είναι ιδιαίτερα ακριβείς. Έτσι, τα σχετικά κυκλώματα μπορούν να υλοποιηθούν αποδοτικά και με μικρό κόστος. Η υλοποίηση δύο συστημάτων για τον FFT 64 σημείων, ένα βασισμένο αποκλειστικά σε γραμμική αναπαράσταση σταθερής υποδιαστολής και ένα υβριδικό που χρησιμοποιεί γραμμική και λογαριθμική αναπαράσταση, δείχνει ότι χωρίς διαφορές όσον αφορά το BER και την καθυστέρηση (delay), η υβριδική προσέγγιση απαιτεί μικρότερη επιφάνεια ολοκλήρωσης και παρουσιάζει σημαντικά χαμηλότερη κατανάλωση ισχύος. / Orthogonal Frequency Division Multiplexing (OFDM) has been established as one of the most prevalent methods for high data rate transmission through wireless channels. In an OFDM communication system, one of the fundamental and most computationally intensive parts is the computation of the Fast Fourier Transform (FFT). The subject of this thesis is to investigate the use of alternative arithmetic representation systems for the implementation of FFT circuits. Typically, these circuits are implemented using linear fixed-point representations. In literature, implementations of the FFT using the Logarithmic Numbering System (LNS) have been proposed and significant gains in quantization errors, chip area and power consumption have been reported. The effectiveness of these proposals in the case of the FFT for OFDM systems is investigated. Focusing on the case of the 64-point FFT for an OFDM modem for an 802.11a wireless network, one of the most efficient architectures proposed is based on the concept of row-column FFT and presents a way of implementing the computation without using any digital (non-fixed input) multiplier. The most important feature of the LNS representation is the fact that multiplication operations turn to mere additions, thus there are significant implementation gains. On the downside though, addition in LNS is very expensive. Combining the aforementioned, it is shown that the implementation of the whole FFT computation in LNS is not a preferable solution. Although the row-column FFT architecture may offer high performance and low implementation cost, it presents a number of deficiencies mainly due to the fact that special purpose circuits are used to perform the multiplications with the complex constants (twiddle factors) that appear in the computation. In order to alleviate these deficiencies, it is proposed to use the LNS representation in some parts of the FFT circuit, thus forming a hybrid-representation system. In hybrid-representation systems two major issues are raised. The first one is how to define equivalence between the arithmetic representation systems used and the second one is related to the cost of the circuits required to perform the conversions between the numbers of the different arithmetic systems. Typically, the equivalence criterion used is mathematically defined and metrics like the Signal-to-Noise Ratio (SNR) or Average Relative Representation Error (ARRE) are commonly used. In this report, a less restrictive metric is used: two arithmetic representations are defined to be equal if the Bit Error Rate (BER) performance of the overall OFDM system is equal. Using this approach, it is shown that short word-length representations may be used and the conversions between the linear and logarithmic systems need not be very accurate. This results in great simplification of the conversion process and the respected circuits can be implemented with low cost. For comparison, two 64-point FFT systems have been implemented, one using a linear fixed-point 2’s complement representation and one using both linear and LNS representation. Without any differences in BER performance and circuit delay, the hybrid-representation system requires less chip area and consumes significantly lower power.

Η άθροιση μεγάλων αριθμών είναι μια χρονοβόρα και ενεργοβόρα διαδικασία. Πολλές μέθοδοι έχουν αναπτυχθεί για να μειωθεί η καθυστέρηση υπολογισμού του αθροίσματος λόγω της μετάδοσης κρατουμένου. Τέτοιες είναι η πρόβλεψη κρατουμένου (carry look ahead) και η επιλογή κρατουμένου (carry select). Αυτές οι αρχιτεκτονικές δεν είναι επαρκώς επεκτάσιμες για μεγάλους αριθμούς (με πολλά bits) ή πολλούς αριθμούς, διότι παράγονται μεγάλα και ενεργοβόρα κυκλώματα. Στην παρούσα εργασία μελετάται η μέθοδος υπολοίπου (RNS), η οποία χρησιμοποιεί συστήματα αριθμών μεγαλύτερα από το δυαδικό. Ορίζεται μια βάση τριών αριθμών και οι αριθμοί αναπαρίστανται στα εκάστοτε τρία συστήματα της βάσης. Η άθροιση γίνεται παράλληλα σε κάθε σύστημα και τέλος οι αριθμοί μετατρέπονται πάλι στο δυαδικό. Τα πλεονεκτήματα αυτής της προσέγγισης είναι η παραλληλία και η απουσία μεγάλων κυκλωμάτων διάδοσης κρατουμένου. Το μειονέκτημα είναι ότι χρειάζονται κυκλώματα μετατροπής από και προς το δυαδικό σύστημα. Αυτού του είδους οι αθροιστές συγκρίνονται για κατανάλωση ενέργειας με τους γνωστούς carry look ahead και carry select. Διαπιστώθηκε ότι οι RNS αθροιστές καταναλώνουν λιγότερη ενέργεια. / The addition of many-bits numbers is a time and power consuming task. Many methods are developed to reduce the sum calculation delay due to carry propagation. Such techniques are Carry Look Ahead and Carry Select, Those techniques are not scalable to many bits numbers or a set of many numbers: the circuits needed are big and power consuming. In this thesis, the the RNS technique is investigated. This technique uses radix bigger than binary. A 3-numbers base is defined and the numbers that participate in the sum are represented uniquely in each element radix. The addition is performed in parallel in each radix. Finally the result is transformed back to the binary numbers system. The advantages of this technique are the parallelization of the process and the lack of carry propagation circuits. The disadvantage is that transformation circuits are need from/to binary system. The RNS adders are compared to CLA and CS for power. Such adders are compared to CLA and CS for power consumption. It is found that RNS adders consume less energy.