In this study the teaching of mathematics has been developed in relation to rational numbers and towards a learning activity. At the same time topic-specific mediated tools have been studied. The iterative model for learning study has been used as research approach. The purpose of the study was to explore what in an algebraic learning activity enables knowledge of rational numbers to develop. The specific questions answered by the study are how an algebraic learning activity can be formed in an otherwise arithmetic teaching tradition, what knowledge is mediated in relation to different mediated tools and what in these tools that enable this knowledge. The result of the study shows how an algebraic learning activity can be developed to support the students to understand rational numbers even in an arithmetic teaching tradition. The important details that developed the algebraic learning activity were to identify the problem to create learning tasks and the opportunity for the students to reflect that are characteristic of a learning activity. The result also shows that the mediating tools, the algebraic symbols and the general model for fractional numbers, have had significant importance for the students' possibilities to explore rational numbers. The conditions for the algebraic symbols seem to be the possibilities for these symbols to include clues to the meaning of the symbol and that the same symbol can be used in relation to several of other mediated tools. The conditions in the general model consisted of that the integer numbers and the rational numbers in the model could be distinguished and that the students could reflect on the meaning of the different parts. The general model consists of the algebraic symbols, developed in the learning activity. The algebraic symbols make the structure of the numbers visible and the general model mediates the structure of additive and multiplicative conditions that are contained in a rational number. The result of the study contributes in part to the field of mathematics education research by examining Elkonin's and Davydov's Mathematical Curriculum in a western teaching practice and in part to a development of the model of Learning study as a didactical research approach by using an activity-theoretical perspective on design and analysis.

Understanding the relationship between fractions and decimals is an important step in developing an overall understanding of rational numbers. Research has demonstrated the feasibility of technology in the form of virtual manipulatives for facilitating students’ meaningful understanding of rational number concepts. This exploratory dissertation study was conducted for the two closely related purposes: first, to investigate a sample of fifth-grade students’ reasoning regarding the relationship between fractions and decimals for fractions with terminating decimal representations while using virtual manipulative incorporating parallel number lines; second, to investigate the affordances of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship. The study employed qualitative methods in which the researcher collected and analyzed data from fifth-grade students’ verbal explanations, hand gestures, and mouse cursor motions. During the course of the study, four fifth-grade students participated in an initial clinical interview, five task-based clinical interviews while using the number line-based virtual manipulatives, and a final clinical interview. The researcher coded the data into categories that indicated the students’ synthetic models, their strategies for converting between fractions and decimals, and evidence of students’ accessing the affordances of the virtual manipulatives (e.g., students’ hand gestures, mouse cursor motions, and verbal explanations). The study yielded results regarding the students’ conceptions of the decimal-fraction relationship. The students’ synthetic models primarily showed their recognition of the relationship between the unit fraction 1/8 and its decimal 0.125. Additionally, the students used a diversity of strategies for converting between fractions and decimals. Moreover, results indicate that the pattern of strategies students used for conversions of decimals to fractions was different from the pattern of strategies students used for conversions of fractions to decimals. The study also yielded results for the affordances of the virtual manipulatives for supporting the students’ reasoning regarding the decimal-fraction relationship. The analysis of students’ hand gestures, mouse cursor motions, and verbal explanations revealed the affordances of alignment and partition of the virtual manipulatives for supporting the students’ reasoning about the decimal-fraction relationship. Additionally, the results indicate that the students drew on the affordances of alignment and partition more frequently during decimal to fraction conversions than during fraction to decimal conversions.

In this study the teaching of mathematics has been developed in relation to rational numbers and towards a learning activity. At the same time topic-specific mediated tools have been studied. The iterative model for learning study has been used as research approach. The purpose of the study was to explore what in an algebraic learning activity enables knowledge of rational numbers to develop. The specific questions answered by the study are how an algebraic learning activity can be formed in an otherwise arithmetic teaching tradition, what knowledge is mediated in relation to different mediated tools and what in these tools that enable this knowledge. The result of the study shows how an algebraic learning activity can be developed to support the students to understand rational numbers even in an arithmetic teaching tradition. The important details that developed the algebraic learning activity were to identify the problem to create learning tasks and the opportunity for the students to reflect that are characteristic of a learning activity. The result also shows that the mediating tools, the algebraic symbols and the general model for fractional numbers, have had significant importance for the students' possibilities to explore rational numbers. The conditions for the algebraic symbols seem to be the possibilities for these symbols to include clues to the meaning of the symbol and that the same symbol can be used in relation to several of other mediated tools. The conditions in the general model consisted of that the integer numbers and the rational numbers in the model could be distinguished and that the students could reflect on the meaning of the different parts. The general model consists of the algebraic symbols, developed in the learning activity. The algebraic symbols make the structure of the numbers visible and the general model mediates the structure of additive and multiplicative conditions that are contained in a rational number. The result of the study contributes in part to the field of mathematics education research by examining Elkonin's and Davydov's Mathematical Curriculum in a western teaching practice and in part to a development of the model of Learning study as a didactical research approach by using an activity-theoretical perspective on design and analysis.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2017-05-19T11:57:59Z No. of bitstreams: 1 Alan Silva dos Santos.pdf: 2565511 bytes, checksum: b1f037051a38c7fab75725edb0d9af53 (MD5) / Made available in DSpace on 2017-05-19T11:57:59Z (GMT). No. of bitstreams: 1 Alan Silva dos Santos.pdf: 2565511 bytes, checksum: b1f037051a38c7fab75725edb0d9af53 (MD5) Previous issue date: 2017-03-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This research aims to analyze, through a didactic sequence, the students' conceptions of an mathematics university course that involve characteristics and properties of rational numbers and irrational numbers, as well as the concept of density of the respective sets in the set of real numbers. The research employs both digital technologies, through GeoGebra software, and non-digital technologies. The studies carried out through a review of the literature and the proposed activities allowed to formulate/refine the problematization around which the investigative procedures developed with a group of undergraduates in Mathematics were developed and which involved questions regarding the conceptions of such subjects about the characteristics and properties relative to rational and irrational numbers, as well as concerning the concept of density of the set of rational numbers and the set of irrational numbers. The research, of a qualitative nature, used data collection instruments designed as digital models, in addition to sequences solved without the use of software, together. The collected data were analyzed using contributions from the Theory of Didactic Situations, as well as theoretical elements related to the use of technologies in Mathematics Education. The analyzes also used references related to numerical representational questions and mathematical knowledge modalities (algorithmic, formal and intuitive). The analyzes carried out suggest that the models and strategies employed were efficient in raising the predominant form of the conceptions of the subjects involved; also indicated that the use of didactic strategies conceived in the research allowed for advances in the re-signification of the mathematical knowledge put into play by the subjects / Esta pesquisa tem por objetivo analisar, por meio de uma sequência didática, as concepções dos alunos do curso de licenciatura em matemática que envolvem características e propriedades dos números racionais e dos números irracionais, bem como o conceito de densidade dos respectivos conjuntos no conjunto dos números reais. A investigação emprega tanto tecnologias digitais, por meio do software GeoGebra, como tecnologias não digitais. Os estudos efetuados por meio da revisão de literatura e das atividades propostas permitiram formular/refinar a problematização em torno da qual se desenvolveram os procedimentos investigativos, levados a efeito com um grupo de licenciandos em Matemática, e que envolvia questões relativas às concepções de tais sujeitos acerca das características e propriedades relativas aos números racionais e irracionais, bem como atinentes ao conceito de densidade do conjunto dos números racionais e do conjunto dos números irracionais. A investigação, de caráter qualitativo, utilizou instrumentos de recolha de dados concebidos como modelos digitais, além de sequências resolvidas sem o uso de software, em conjunto. Os dados coletados foram analisados empregando aportes da Teoria das Situações Didáticas, além de elementos teóricos relativos ao uso de tecnologias em Educação Matemática. As análises empregaram, também, referenciais ligados às questões representacionais numéricas e às modalidades de conhecimento matemático (algorítmico, formal e intuitivo). As análises realizadas sugerem que os modelos e estratégias empregadas foram eficientes em levantar a forma predominante das concepções dos sujeitos envolvidos; indicaram, também, que o uso das estratégias didáticas concebidas na investigação possibilitou avanços na ressignificação do conhecimento matemático posto em jogo pelos sujeitos

This qualitative study explored a mathematics workshop for parents and the impact on a parent’s mathematical content knowledge in rational numbers, perceptions of current instructional practices, and parental beliefs in supporting their children in learning mathematics. A 6-week parent workshop on rational numbers was offered in a rural middle school. Data sources included interviews and workshop audio transcriptions. This study concluded that a mathematics workshop supports parents in developing a conceptual understanding of rational numbers and rational number operations. Furthermore, parents recognized the importance of discourse, representation, and justification for building conceptual understanding in mathematics. Parents, who participated in the workshops, were more open to the use of standards based instructional practices for developing conceptual understanding. Parental engagement in mathematics should include discourse at home to help students justify and explain their thinking. Questions related to the teaching of non-standard procedures without building a conceptual understanding hindered many parents from completely accepting new instructional practices.

Fractions are an essential skill for students to master, and one students struggling in mathematics face particular difficulty with (National Mathematics Advisory Panel, 2008; Mazzocco & Devlin, 2008). This study employed the multi-probe multiple baseline design to examine the effectiveness of the concrete-representational-abstract (CRA) approach and explicit teaching practices to teach fraction equivalence to students struggling in mathematics. The study was conducted across four students, and replicated simultaneously across four more. The CRA approach included concrete aids such as fraction circles and fraction strips, representations such as pictures of fraction circles and polygons, and algorithms. Explicit teaching involved following a model-lead-test sequence and included an advanced organizer, corrective feedback and cumulative reviews. Results of this study indicated that the intervention program was effective to improve students‟ performances in fraction equivalence tasks. In particular, the use of vii representations was seen to impact performance and concrete aids alone may not be sufficient to improve performance. With regards types of representational and concrete aids employed, results of this study tended to favor the use of linear versus circular aids. Results indicated that students whose performances tend to vary may not benefit to the same extent as those who have stable profiles. Students who demonstrate variable profiles may require additional practice to master skills being taught. This study also examined transfer of skills to word problems and, results demonstrated that the CRA and explicit teaching approaches were beneficial in helping aiding transfer. Several aspects of the program may have contributed to aiding transfer including, minimal exposure to word problems during intervention, drawing connections between representations and abstract information, and incorporating the fair sharing understanding or quotient interpretation of fractions. This program concluded that students were able to maintain performances over time, and that representations in particular appeared to aid conceptual understanding and promote maintenance of skills. / text

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-30T13:15:08Z No. of bitstreams: 1 arquivototal.pdf: 799210 bytes, checksum: 8de2ace5434a5d92b8604de7573abfc4 (MD5) / Approved for entry into archive by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-30T13:17:30Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 799210 bytes, checksum: 8de2ace5434a5d92b8604de7573abfc4 (MD5) / Made available in DSpace on 2017-08-30T13:17:30Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 799210 bytes, checksum: 8de2ace5434a5d92b8604de7573abfc4 (MD5) Previous issue date: 2016-02-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The study of ontinued fra tions will start with some histori al fa ts, aiming at a better understanding of the subje t. We will bring the de nition of ontinued fra tions for a number α real, with the de nition for α rational and α irrational. The dis ussion will fo us on meaning results for the al ulation of redu ed and good approximations of irrational numbers, also aimed at determining the error between the redu ed and the irrational number. We will bring a study of the periodi ontinued fra tions, with emphasis on Lagrange theorem, whi h relates a periodi ontinued fra tion and a quadrati equation. Finishing with a fo us on problem solving, as the al ulation of ontinued fra tions of irrational numbers of the form √a2 + b, as well as proof of the irrationality of e by al ulating its ontinued. / O estudo das frações ontínuas terá ini io om alguns fatos históri os, visando uma melhor ompreensão do tema. Traremos a de nição de frações ontínuas para um erto número α real, apresentando a de nição para α ra ional e para α irra ional. A dis ussão será entrada em resultados importantes para o ál ulo de reduzidas e boas aproximações de números irra ionais, visando também a determinação do erro entre a reduzida e o número irra ional. Traremos um estudo sobre as frações ontínuas periódi as, om enfase ao teorema de Langrange, que rela iona uma fração ontínua periódi a e uma equação do segundo grau. Finalizando om enfoque na resolução de problemas, omo o ál ulo de frações ontínuas de números irra ionais da forma √a2 + b, assim omo a prova da irra ionalidade de e através do ál ulo de sua fração ontínua.

Made available in DSpace on 2014-12-17T15:04:48Z (GMT). No. of bitstreams: 1 TacioVS.pdf: 1674095 bytes, checksum: d9c147d25ee2ae51bde593f7d628ca51 (MD5) Previous issue date: 2007-04-26 / The awareness of the difficulty which pupils, in general have in understanding the concept and operations with Rational numbers, it made to develop this study which searches to collaborate for such understanding. Our intuition was to do with that the pupils of the Education of Young and Adults, with difficulty in understanding the Rational numbers, feel included in the learning-teaching process of mathematics. It deals with a classroom research in a qualitative approach with analysis of the activities resolved for a group of pupils in classroom of a municipal school of Natal. For us elaborate such activities we accomplished the survey difficulties and obstacles that the pupils experience, when inserted in the learning-teaching process of the Rational numbers. The results indicate that the sequence of activities applied in classroom collaborated so that the pupils to overcome some impediments in the learning of this numbers / A consci?ncia da dificuldade que alunos, em geral, t?m em compreender o conceito e opera??es com N?meros Racionais, nos fez desenvolver este estudo que busca colaborar para tal compreens?o. Nosso intuito foi fazer com que os alunos da Educa??o de Jovens e Adultos, com dificuldade em compreender os N?meros Racionais, sintam-se inclusos no processo ensinoaprendizagem de matem?tica. Trata-se de uma pesquisa em sala de aula, numa abordagem qualitativa com an?lise das atividades resolvidas por um grupo de alunos, em sala de aula de uma escola municipal de Natal. Para elaborarmos tais atividades, realizamos o levantamento de dificuldades e obst?culos que os alunos t?m, quando inseridos no processo de ensinoaprendizagem dos N?meros Racionais. Os resultados indicam que a seq??ncia de atividades aplicadas em sala de aula colaboraram para que os alunos superassem alguns entraves na aprendizagem destes n?meros

Submitted by Jean Medeiros (jeanletras@uepb.edu.br) on 2017-11-23T14:03:34Z No. of bitstreams: 1 PDF - Paulo Henrique Freitas Silva.pdf: 42823648 bytes, checksum: 5039836455fff1766f55b2b0e0ef154b (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2017-12-06T18:43:49Z (GMT) No. of bitstreams: 1 PDF - Paulo Henrique Freitas Silva.pdf: 42823648 bytes, checksum: 5039836455fff1766f55b2b0e0ef154b (MD5) / Made available in DSpace on 2017-12-06T18:43:49Z (GMT). No. of bitstreams: 1 PDF - Paulo Henrique Freitas Silva.pdf: 42823648 bytes, checksum: 5039836455fff1766f55b2b0e0ef154b (MD5) Previous issue date: 2017-05-04 / This work has objective identify how is the teaching and learning of fractions in the classroom and researches, and what the approximations between classrooms and researches. For this, were analyzed researches about the fractions theme and informations provided by 25 elementary school teachers, collected through an open questionnaire composed for 10 questions. The data collected are of the qualitative type and were analyzed according to the method qualitative research, in a Discurso do Sujeito Coletivo approach. All the research analyzed in this work, when they propose activitys about the fractions theme, they show, directly or indirectly, that is important to work this theme with others representations - like geometrics figures and manipulables materials - beyond of fractional bar notation, for the students understand this content. Beyond this, the searchers shows alternatives for decrease the problems cited in the literature, like lack of attention and difficulties of understand the equivalence and comparation ideas, and operations with fractions. They also affirm that, in classroom, there are a preocupation with memorization of formulas and procedures and correct answer, too, but this does not necessaryli imply in understanding to conteude studed. This shows a distance between classroom and the researches, since the researches, usually, they propose alternatives and discussions whose main focus is a concern in learning with understand of the fraction theme. / Este trabalho tem como objetivo identificar como tem sido o ensino-aprendizagem de frações na sala de aula e nas pesquisas, e quais as possíveis aproximações das pesquisas com a sala de aula. Para isso, foram analisadas pesquisas sobre o referido tema e analisados dados fornecidos por 25 professores do Ensino Fundamental, que foram levantados por meio de um questionário aberto composto por 10 perguntas. Os dados obtidos são do tipo qualitativo e foram analisados conforme o método qualitativo de pesquisa, na abordagem do Discurso do Sujeito Coletivo (DSC). Todas as pesquisas analisadas neste trabalho, quando propõem atividades sobre o tema frações, mostram, direta ou indiretamente, que é importante trabalhar com outras representações de frações, como figuras geométricas e materiais manipuláveis - além da notação barra-fracionária - para que os alunos possam compreender esse conteúdo. Além disso, os pesquisadores mostram alternativas para amenizar problemas citados pela literatura, como falta de atenção e dificuldade para compreender as ideias de equivalência, comparação e operações com frações, e afirmam que em sala de aula há uma preocupação mais voltada para a memorização de fórmulas e de procedimentos e também com a resposta correta, o que não implica, necessariamente, em compreensão do conteúdo estudado. Isso evidencia um distanciamento da sala de aula e as pesquisas, já que essas, geralmente, trazem alternativas e discussões cujo foco principal é a promoção de uma aprendizagem com compreensão desse conteúdo.

Abstrakt Svensk titel: Delar av det hela. En fallstudie om hur multisensoriskt material påverkar SUM-elevers grundläggande förståelse för bråkräkning.   English title: Part of the whole. A case study on how multisensory material affects SEN-students' basic understanding of fractions.   Syftet med denna studie var att undersöka hur elevers förståelse för rationella tal i matematiken påverkades i en interventionsstudie med fokus på arbete med multisensoriskt material. Syftet med studien var dels att pröva en intensivundervisning med multisensorisk träning för några elever i matematiksvårigheter, samt att undersöka om denna intensivundervisning bidrog till ökad förståelse av rationella tal.   Studien har genomförts med fyra elever i årskurs 5 likt en fallstudie med ambitionen att fånga in och analysera undervisningssituationerna samt hur elevernas inre bilder kan komma till uttryck. I analysen har Vygotskijs sociokulturella perspektiv och Bruners teori kring representationer använts, eftersom de kompletterar varandra vad gäller att synliggöra lärandet som sker och den förståelse som utvecklas hos elever vid en intensivundervisning med ett multisensoriskt material. Vygotskijs teori används i analysen av elevernas resultat med fokus på det stöttande lärandet i intensivundervisningen och Bruners teori synliggör hur elevernas grundläggande förståelse och deras inre representationer för rationella tal har påverkats.   Studiens resultat visar att intensivundervisning med multisensoriskt material inverkar positivt på elevers förståelse för rationella tal. En utveckling av elevernas inre bilder var också synlig i studien. / Abstract The aim of this study was to investigate students’ understanding of rational numbers and how it could be influenced in an intervention study with multisensory material. The aim of the study was to study intensive teaching with multisensory training for some students in mathematical difficulties, as well as to study whether this intensive teaching contributed to increased understanding of rational numbers.   The study has been conducted with four students in grade five as a case study with  ambition to capture and analyze the teaching situations and how the students’ inner images can be expressed. In the analysis, Vygotsky's sociocultural perspective and Bruner's theory of representations have been used, as they complement each other in making the learning that occurs visible and the understanding developed by students in intensive teaching with a multisensory material. Vygotsky's theory is used in the analysis of the students outcomes, focusing on the supportive learning in intensive teaching, and Bruner's theory shows how the students' basic understanding and their inner representations for rational numbers have been influenced.   The study’s findings show that intensive teaching with multisensory material had a positive impact on the students’ understanding of rational numbers. A development of the students’ inner images was also visible in this study.