Syftet med studien är att erhålla en djupare förståelse av elevers kunskaper om det decimala talsystemet. Frågeställningen som undersöks är; vilka olika tillvägagångsätt kan identifieras när eleverna beskriver hur de har löst operationer som behandlar det decimala talsystemet? Den här studien har använt sig av en kvalitativ innehållsanalys, där elevernas olika tillvägagångsätt att lösa uppgifter har analyserats. Studien har lånat ord som förståelse och kvalitativa skillnader från den fenomenografiska forskningsansatsen. I studien har 17 elever deltagit och genomfört ett arbetsblad. Därefter valdes 10 elever slumpmässigt ut för vidare intervjuer. Resultatet som presenteras baseras på de uppgifter där det förekommer skillnader i elevsvaren. I uppgifterna och i de efterföljande elevintervjuerna har flertalet tillvägagångsätt kunnat identifieras. Det huvudsakliga resultatet visar att eleverna löste uppgifterna med olika tillvägagångsätt. Elevsvaren har i diskussionsdelen jämförts mot tidigare forskning för att kunna behandla studiens syfte. Vidare problematiseras även resultatets relevans för yrkesrollen och hur matematiklärare kan använda sig av resultatet för att planera och genomföra sin undervisning. / The aim of the study is to obtain a deeper understanding of students' knowledge of the decimal number system. The subject of interest is what different approaches can be identified when students describe how they have solved operations that deal with the decimal number system? This study has used a qualitative content analysis, where the students' different approaches to solving tasks have been analyzed. The study has borrowed words such as understanding and qualitative differences from the phenomenographic research approach. In the study, 17 students participated and completed a worksheet. Subsequently, 10 students were randomly selected for further interviews. The presented results are based on the data where there are differences in student responses. Through the task and the subsequent student interviews different approaches were identified. The main result is that the students solved the tasks with different approaches. In the discussion section, the student responses have been compared with previous research. Furthermore, the relevance of the result for the professional role and how the mathematics teacher is also problematized can use the results to plan and carry out their teaching.

Syftet med detta arbete är att utveckla ett ramverk för analys av variationsmönster som förekommer inom decimala talsystemet i matematikläromedel, för att därefter gå över till att använda detta ramverk för analys av hur tre vanligt förekommande läromedel i Sverige behandlar det decimala talsystemet i årskurs 3. Studien utgår därmed från följande frågeställning: Hur kan kritiska aspekter inom området decimala talsystemet, som framkommer i tidigare forskning, användas för läromedelsanalys? På vilka sätt behandlar det tre utvalda läromedlen för årskurs 3 de kritiska aspekterna i det decimala talsystemet? Studiens fokus är det matematiska innehåll som eleven möter i läroboken vid inlärning av det decimala talsystemet och den variation som finns i de kritiska aspekterna som forskningen visar på. Forskningen visar nämligen att det är vanligt förekommande att elever har problem med inlärning av det decimala talsystemet och att detta kan leda till problem senare under deras skolgång. Metoden som används är en kvalitativ läromedelsanalys där tre läromedel för årskurs 3 analyserades med syftet att undersöka vilka variationsmönster som förekommer i de kritiska aspekterna som forskningen lyfter. De kritiska aspekter som granskades var platsvärde, gruppering, språk och variationsmönster. Analysen visar att varitionsmönsters förekomst skiljer sig mycket mellan de olika böckerna. De kritiska aspekterna förekom i alla läromedlen men i olika omfattning. Ett av lärmedlen hade ett mer framstående variationsmönster, särskilt inom området platsvärde. Utifrån denna studie är det rimligt att dra slutsatsen att det inte går att förlita sig enbart på läromedlen i undervisning av det decimala talsystemet, samt att läraren bör kritiskt granska dessa före användning. Det vill säga användandet av enbart läromedel kan utgöra problem vid lärandet av det decimala talsystemet och lärare behöver bli mer medvetna om sitt ansvar när det kommer till val av läromedel. / The purpose of this work is to develop a framework for analysis of variation patterns occurring within the decimal number system in mathematic textbooks, and then move on to using this framework for analysis of how three commonly used textbooks, in Sweden deal with the decimal number system in Year 3. This purpose is achieved by answering the following questions. In what way can critical aspects concerning the field of the decimal number system, which emerge from previous research, be used for textbook analysis? In what ways do the three selected textbooks for Year 3 deal with the critical aspects of the decimal number system? The focus of the study is the mathematical content that students encounter in textbooks when learning the decimal number system and the variation that exists in the critical aspects revealed by research. Research shows that it is common for students to have problems with learning the decimal number system and that this can lead to issues later in their schooling. Qualitative analysis was used to analyze three teaching materials for Year 3 with the aim of examining which variation patterns occur in the critical aspects highlighted by the research. The critical aspects examined were site value, grouping, and language. The analysis showed that the occurrence of variation patterns varies greatly among different books. The critical aspects occurred in all the teaching materials, but to varying degrees. One of the teaching aids had more prominent variation patterns, especially concerning place value. From this study, it can be concluded that it is not possible to rely solely on textbooks when teaching the decimal number system, and teachers should critically examine teaching aids before using them. Relying on textbooks alone can be a problem in teaching the decimal number system, and teachers need to become more aware of their responsibilities when choosing textbooks in mathematics.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-07-27T13:29:56Z No. of bitstreams: 1 Aline Tafarelo Tracanella.pdf: 2762008 bytes, checksum: a0ecbfb9e128d24bccdf1a07c2c5e734 (MD5) / Made available in DSpace on 2018-07-27T13:29:57Z (GMT). No. of bitstreams: 1 Aline Tafarelo Tracanella.pdf: 2762008 bytes, checksum: a0ecbfb9e128d24bccdf1a07c2c5e734 (MD5) Previous issue date: 2018-05-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / As soon as children begin their school life, they already carry with them an idea about the numbers and operation of the Decimal Number System (DNS). However, this knowledge need to be systematized, extended and deepened appropriately in order to assist in the construction of other mathematical concepts. Given this problem, the present research aims to investigate the mobilized knowledge of the positional value in the DNS and the understanding of the characteristics of number zero in the same system by students of the fourth year of Elementary School. Therefore, it is done a brief historical context to rescue how the development of this kind of knowledge by ancient people has developed over time. As theoretical contributions, it is used the researches of Piaget & Szeminska, and of Kamii on the constructions of the number concept by the students. Regarding to the acquisition of the properties of the DNS, it is discussed the researches of Fayol, Lerner & Sadovsky as well as Zunino, who also studies the issue of the number zero in this system. To achieve the research objective, it is adopted the qualitative methodology, since the focus of it is on the mobilized knowledge by the students in the search for a solution to proposed activities. It was also developed an instrument with six exercises involving the positional value and the number zero, based on the proposed sequence in the Brandt version. One week after an application of the instrument, it was conducted a semistructured interview, which was of very important to understand the answers provided by the students. In the analysis and discussion of the obtained data, it is understand that the students mobilized knowledge about the numerical sequence and the criteria of comparison pointed out by Lerner & Sadovsky. In addition to these mobilized knowledge, the participants also used the contextualization of activities to justify their responses, using a comparison with everyday situations, such as, for example, age observation among children. Regarding the number zero, it was analyzed the meanings attributed to this number by the students during interviews. During the research phases, all students stated that zero “worth nothing”, but they have provided justifications that meet the historical facts pointed out in the brief contextualization carried out in the third chapter of the research. It is also noted that the participants are building their knowledge about DNS, presenting an unstable knowledge that changes according to the question asked regarding the proposed situation. The results found in this research indicate that the work with DNS needs to be continuous throughout the initial years of Elementary School, as the students continue to build their knowledge about DNS and expand their understanding of the number zero in the years after the literacy cycle / Assim que as crianças iniciam sua vida escolar, já carregam consigo alguma ideia sobre os números e sobre o funcionamento do Sistema de Numeração Decimal (SND). Todavia esses conhecimentos precisam ser sistematizados, ampliados e aprofundados adequadamente, para auxiliar na construção de outros conceitos matemáticos. Diante dessa problemática, a presente pesquisa tem por objetivo investigar que conhecimentos são mobilizados por alunos do quarto ano do Ensino Fundamental acerca do valor posicional no SND e sobre a compreensão do número zero nesse mesmo sistema. Para isso, buscamos em uma breve contextualização histórica resgatar como se deu o desenvolvimento desses saberes por povos antigos no decorrer do tempo. Como aportes teóricos, nos baseamos nas pesquisas de Piaget e Szeminska e de Kamii sobre a construção do conceito de número pelos alunos. Com relação à aquisição das propriedades do SND, discorremos sobre as pesquisas de Fayol e de Lerner e Sadovsky, bem como de Zunino, que aborda também a questão do número zero nesse sistema. Para atender ao objetivo da pesquisa, adotamos a metodologia de cunho qualitativo, pois o foco da investigação está nos conhecimentos mobilizados pelos educandos na busca por uma solução para as atividades propostas. Elaboramos um instrumento com seis exercícios envolvendo o valor posicional e o número zero, baseado na sequência proposta na tese de Brandt. Uma semana após a aplicação do instrumento, realizamos uma entrevista semiestruturada, que foi de suma importância para compreender com maior clareza as respostas fornecidas pelos alunos. Na análise e discussão dos dados obtidos, compreendemos que os estudantes mobilizaram conhecimentos acerca da sequência numérica e dos critérios de comparação apontados por Lerner e Sadovsky. Além desses conhecimentos mobilizados, os participantes também recorreram à contextualização das atividades para justificar suas respostas, usando a comparação com situações cotidianas, como, por exemplo, a observação da idade entre crianças. Com relação ao número zero, analisamos os significados atribuídos a esse número pelos alunos durante as entrevistas. Durante as fases da pesquisa, todos os educandos afirmaram que o zero “não vale nada”, mas trouxeram justificativas que vão ao encontro dos fatos histórico apontados na breve contextualização realizada no primeiro capítulo da investigação. Notamos também que os participantes estão construindo seus conhecimentos acerca do SND, apresentando um conhecimento não estável, ou seja, que se altera de acordo com a pergunta feita referente à situação proposta. Os resultados encontrados nessa pesquisa apontam que o trabalho com o SND precisa ser contínuo, durante todos os anos iniciais do Ensino Fundamental, pois os alunos continuam construindo seus conhecimentos acerca do SND e ampliando sua compreensão sobre o número zero nos anos posteriores ao ciclo de alfabetização

Made available in DSpace on 2016-04-27T16:57:31Z (GMT). No. of bitstreams: 1 Mariana Campioni Morone Cardoso.pdf: 1708711 bytes, checksum: f36e3e1fad86d85e0e2252fcf18410f9 (MD5) Previous issue date: 2014-07-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This paper aims at analyzing dissertations and theses that focus on teaching the first years and the comprehension of the decimal number system. On CAPES platform we have used the following key words: Decimal number system and first years; Decimal number system and teacher formation between 2006 and 2010. Having chosen the researches, in preliminar analysis we collected and presented their summaries, objectives, theoretical references, methodology and results through text recording. We established categories, in order to relate the selected researches to each other, searching for answers to the following questions: which objectives have been pursued and which results have been reached in relation to the decimal number system relatively to teaching such topic in the first years of school? To do that, we have decided on using the content analysis on a qualitative basis. We deconstructed the structure, recognizing its main characteristics and extracted from that its meaning following Laville and Dionne. The analysis of the researches aims at the need to be alert to the necessary command of the content for effective teaching. So, we consider urgent the adoption of a perspective of formation in which the future teacher involves himself in teaching reflexive processes during his learning / Neste trabalho buscamos analisar dissertações e teses que focalizassem a docência dos anos iniciais e a compreensão do sistema de numeração decimal. Na plataforma CAPES, utilizamos as seguintes palavras-chaves: Sistema de numeração decimal e anos iniciais; Sistema de numeração decimal e formação de professores entre os anos de 2006 e 2010. Após a escolha das pesquisas, em análises preliminares recolhemos e apresentamos seus resumos, seus objetivos, referencial teórico, metodologia e resultados, através de fichamentos. Nas análises, definimos categorias, a fim de estabelecer relações entre essas pesquisas selecionadas, buscando repostas às questões: que objetivos têm sido buscados e que resultados vêm sendo alcançados em relação ao sistema de numeração decimal relativamente à docência deste assunto nos anos iniciais do Ensino Fundamental? Para tal, optamos pela análise de conteúdo, de caráter qualitativo. Desmontamos a estrutura, reconhecendo suas principais características, e disso extraímos sua significação, seguindo Laville e Dionne. A análise das pesquisas selecionadas, aponta para a necessidade de se atentar ao necessário domínio do conteúdo para que se efetive o ensino. Logo, consideramos urgente a adoção de uma perspectiva de formação na qual o futuro professor se envolva em processos reflexivos da docência, na sua aprendizagem

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-01-23T11:30:17Z No. of bitstreams: 1 Ivonildes dos Santos Milan.pdf: 1457633 bytes, checksum: 5afb9339e38d655529cb405ab53dc782 (MD5) / Made available in DSpace on 2018-01-23T11:30:17Z (GMT). No. of bitstreams: 1 Ivonildes dos Santos Milan.pdf: 1457633 bytes, checksum: 5afb9339e38d655529cb405ab53dc782 (MD5) Previous issue date: 2017-11-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / On this research, we aim to reflect upon the teaching and learning of the Decimal Number System on the second grade of elementary School, more specifically to analyze the didactical conditions that allow the comprehension of what is hidden - the positional system. We used some of the contributions of Mathematics’ Didactics that defends the usage of didactical situations that raise, in students, the use of previous knowledge to select, organize, interpret information, and make decisions, in order to allow them to find different ways to build mathematical knowledge. Our methodology is inspired by Didactical Engineering, which comprehends the usage/elaboration of didactical situations that ensemble a meaningful learning board in the classroom. The didactical sequence, elaborated by Argentinian researchers from the Didactical Situations Theory, integrates an investigation project - developed in the city of Buenos Aires with second grade students – and has, as a starting point, the interaction with written numbers. We applied the didactical sequence twice to second grade students from the same elementary school, located in São Paulo. Our research has brought relevant contributions, such as: the way students relate, think and comprehend the positional value; promote successive approximations on the value of algorithms that represent the first grouping of ten basis; justify the efficiency of didactical sequences in mathematical learning; identify variables, in teaching and learning, that secure the process of both successive conceptualization to new knowledge and also variables present in the usual teaching which unfeasible the construction process of knowledge by students, and yet, confirms the potential of the group discussions to Mathematical learning / Nessa pesquisa, objetivamos refletir sobre o ensino e a aprendizagem do Sistema de Numeração Decimal no segundo ano do Ensino Fundamental, mais especificamente analisar as condições didáticas que possibilitam a compreensão daquilo que está oculto – o sistema posicional. Utilizamos algumas contribuições da Didática da Matemática, que defende a utilização de situações didáticas que suscitem, nos alunos, ações que mobilizem conhecimentos já adquiridos, para que selecionem, organizem, interpretem informações e tomem decisões que os permitam encontrar diferentes formas de construir conhecimentos matemáticos. Nossa metodologia se inspira na Engenharia Didática, que compreende a utilização/elaboração de situações didáticas que configurem um quadro de aprendizagem significativa em sala de aula. A sequência didática, elaborada por pesquisadoras argentinas a partir da Teoria das Situações Didáticas, integra um projeto de investigação – desenvolvido na Província de Buenos Aires com alunos do segundo ano –, cujo ponto de partida é a interação com a numeração escrita. Aplicamos a sequência didática duas vezes a alunos do segundo ano do Ensino Fundamental, numa mesma escola, localizada em São Paulo. Nossa pesquisa trouxe contribuições relevantes, tais como: o modo como os alunos se relacionam, pensam e entendem o valor posicional; promover aproximações sucessivas sobre o valor dos algarismos que representam o primeiro agrupamento da base dez; justificar a eficácia das sequências didáticas na aprendizagem matemática; identificar variáveis, no ensino e aprendizagem, que asseguram o processo tanto de conceitualizações sucessivas a novos conhecimentos quanto de variáveis presentes no ensino usual, as quais inviabilizam o processo de construção dos conhecimentos pelos alunos; e, ainda, confirmar o potencial das discussões coletivas para a aprendizagem matemática

Visual Type Instrument - VTI, is a tool containing a typeface and a set of geometric calculations that generates numerical data in spreadsheets. The data is determined from the typefaces visual dimensions and is used to calculate leading, type size, margins and format. The aim is to offer, for all of those working with typography and design, a practical way to manage text, image and format in relationship to visual size. Instead of using point measurements, VTI uses a new device called Edo. An Edo is the same as one twelfth of a millimetre. This makes VTI compatible with the metric system and helps us to set text in relation to our own formats which applies namely in millimetres.    VTI uses logical arithmetic and geometry that is set in an automated process, and many choices that normally postpone the working process can be excluded. Simultaneously VTI is meant to discreetly make the user aware that more settings can be activated in line with the users own development, which makes the creative flow unaffected.    Using a metrics based on simple fractions, a mnemonic awareness is developed with the user and an understanding of how every detail fits together becomes more obvious. The typography becomes the key to any design choices that follows. The user becomes aware of the visual size, and in a cognitive way stimulated to recreate a font size regardless of the fonts limitations.