As escritas numéricas e o valor posicional do número são o tema desta pesquisa que tem como objetivos: investigar concepções construídas na interação criança-escrita numérica que contribuem para a construção do valor posicional característico do sistema de numeração decimal; identificar contribuições das notações de números multidígitos à conceituação do valor posicional do número, e verificar as formas cognitivas por meio das quais estas contribuições se efetivam. Consiste num estudo de análise qualitativa.Fundamenta-se na concepção epistemológica construtivista, e volta-se para as microaprendizagens presentes no processo de construção da compreensão do valor posicional e de apropriação da escrita numérica convencional de multidígitos. Envolveu nove alunos da segunda série do Ensino Fundamental de uma escola estadual do município de Erechim-RS, selecionados por sorteio a partir da realização de pré-testes com 44 crianças de três turmas nos quais foram avaliadas noções de composição aditiva, domínio da escrita numérica de multidígitos e do valor posicional. Os critérios utilizados para a seleção foram: compreensão da composição aditiva, não-compreensão do valor posicional e o não-domínio das escritas numéricas de multidígitos. Os alunos participantes da pesquisa, organizados em trios, participaram de onze situações didáticas ao longo do segundo semestre letivo do ano de 2005. A pesquisa evidenciou um processo construtivo não linear, no qual as crianças construíram concepções próximas ao valor posicional, à medida que as situações didáticas provocavam reflexões e sucessivas tomadas de consciência sobre as notações em si e sobre as relações entre escritas e agrupamentos. No que se refere ao valor posicional, a escrita numérica não sugeriu, de imediato, a possibilidade de formarem agrupamentos de acordo com as propriedades do sistema de numeração decimal, do mesmo modo que os agrupamentos (organizados dentro destes princípios) não sugeriram, num primeiro momento, a escrita numérica correspondente. Ambos, escritas e agrupamentos, num primeiro momento, significaram para as crianças a quantidade total representada tanto pelas escritas numéricas quanto pelos agrupamentos. Observou-se uma construção incipiente e progressiva dos princípios aditivo e multiplicativo do sistema, bem como da compreensão da organização do sistema em potências de dez, elementos fundamentais ao reconhecimento do valor relativo do algarismo. As notações, por si só, não foram “transparentes” às crianças e o aprendizado do valor posicional ocorreu a partir da reflexão sobre as relações entre os resultados das ações e sobre as próprias reflexões realizadas por abstrações reflexionantes e tomadas de consciência. / Numeric writings and the number place value are the theme of this research, which aims: to investigate conceptions built on the children-numeric writing interaction that contributes for the construction of place value characteristic of the decimal system; to identify contributions of multidigit numbers to the conceptualization of the number place value and; to verify the cognitive forms through which these contributions are effective. It is a study of qualitative analysis. It is grounded on the constructivist epistemological conception, and it turns to micro-learnings present in the process of construction of place value comprehension and of appropriation of the conventional numeric writing of multidigits. Nine fundamental second grade students from a public state school in the town of Erechim/RS were involved in the research, selected by assortment after the accomplishment of pre-tests with 44 children from three classes in which notions of additive composition, the understanding of numeric writing of multidigit and place value were evaluated. The criteria used for the selection were: comprehension of additive composition, the non-comprehension of place value and the non-dominance of numeric writings of multidigits. The participants of the research, organized in triads, participated in eleven didactic situations during the second school semester of 2005. The research evidenced a constructive, non-linear process, in which children built conceptions close to the place value, as the didactic situations provoked reflections and successive consciousness about notations and about the relations between writings and groupings. Regarding place value, the numeric writing did not suggest, at first, the possibility of forming groups according to the properties of decimal system, in the same manner that the groupings (organized within these principles) did not suggest immediately the correspondent numeric writing. In a first moment, both, writings and groupings, meant for the children the total quantity represented as for numeric writings as for groupings. An incipient and progressive construction of additive and multiplicative principles of the system was observed, as well as the comprehension of the system organization in potencies of tens, fundamental elements for the recognition of the number relative value. The notations, by themselves, were not “transparent” for the children and the learning of place value took place from the reflection about relations between action results and about the reflections accomplished by reflexive abstractions and consciousness. / Las escritas numéricas y el valor posicional del número son temas de esta pesquisa, que tiene como objetivos: investigar concepciones construidas en la interacción niñoescrita numérica que contribuyen para la construcción del valor posicional característico del sistema de numeración decimal; identificar contribuciones de las notaciones de números multidígitos a la conceptuación del valor posicional del número y verificar las formas cognitivas a través de las cuales estas contribuciones se efectúan. Consiste en un estudio de análisis cualitativo. Se fundamenta en la concepción epistemológica constructivista y se vuelve para los microaprendizajes presentes en el proceso de construcción de la comprensión del valor posicional y de apropiación de la escrita numérica convencional de multidígitos. Involucró nueve alumnos del segundo año de la Enseñanza Primaria – 2ª série do Ensino Fundamental – de una escuela estadual del municipio de Erechim-RS, seleccionados, por sorteo, a partir de la realización de pré-testes con 44 niños de tres grupos en los cuales fueron evaluados nociones de composición aditiva, el dominio de la escrita numérica de multidígitos y del valor posicional. Los criterios utilizados para la selección fueron: comprensión de la composición aditiva, no comprensión del valor posicional y el no dominio de las escritas numéricas de multidígitos. Los alumnos participantes de la pesquisa, organizados en triades, participaron de once situaciones-didácticas a lo largo del segundo semestre lectivo del año de 2005. La pesquisa evidenció un proceso constructivo, no linear, en el cual los niños construyeron concepciones próximas al valor posicional, a la medida que las situaciones didácticas provocaban reflexiones y sucesivas tomadas de conciencia sobre las notaciones en si y sobre las relaciones entre escritas y agrupamientos. A lo que se refiere al valor de posición, la escrita numérica no sugirió, de inmediato, la posibilidad de formar agrupamientos de acuerdo con las propiedades del sistema de numeración decimal, del mismo modo que las agrupaciones (organizados dentro de estos principios) no sugirieron de inmediato la escrita numérica correspondiente. Ambos, escritas y agrupamientos, en un primer momento, significaron para los niños la cuantidad total representada tanto por las escritas numéricas, cuanto por los agrupamientos. Se observó una construcción incipiente y progresiva de los principios aditivo y multiplicativo del sistema, bien como de la comprensión de la organización del sistema en potencias de diez, elementos fundamentales al reconocimiento del valor relativo del guarismo. Las notaciones, tan sólo, no fueron “transparentes” los niños y el aprendizaje del valor posicional ocurrió a partir de la reflexión sobre las relaciones entre los resultados de las acciones y sobre las propias reflexiones realizadas por abstracciones reflexionantes y tomadas de conciencia.

As escritas numéricas e o valor posicional do número são o tema desta pesquisa que tem como objetivos: investigar concepções construídas na interação criança-escrita numérica que contribuem para a construção do valor posicional característico do sistema de numeração decimal; identificar contribuições das notações de números multidígitos à conceituação do valor posicional do número, e verificar as formas cognitivas por meio das quais estas contribuições se efetivam. Consiste num estudo de análise qualitativa.Fundamenta-se na concepção epistemológica construtivista, e volta-se para as microaprendizagens presentes no processo de construção da compreensão do valor posicional e de apropriação da escrita numérica convencional de multidígitos. Envolveu nove alunos da segunda série do Ensino Fundamental de uma escola estadual do município de Erechim-RS, selecionados por sorteio a partir da realização de pré-testes com 44 crianças de três turmas nos quais foram avaliadas noções de composição aditiva, domínio da escrita numérica de multidígitos e do valor posicional. Os critérios utilizados para a seleção foram: compreensão da composição aditiva, não-compreensão do valor posicional e o não-domínio das escritas numéricas de multidígitos. Os alunos participantes da pesquisa, organizados em trios, participaram de onze situações didáticas ao longo do segundo semestre letivo do ano de 2005. A pesquisa evidenciou um processo construtivo não linear, no qual as crianças construíram concepções próximas ao valor posicional, à medida que as situações didáticas provocavam reflexões e sucessivas tomadas de consciência sobre as notações em si e sobre as relações entre escritas e agrupamentos. No que se refere ao valor posicional, a escrita numérica não sugeriu, de imediato, a possibilidade de formarem agrupamentos de acordo com as propriedades do sistema de numeração decimal, do mesmo modo que os agrupamentos (organizados dentro destes princípios) não sugeriram, num primeiro momento, a escrita numérica correspondente. Ambos, escritas e agrupamentos, num primeiro momento, significaram para as crianças a quantidade total representada tanto pelas escritas numéricas quanto pelos agrupamentos. Observou-se uma construção incipiente e progressiva dos princípios aditivo e multiplicativo do sistema, bem como da compreensão da organização do sistema em potências de dez, elementos fundamentais ao reconhecimento do valor relativo do algarismo. As notações, por si só, não foram “transparentes” às crianças e o aprendizado do valor posicional ocorreu a partir da reflexão sobre as relações entre os resultados das ações e sobre as próprias reflexões realizadas por abstrações reflexionantes e tomadas de consciência. / Numeric writings and the number place value are the theme of this research, which aims: to investigate conceptions built on the children-numeric writing interaction that contributes for the construction of place value characteristic of the decimal system; to identify contributions of multidigit numbers to the conceptualization of the number place value and; to verify the cognitive forms through which these contributions are effective. It is a study of qualitative analysis. It is grounded on the constructivist epistemological conception, and it turns to micro-learnings present in the process of construction of place value comprehension and of appropriation of the conventional numeric writing of multidigits. Nine fundamental second grade students from a public state school in the town of Erechim/RS were involved in the research, selected by assortment after the accomplishment of pre-tests with 44 children from three classes in which notions of additive composition, the understanding of numeric writing of multidigit and place value were evaluated. The criteria used for the selection were: comprehension of additive composition, the non-comprehension of place value and the non-dominance of numeric writings of multidigits. The participants of the research, organized in triads, participated in eleven didactic situations during the second school semester of 2005. The research evidenced a constructive, non-linear process, in which children built conceptions close to the place value, as the didactic situations provoked reflections and successive consciousness about notations and about the relations between writings and groupings. Regarding place value, the numeric writing did not suggest, at first, the possibility of forming groups according to the properties of decimal system, in the same manner that the groupings (organized within these principles) did not suggest immediately the correspondent numeric writing. In a first moment, both, writings and groupings, meant for the children the total quantity represented as for numeric writings as for groupings. An incipient and progressive construction of additive and multiplicative principles of the system was observed, as well as the comprehension of the system organization in potencies of tens, fundamental elements for the recognition of the number relative value. The notations, by themselves, were not “transparent” for the children and the learning of place value took place from the reflection about relations between action results and about the reflections accomplished by reflexive abstractions and consciousness. / Las escritas numéricas y el valor posicional del número son temas de esta pesquisa, que tiene como objetivos: investigar concepciones construidas en la interacción niñoescrita numérica que contribuyen para la construcción del valor posicional característico del sistema de numeración decimal; identificar contribuciones de las notaciones de números multidígitos a la conceptuación del valor posicional del número y verificar las formas cognitivas a través de las cuales estas contribuciones se efectúan. Consiste en un estudio de análisis cualitativo. Se fundamenta en la concepción epistemológica constructivista y se vuelve para los microaprendizajes presentes en el proceso de construcción de la comprensión del valor posicional y de apropiación de la escrita numérica convencional de multidígitos. Involucró nueve alumnos del segundo año de la Enseñanza Primaria – 2ª série do Ensino Fundamental – de una escuela estadual del municipio de Erechim-RS, seleccionados, por sorteo, a partir de la realización de pré-testes con 44 niños de tres grupos en los cuales fueron evaluados nociones de composición aditiva, el dominio de la escrita numérica de multidígitos y del valor posicional. Los criterios utilizados para la selección fueron: comprensión de la composición aditiva, no comprensión del valor posicional y el no dominio de las escritas numéricas de multidígitos. Los alumnos participantes de la pesquisa, organizados en triades, participaron de once situaciones-didácticas a lo largo del segundo semestre lectivo del año de 2005. La pesquisa evidenció un proceso constructivo, no linear, en el cual los niños construyeron concepciones próximas al valor posicional, a la medida que las situaciones didácticas provocaban reflexiones y sucesivas tomadas de conciencia sobre las notaciones en si y sobre las relaciones entre escritas y agrupamientos. A lo que se refiere al valor de posición, la escrita numérica no sugirió, de inmediato, la posibilidad de formar agrupamientos de acuerdo con las propiedades del sistema de numeración decimal, del mismo modo que las agrupaciones (organizados dentro de estos principios) no sugirieron de inmediato la escrita numérica correspondiente. Ambos, escritas y agrupamientos, en un primer momento, significaron para los niños la cuantidad total representada tanto por las escritas numéricas, cuanto por los agrupamientos. Se observó una construcción incipiente y progresiva de los principios aditivo y multiplicativo del sistema, bien como de la comprensión de la organización del sistema en potencias de diez, elementos fundamentales al reconocimiento del valor relativo del guarismo. Las notaciones, tan sólo, no fueron “transparentes” los niños y el aprendizaje del valor posicional ocurrió a partir de la reflexión sobre las relaciones entre los resultados de las acciones y sobre las propias reflexiones realizadas por abstracciones reflexionantes y tomadas de conciencia.

In recent years, the idea of language influencing the cognitive development of an understanding of place value has received increasing attention. This study explored the influence of using explicit number names on prekindergarten and kindergarten students’ ability to rote count, read two-digit numerals, model two-digit numbers, and identify the place value of individual digits in two-digit numerals. Through individual student interviews, preand post-assessments were administered to evaluate rote counting, reading five two-digit numerals, modeling five two-digit numbers, and identifying place value in two two-digit numerals. Chi-square tests for independence showed two significant relations: (1) the relationship between the control and treatment group membership on the postassessment of modeling two-digit numbers and (2) the relationship between place value identifications and group membership. Analysis of the children’s performance and error patterns revealed interesting differences between children taught with explicit number names and children taught with traditional number names. The improvement of the treatment group overall exceeded the improvement of the control group. This study indicates that teaching children to use explicit number names can, indeed, have a positive influence on their understanding of place value.

Since becoming an educator and gaining extensive classroom experience, I have concluded that it would be beneficial to elementary school children to learn other number bases and their basic functions and operations. In this thesis, I have developed five units involving lesson plans for incorporating various number bases into the existing curriculum. The units are Decimal (Base 10), Duodecimal (Base 12), Quinary (Base 5), Binary (Base 2), and Octal (Base 8) which are all appropriate for the elementary level.

This qualitative study documents a classroom teaching experiment in a semester-long undergraduate mathematics education course for 16 prospective elementary school teachers. The purpose of this study was to investigate how social aspects of the classroom environment facilitated the collective mathematical learning of place value and whole number operations by preservice elementary school teachers. Design-based research methodology was used for formulating the study. A hypothetical learning trajectory and instructional sequence related to place value and operations were created and refined in the two semesters prior to this study. The instructional sequence was in its third iteration for this study. The developmental levels that children progress through in learning place value and operations were used in identifying the learning trajectory and supporting tasks in which the preservice teachers were asked to engage. A large portion of the instructional sequence involved a setting of base eight instead of base ten. The sequence returned to base ten in order to discuss whole number operations and alternative strategies for operations in an effort to further develop the preservice teachers' conceptual understandings of place value and operations and to examine children's thinking strategies. Data were collected through video-taped recordings of class sessions, audio-taped recordings of table discussions and research team meetings, field notes, and journals written by the research team. Sixteen preservice teachers participated in the study which lasted over 5 class sessions of 3 hours and 10 minutes each. The emergent perspective which attempts to coordinate the individual learning and the social aspects of the classroom that support collective learning was used as an interpretive lens for data collection and analysis. The social aspects along with some aspects of individual student understandings together give an indication of collective mathematical understandings of the students as a whole group. Social norms established were: a) the expectation of providing explanations and justifications for solutions and solution methods, b) making sense of each other's solutions and c) asking questions of classmates or the instructor. Sociomathematical norms that were valued but not fully established were: a) criteria for different solutions and solution methods and b) criteria for what constituted a good explanation. Data analysis for the establishment of classroom mathematical practices was conducted using Toulmin's argumentation model (Toulmin, 1969). A three phase approach described by Rasmussen and Stephan (in press) was used in determining what constituted a classroom mathematical practice. The classroom mathematical practices that facilitated student learning in this study were: a) unitizing, b) flexibly representing numbers, and c) reasoning about operations. This study led to the refinement of the hypothetical learning trajectory and further progress in defining an instructional theory of how preservice teachers may come to understand place value and whole number operations.

The purpose of this study was to find some remedy to frustrations engendered when children fall to grasp the essential principle of place value after several attempts at reteaching. It was hypothesized that these children must have failed to acquire understanding of some numerical principle(s) prerequisite to understanding the place value aspect of the numeration system. Four plausible prerequisite principles were identified (1) synthesis of ordinal and cardinal properties of the numeration system, (2) both the addition and subtraction operations, (3) understanding of counting by groups, and (4) understanding of exchange equivalences such as one ten for ten ones, etc. It was hypothesized that understanding of analog clock reading was also dependent upon understanding of the same four prerequisite principles. By conducting four pilot studies, six interview protocol instruments were developed to measure levels of understanding for the four prerequisite principles and the place value and clock reading criterion principles. Three levels of understanding: no understanding, transitional understanding, and competence were designated to correspond with Plagetian stages in the development of a new operation. Forty-eight children, twenty with second grade completed and twenty-eight with third grade completed, were tested on all six instruments. Hypotheses tested were: (1) if the four identified prerequisite principles are necessary to understanding of place value, then subjects will demonstrate a level of understanding on the place value measure no higher than their lowest level of understanding achieved on the four prerequisite measures; and (2) if the four identified prerequisite . principles are necessary to understanding of clock reading, then subjects will demonstrate a level of understanding on the clock reading measure no higher than their lowest level of understanding achieved on . the four prerequisite measures. The findings were that both hypotheses were supported at the .01 probability level. Analysis of the research design and examiner observations suggested possible explanations for anomalous aspects of the obtained data. Limitations, directions for further research, and implications for teachers were also discussed. / Ph. D.

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.

In an attempt to examine the effects of journaling and vocabulary strategies on elementary students’ attitudes towards mathematical performance, I embedded reflective journaling and vocabulary strategies into my fourth grade mathematics curriculum. The mathematics content focused on whole number place value, multiplication, and division. My study revealed the positive effects these interventions can have on elementary students’ attitudes towards mathematics.

This diploma thesis deals with teaching of place-value systems in secondary school with use of created computer program. The theoretical part of the thesis deals, apart from other things, with some good reasons to teach non-base ten systems. The main part of the thesis deals with created computer program that contains teaching of binary number system and with created computer model of a counting device. These programs are also accessible from the internet. Results of experimental teaching that took place in two secondary schools are included in the thesis as well.