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.

A classroom teaching experiment was conducted to document prospective teachers' development of whole number concepts and operations. The purpose of this mixed-methods study was to examine the collective understanding of prospective teachers in an elementary mathematics content course. Design research methodology, specifically a classroom teaching experiment was the methodology selected for this study since it allows learning to be documented in a classroom environment and is iterative in nature. A revised hypothetical learning trajectory and instructional tasks from a previous classroom teaching experiment were used in this study (Andreasen, 2006). Research about children's development of whole number concepts and operations was used in developing instructional learning goals. In addition, research regarding prospective teachers' development supported the instructional modification that all tasks would be presented and expected to be reasoned about in base-8. Both qualitative data and quantitative data were collected. Qualitative data included whole class dialogue that was videotaped and transcribed, as well as student work samples. Quantitative data included items from the Content Knowledge for Teaching Mathematics database that were administered prior to and subsequent to the instructional sequence in base-8 (Hill, Schilling, & Ball, 2005). It should be noted that the items selected from the database were in base-10. The emergent perspective served as the interpretive framework of the collected qualitative data. This perspective reflexively coordinates the social or group perspective simultaneously with psychological or individual perspective. As stated, this study sought to describe the communal mathematics understanding of prospective teachers in an elementary mathematics content course. Toulmin's (1969) model of argumentation and Rasmussen and Stephan's three-phase methodology served to document normative ways of group reasoning called classroom mathematical practices. The following classroom mathematical practices were identified as taken-as-shared by prospective teachers: (a) developing small number relationships using Double 10-Frames, (b) developing two-digit thinking strategies using the open number line, (c) flexibly representing equivalent quantities using pictures or Inventory Forms, and (d) developing addition and subtraction strategies using pictures or an Inventory Form. Quantitative results indicated that prospective teachers were able to apply mathematical understandings grounded in base-8 to whole number concepts in base-10. In the end, counting and calculating in base-8 provides a meaningful context for prospective teachers to reconstruct their knowledge of whole number concepts and operations.

Essa dissertação apresenta um conjunto de Objetos Digitais de Aprendizagem (ODAs) que foram desenvolvidos com o objetivo de promover a aprendizagem das operações com números positivos e negativos sob a perspectiva da teoria dos Campos Conceituais de Gerard Vergnaud. Além disso, também foi desenvolvida uma proposta didática para auxiliar o professor que desejar utilizá-lo nas suas aulas. Nossa pesquisa ainda apresenta a construção histórica do conjunto dos números positivos e negativos, uma discussão sobre o uso das tecnologias de informação e comunicação (TICs) em Educação e uma revisão de propostas voltadas para o ensino dos números positivos e negativos. De caráter experimental, nossa proposta foi aplicada em dois momentos diferentes: no final de 2008 numa turma de 6º série do Ensino Fundamental do Colégio de Aplicação da UFRGS e durante o primeiro semestre de 2010 numa escola da rede privada do município de Guaíba/RS. A análise dos resultados obtidos serviu como subsídio para a implementação de modificações no ODA e na proposta didática, bem como para a reflexão do desenvolvimento de ODAs que promovam o desenvolvimento do raciocínio aditivo e multiplicativo através de problemas que envolvam operações com números positivos e negativos. / This master's research shows a set of Digital Learning Objects (DLO) which were developed to promote learning of operations with whole numbers through Vergnaud's theory of conceptual fields. Furthermore, a didactical proposal was also developed in order to aid the teacher who wishes to use it in the classroom. Our research also presents the historical construction of the set of positive and negative numbers, a discussion of the use of information and communication technologies (ICTs) in Education and a review of proposals aimed at whole numbers teaching. Having an experimental character, our proposal was applied on two different moments: in the end of 2008 on an 6th grade of Colégio de Aplicação da UFRGS in the city of Porto Alegre and during the first semester of 2010 in a private school in the district of Guaíba/RS. The analysis of the results gathered served as foundation to implement a few modifications on the DLO and on the didactical proposal, as well to reflect over the development of DLOs which can promote the development of the additive and multiplicative reasoning through problems involving operations with whole numbers.

Essa dissertação apresenta um conjunto de Objetos Digitais de Aprendizagem (ODAs) que foram desenvolvidos com o objetivo de promover a aprendizagem das operações com números positivos e negativos sob a perspectiva da teoria dos Campos Conceituais de Gerard Vergnaud. Além disso, também foi desenvolvida uma proposta didática para auxiliar o professor que desejar utilizá-lo nas suas aulas. Nossa pesquisa ainda apresenta a construção histórica do conjunto dos números positivos e negativos, uma discussão sobre o uso das tecnologias de informação e comunicação (TICs) em Educação e uma revisão de propostas voltadas para o ensino dos números positivos e negativos. De caráter experimental, nossa proposta foi aplicada em dois momentos diferentes: no final de 2008 numa turma de 6º série do Ensino Fundamental do Colégio de Aplicação da UFRGS e durante o primeiro semestre de 2010 numa escola da rede privada do município de Guaíba/RS. A análise dos resultados obtidos serviu como subsídio para a implementação de modificações no ODA e na proposta didática, bem como para a reflexão do desenvolvimento de ODAs que promovam o desenvolvimento do raciocínio aditivo e multiplicativo através de problemas que envolvam operações com números positivos e negativos. / This master's research shows a set of Digital Learning Objects (DLO) which were developed to promote learning of operations with whole numbers through Vergnaud's theory of conceptual fields. Furthermore, a didactical proposal was also developed in order to aid the teacher who wishes to use it in the classroom. Our research also presents the historical construction of the set of positive and negative numbers, a discussion of the use of information and communication technologies (ICTs) in Education and a review of proposals aimed at whole numbers teaching. Having an experimental character, our proposal was applied on two different moments: in the end of 2008 on an 6th grade of Colégio de Aplicação da UFRGS in the city of Porto Alegre and during the first semester of 2010 in a private school in the district of Guaíba/RS. The analysis of the results gathered served as foundation to implement a few modifications on the DLO and on the didactical proposal, as well to reflect over the development of DLOs which can promote the development of the additive and multiplicative reasoning through problems involving operations with whole numbers.

Essa dissertação apresenta um conjunto de Objetos Digitais de Aprendizagem (ODAs) que foram desenvolvidos com o objetivo de promover a aprendizagem das operações com números positivos e negativos sob a perspectiva da teoria dos Campos Conceituais de Gerard Vergnaud. Além disso, também foi desenvolvida uma proposta didática para auxiliar o professor que desejar utilizá-lo nas suas aulas. Nossa pesquisa ainda apresenta a construção histórica do conjunto dos números positivos e negativos, uma discussão sobre o uso das tecnologias de informação e comunicação (TICs) em Educação e uma revisão de propostas voltadas para o ensino dos números positivos e negativos. De caráter experimental, nossa proposta foi aplicada em dois momentos diferentes: no final de 2008 numa turma de 6º série do Ensino Fundamental do Colégio de Aplicação da UFRGS e durante o primeiro semestre de 2010 numa escola da rede privada do município de Guaíba/RS. A análise dos resultados obtidos serviu como subsídio para a implementação de modificações no ODA e na proposta didática, bem como para a reflexão do desenvolvimento de ODAs que promovam o desenvolvimento do raciocínio aditivo e multiplicativo através de problemas que envolvam operações com números positivos e negativos. / This master's research shows a set of Digital Learning Objects (DLO) which were developed to promote learning of operations with whole numbers through Vergnaud's theory of conceptual fields. Furthermore, a didactical proposal was also developed in order to aid the teacher who wishes to use it in the classroom. Our research also presents the historical construction of the set of positive and negative numbers, a discussion of the use of information and communication technologies (ICTs) in Education and a review of proposals aimed at whole numbers teaching. Having an experimental character, our proposal was applied on two different moments: in the end of 2008 on an 6th grade of Colégio de Aplicação da UFRGS in the city of Porto Alegre and during the first semester of 2010 in a private school in the district of Guaíba/RS. The analysis of the results gathered served as foundation to implement a few modifications on the DLO and on the didactical proposal, as well to reflect over the development of DLOs which can promote the development of the additive and multiplicative reasoning through problems involving operations with whole numbers.