While there is a growing body of research regarding which instructional strategies are beneficial for English Learners (ELs) in mathematics classes (e.g., Chval & Chávez, 2012; Khisty & Chval, 2002; Moll 1988; Moschkovich, 2002), there are few detailed descriptions of what resources ELs draw on to enable their mathematical discourse. Therefore, this dissertation offers a detailed description of how EL students leverage resources contributed by themselves, their teacher, and the mathematical task in order to enable mathematical discourse. At its core, it strives to be an existence proof of how a linguistically diverse classroom with EL students can be a place of rich, meaningful discourse that supports the learning of mathematics. Using data from a teaching experiment designed to create an ideal classroom environment that supports rich mathematical discourse for EL students, I performed an exploratory case study (Yin, 2017) to investigate the following research question: In a specially-designed setting, how can different resources enable discourse for 9th-grade English Learners during mathematics lessons? In particular, I sought to describe the resources contributed by the students, teacher, and task that worked to enable discourse. In Chapter One, I present a statement of the problem, as well as an overview of statistical data regarding English Learners in the United States. Chapter Two is a review of relevant literature. Chapters Three, Four, and Five are each a stand-alone article intended to be submitted for publication. Chapter Three is an article which theorizes resources for enabling discourse. Chapter Four presents results of an empirical study that describes how resources from the student, the teacher, and the task are leveraged by students to enable their discourse. Chapter Five is intended for a practitioner journal and focuses on how the temporizing (intentional delay) of vocabulary can enable discourse for English Learners. Finally, Chapter 6 ties all chapters together and discusses implications and ideas for future research. The benefits of discourse in mathematics classes have been widely studied and accepted. However, there are few studies around how resources are leveraged in order to enable discourse specifically for English Learners studying mathematics in high school classrooms. In addition, there is a pervasive, misguided deficit view of English Learners which may serve as an obstacle for teachers to notice student resources or implement tasks which have the potential for discourse. Small changes in a task or instruction can lead to more opportunities for English Learners to communicate mathematically. Perhaps more important, however, is the need to recognize the abundance of resources that EL students themselves bring into the classroom. By purposefully honoring the multitude of ways students leverage resources to talk about mathematics, access to meaningful mathematics in the classroom can be improved. This can have a lasting impact for students during their time in high school and beyond. / 2023-01-27T00:00:00Z

Syftet med studien är att bidra med kunskap om matematikdidaktiska strategier för elever inom AST. Den kvalitativa studien bygger på intervjuer av lärare i verksamheter som är speciellt inriktade mot elever inom AST. Som teoretiska ramverk i studien har vi varit inspirerade av grundad teori och sociokulturell perspektiv.Vi har i vår undersökning identifierat matematikdidaktiska strategier som är gynnsamma i undervisningen för att utveckla kommunikationsförmågan. Samtliga intervjuade lärare framhåller att det inte är lika stor utmaning för eleverna att kommunicera muntligt som skriftligt i matematikundervisningen. När det gäller den muntliga kommunikationen visar undersökningen att det är viktigt att skapa trygghet så att eleverna vågar prata samt hitta material som uppmuntrar samtal. Den skriftliga kommunikationen i matematiken hos elever inom AST gynnas av att lärare presenterar uppgifter som eleverna tolkar korrekt samt skapar aktiviteter som lockar till redovisning. Utifrån undersökningens empiri och den tidigare forskningen har vi lyckats identifiera gynnsamma matematikdidaktiska strategier för att stimulera elevernas kommunikationsförmåga. Med stöd av detta kan specialläraren handleda lärarna för att förebygga och överbrygga de utmaningar de står inför i undervisningen av elever inom autismspektrumtillståndet.

The Swedish curriculum points out mathematical communication as one of the importantabilities that students need to develop. Previous studies show that students have not been given the right conditions to develop this ability sufficiently. The purpose of this study is to investigate how some Swedish teachers interpret mathematical communication and how the work with this takes place in their classrooms. The investigation consists of qualitative interviews with six teachers in the grades of Fk-3. The interviews have been focused around the teacher's interpretation regarding mathematical communication, how this is reflected in their work in the classroom and also if they see any advantages or difficulties in working with mathematical communication with their pupils. The result of the study shows that the teachers are well aware of the importance of communication and they have a will and an intention to let the pupils communicate mathematically. Mathematical communication is emphasized mainly as important for the pupil's learning and understanding. A difficulty some of the teachers mention is that communication needs time and there is a risk that pupils possibility to develop communicative abilities are limited and do not get the focus the teacher actually think is needed. / Matematisk kommunikation framhålls som en viktig förmåga i våra styrdokument samtidigtsom granskningar har visat att eleverna inte alltid har givits tillräckliga möjligheter attutveckla denna förmåga. Syftet med detta arbete är att, utifrån en fenomenologisk ansats, undersöka hur några lärare tolkar och arbetar med matematisk kommunikation i årskurserna Fk-3. För att uppnå detta har kvalitativa intervjuer genomförts med sex yrkesverksammalärare som arbetar inom årskurserna Fk-3. Intervjuerna har berört lärarnas syn på matematisk kommunikation, hur detta tar sig i uttryck i deras klassrum, samt vilka möjligheter och svårigheter lärarna ser med att arbeta med matematisk kommunikation med eleverna. Resultatet visar att de intervjuade lärarna är väl medvetna om vikten av kommunikation inom matematik och att de har en vilja och avsikt att låta eleverna arbeta tillsammans inom matematiken. Den matematiska kommunikationen ses framförallt som viktig för elevernas förståelse och lärande. En svårighet några av lärarna upplever är att kommunikation kräver tid och några uttrycker att det, då det är många delar som ska hinnas med, finns en risk att elevernas möjlighet till kommunikation begränsas.

Promoting Mathematical Understanding Through Open-Ended Tasks; Experiences of an Eighth-Grade Gifted Geometry Class by Carol H. Taylor Gifted students of mathematics served through acceleration often lack the opportunities to engage in challenging, complex investigations involving higher-level thinking. This purpose of this study was to examine the ways mathematically gifted students think about and do mathematics creatively as indicators of deep understanding through collaborative work on four open-ended tasks with high-level cognitive demand. The study focused on the mathematical thinking involved in students’ construction of mathematical understanding through the social interaction of group problem solving. This case study used ethnographic methodology within a social constructivist frame with gifted education and sociocultural contextual influences. Participants were 15 gifted students in an 8th-grade gifted geometry class. Data collection included field notes, student artifacts, student journal entries, audio recordings, and reflections. Transcribed audio recordings were segmented (Tesch, 1990) into phases of interaction, coded by function, then coded by levels of exhibited mathematical thinking from observable cognitive actions (Dreyfus, Hershkowitz, & Schwarz, 2001; Williams, 2000; Wood, Williams, & McNeal, 2006), and analyzed for maintenance or decline of high-level cognitive demand (Stein, Smith, Henningsen, & Silver, 2000). Interpretive data analysis was connected to data analysis of transcribed recordings. Results indicated social interaction among students enabled them to talk through the mathematics to understand mathematical concepts and relationships, to construct more complex meaning, and exhibit mathematical creativity, inventiveness, flexibility, and originality. Students consistently exhibited these characteristics indicating mathematical thinking at the levels of building-with analyzing, building-with synthetic-analyzing, building-with evaluative-analyzing, constructing synthesizing, and occasionally constructing evaluating (Dreyfus et al., 2001; Williams, 2000; Wood et al., 2006). The results of the study support the claim of a relationship between mathematical giftedness and the ability to abstract and generalize (Sriraman, 2003), provide evidence that given the opportunity, students can construct deep mathematical understanding, and indicate the importance of social interaction in the construction of knowledge. This study adds to the body of knowledge needed in research on gifted education, problem solving, small-group interaction, mathematical thinking, and mathematical understanding, through empirically assessed classroom practice (Friedman-Nima et al., 2005; Good, Mulryan, & McCaslin, 1992; Hiebert & Carpenter, 1992; Lester & Kehle, 2003; Phillipson, 2007; Wood, Williams, & McNeal, 2006).

Submitted by Stéfany Moreira (stemellra@yahoo.com.br) on 2013-03-04T13:34:29Z No. of bitstreams: 1 DISSERTAÇÃO_EnsinandoAprendendoAnálise.pdf: 2234228 bytes, checksum: 25295fe6ac4fa3c1ebf8cd2f7a31c3dc (MD5) / Approved for entry into archive by Neide Nativa (neide@sisbin.ufop.br) on 2013-03-13T18:52:19Z (GMT) No. of bitstreams: 1 DISSERTAÇÃO_EnsinandoAprendendoAnálise.pdf: 2234228 bytes, checksum: 25295fe6ac4fa3c1ebf8cd2f7a31c3dc (MD5) / Made available in DSpace on 2013-03-13T18:52:20Z (GMT). No. of bitstreams: 1 DISSERTAÇÃO_EnsinandoAprendendoAnálise.pdf: 2234228 bytes, checksum: 25295fe6ac4fa3c1ebf8cd2f7a31c3dc (MD5) Previous issue date: 2010 / A Análise Combinatória é um dos núcleos da matemática discreta e parte importante da Probabilidade. Contudo, percebemos, ao longo de nossas experiências como professoras, no contato com os colegas e na literatura, que é comum o ensino da Análise Combinatória exclusivamente por meio de manipulação de fórmulas ou resoluções padronizadas e que os resultados em avaliações nacionais e regionais não são bons. Por outro lado, existem estudos sobre o desenvolvimento do pensamento combinatório e os principais erros e dificuldades enfrentados por alunos e professores que trazem contribuições para o processo. Aliamos nesta pesquisa um estudo sobre pensamento combinatório e comunicação matemática para construir uma proposta de ensino de Análise Combinatória. Nosso propósito era responder à seguinte questão: “Que contribuições uma proposta de ensino que enfatiza a Comunicação Matemática pode trazer para o ensino e a aprendizagem de Análise Combinatória em uma turma do 2º ano do Ensino Médio de uma escola pública de Itabirito (MG)?”. Para isso, desenvolvemos e aplicamos uma proposta de ensino de Análise Combinatória, fundamentada nos estudos sobre desenvolvimento do pensamento combinatório e um ambiente de estímulo à argumentação e discussão de situações- problema em pequenos e grande grupos. A coleta de dados se deu por meio de notas de campo (diário da pesquisadora), gravações em áudio e vídeo de todas as aulas, registros produzidos pelos alunos ao longo das aulas, questionários e testes diagnósticos. A análise dos resultados evidencia que a maioria dos alunos participou com interesse da proposta e, gradativamente, passou a se expressar mais e com maior segurança e propriedade sobre os conceitos estudados e alcançou uma compreensão mais profunda dos mesmos, desenvolvendo tanto o pensamento combinatório quanto a argumentação. A comparação entre os resultados dos testes diagnósticos evidencia – em todos os participantes do estudo – um significativo crescimento na compreensão dos conceitos e na resolução de problemas combinatórios. Além disso, a análise revela que a ênfase na comunicação matemática foi fundamental para os bons resultados da proposta. Os dados sugerem que as discussões em pequenos e grandes grupos, quando realizadas de modo organizado e mediadas pelo professor, em um clima de respeito mútuo e estímulo à argumentação, trazem contribuições para o desenvolvimento do pensamento combinatório. Tal estudo gerou um produto educacional – um livreto com a descrição completa e comentada das atividades realizadas – destinado a professores de Matemática. __________________________________________________________________________________________ / ABSTRACT: Combinatorial analysis is one of the centralcores of discrete mathematics and an important part of probability. However, we perceive through our teaching experiences, conversations with colleagues and in the literature; that combinatorial analysis is commonly taught exclusively through manipulation of formulas or the resolution of standardized problems and that the results of national and regional assessments do not correspond to our expectations. On the other hand, thare are atudies, on the development of combinatorial thinling and the principal errors and difficulties faced by students and teachers, which provide contributions to the teaching and learning process. We bring to our study a study on combinatorial thinking and mathematical comminication in order to develop a teaching proposal for combinatorial analysis. Our purpose was to answer the following question: "What are possible contributions of a teaching proposal that emphasizes mathematics communications for the teaching and learning of combinatorial analysis for juniors in a public high school in Itabirito (MG)?. In order to obtain an answer to the question we developed and applied a teaching proposal for combinatorial analysis based on studies regarding the development of combinatorial thinking and an environment to stimulate debate and discussion of simuated problems in small and large groups. Data collection was realized through field notes (the researcher's diary), audio and video recordings of all classes, natations produced by the students during the lessons, questionnaires and diagnostic tests. the analysis of the results shows that most students participated with interest, gradually came to express themselves more and with more confidence and ownership of the concepts studies; reaching a deeper understanding of the concepts and developing both combinatorial thinking and debating skills. The comparison with the results of the diagnostic tests shows a significantgrowth in the understanding of the concepts and in solving combinatorial problems for all participants in the study. Furthermore, the analysis reveals that the emphasis on mathematical communication was fundamental to the succes of the proposal. The data suggests that the discussions in small and large groups, when performed in an organized manner, mediated by the teacher in a climate of mutual respect, in a way that encourages debate, contributes to the development of combinatorial thinking. This study generated an educational product - a booklet destined for mathematics teachers with a complete commented description of the actiities realized.

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This design experiment research introduces a mathematical inquiry titled Pattern Math. The Pattern Math activities create an atmosphere where students can think mathematically, communicate mathematically and make connections between different mathematical concepts. Based on simple patterns with complex explanations, the Pattern Math activities provide students with the opportunity to develop their conceptual understanding of mathematics. Through reflections on the activities, students are able to reexamine their views of learning mathematics. This design experiment research has a narrative approach and incorporates the teaching and research technique of interactive writing. The research highlights the power of inquiry. By providing students with the opportunity to work within their zone of proximal development, the Pattern Math activities provide students with the opportunity to make mathematical discoveries and come to understand algebra and arithmetic with conceptual understanding.

This design experiment research introduces a mathematical inquiry titled Pattern Math. The Pattern Math activities create an atmosphere where students can think mathematically, communicate mathematically and make connections between different mathematical concepts. Based on simple patterns with complex explanations, the Pattern Math activities provide students with the opportunity to develop their conceptual understanding of mathematics. Through reflections on the activities, students are able to reexamine their views of learning mathematics. This design experiment research has a narrative approach and incorporates the teaching and research technique of interactive writing. The research highlights the power of inquiry. By providing students with the opportunity to work within their zone of proximal development, the Pattern Math activities provide students with the opportunity to make mathematical discoveries and come to understand algebra and arithmetic with conceptual understanding.

<p>Undersökningen beskriver den matematiska kommunikationen mellan speciallärare och matematiksvaga elever. Den empiriska datan består av två matematiklektioner från år 9 i grundskolan. Datan analyserades för att finna olika typer av kommunikation.</p><p>Lärarna var i många delar medvetna i sitt sätt att kommunicera med eleverna och undervisningsmiljön kändes trygg. Under lektionerna observerades två parallella språk, ett vardagsanknutet och ett matematiskt språk. Det framkom att ett tydligt förhållningssätt beträffande matematiskt språk kan underlätta för elevens inlärning samt medvetandegöra läraren om den kommunikativa betydelsen i undervisningen med styrdokumentens mål i fokus.</p><p> </p> / <p>The present investigation describes the mathematical communication between the special needs teacher and pupils with mathematical disabilities. The empirical data consists of two mathematics lessons from grade 9 in the compulsory school. The data was analyzed in order to find different kinds of communication. The teachers were in many ways aware of their way to communicate with pupils and the teaching environment felt confident. During the lessons two parallel languages were observed, one everyday related and one mathematical language. It was found that keeping a distinct approach to the mathematical language can facilitate student learning and make the teacher aware about teaching communicatively with steering documents goals in focus.</p><p> </p>

Esta pesquisa tem por base a observação da realidade da sala de aula de quatro professoras dos anos iniciais (duas brasileiras e duas portuguesas), sendo seu objetivo investigar possíveis influências no seu desenvolvimento profissional a partir de sua participação num programa de formação contínua em Matemática o Pró-Letramento Matemática, no Brasil; e o Programa de Formação Contínua em Matemática para Professores dos 1.º e 2.º ciclos, em Portugal. Considerando as concepções e dinâmicas de formação dos professores de matemática desses países, tem-se por hipótese que os trabalhos conjuntos criados nos ciclos de conhecimento e a troca de ideias interpretadas em contextos colaborativos podem ser geradores de práticas profissionais transformadoras. A análise tem como referencial teórico a formação contínua de professores e o desenvolvimento profissional docente, o conhecimento profissional do professor, e as práticas letivas dos professores em matemática. Em termos metodológicos o estudo realiza uma abordagem qualitativa de natureza interpretativa, incluindo quatro estudos de caso, cujos instrumentos se pautam em entrevistas, questionários, observação de aulas, notas de campo e a análise documental dos materiais e planos de aula produzidos pelas docentes durante e após o curso, além dos documentos legais relativos aos processos de formação nos dois países. Os resultados evidenciam que os programas em causa contribuíram de forma significativa e diferenciada para o desenvolvimento profissional de cada uma das professoras analisadas, destacando-se três mudanças importantes nas suas práticas letivas: (i) o modo como passaram a explorar a apresentação e a resolução das tarefas atentando para o respectivo grau de desafio, (ii) a valorização da comunicação matemática por meio do estabelecimento constante de questionamentos, e (iii) a organização dos alunos na sala de aula. Contudo, sugere-se que cursos dessa natureza devem valorizar a planificação das aulas e procurar modos práticos de auxiliar o professor nesse processo. / This research is based on observing the classroom reality of four teachers of early years (two Brazilian and two Portuguese). Its aim is to investigate possible influences on teachers professional development from their participation in an in-service teacher education program in mathematics Mathematical Pro-Literacy in Brazil; and In-service Teacher Education Programme for Mathematics Teachers of the First and Second Cycles, in Portugal. Considering the conceptions and teacher education dynamics for mathematics teachers in these countries, it is conjectured that the joint work created in the cycles of knowledge and the exchange of ideas interpreted in collaborative contexts can be generators of transformative professional practices. The analysis has as its theoretical frame in the in-service education of teachers and teacher professional development, teacher professional knowledge, and mathematics teachers teaching practice. In methodological terms, the study follows a qualitative and interpretative approach, including four case studies, whose data collection instruments are interviews, questionnaires, classroom observations, field notes, and document analysis of materials and lesson plans produced by teachers during and after the course, in addition to the legal documents related to teacher education processes in both countries. The results show that the programs contributed in a significant and differentiated way for the professional development of all teachers analyzed, highlighting three important changes in their teaching practice: (i) how they began to explore the explanation and solution of tasks paying attention to their degree of challenge, (ii) the appreciation of mathematical communication through the establishment of constant questioning, and (iii) the organization of the students in the classroom. However, it is suggested that such courses should enhance the attention to lesson planning and look for practical ways to help teachers in this process.