Student Number : 0008975N - M Ed research report - School of Education - Faculty of Humanities / This qualitative study was conducted in one school in an informal settlement, West of Johannesburg. The study explored how a grade 7 teacher promoted mathematical reasoning in multilingual mathematics class of English second language learners. The focus of the research was on how a Grade 7 mathematics teacher interacts with the learners to encourage mathematical reasoning during his teaching in a multilingual class. The study also looked at the kind of tasks the teacher used to promote mathematical reasoning and how he uses language to enable mathematical reasoning. The study was informed by a theory of learning which emphasises the importance of social interaction in the classroom where the teacher encourages learners to interact with each other to explain their thinking and to justify their answers. Data was collected through lesson and teacher interviews. The study shows the teacher focused more on developing the learners’ procedural fluency. This focus on procedural fluency was accompanied by the dominance of the use of English by the learners.

The nature of geometric tasks that students engage with in classrooms influences the development of their geometric thinking. Although mathematics standards emphasize formal proofs and mathematical reasoning skills, geometric tasks in classrooms remain focused on students’ abilities to recall mathematical facts and use simple procedures rather than conceptual understanding. In order to facilitate students’ high-level mathematical thinking, teachers need to provide sufficient opportunities for students to engage in cognitively demanding mathematical tasks. The use of dynamic geometry software (DGS) in classrooms facilitates conceptual understanding of geometric proofs. The black box approach is a new type of task in which students interact with pre-constructed figures to explore mathematical relationships by dragging and measuring geometric objects. This approach is challenging to students because it “requires a link between the spatial or visual approach and the theoretical one” (Hollebrands, Laborde, & Sträßer, 2008, p. 172). This study examined how preservice secondary mathematics teachers make choices or create geometric tasks using DGS in terms of cognitive demand levels and how the black box approach influences the way preservice teachers conceptualize their roles in their lesson designs. Three preservice secondary mathematics teachers who took a semester-long mathematics teaching course participated in this qualitative case study. Data include two lesson plans, before and after instructions for geometric DGS tasks, pre- and post-interview transcripts, electronic files of geometric tasks, and reflection papers from each participant. The Mathematical Task Framework (Stein, Smith, Henningsen, & Silver, 2009) was used to characterize mathematical tasks with respect to level of cognitive demand. A Variety of geometric task types using DGS was introduced to the participants (Galindo, 1998). The dragging modalities framework (Arzarello, Olivero, Paola, & Robutti, 2002; Baccaglini-Frank & Mariotti, 2010) was employed to emphasize the cognitive demand of geometric tasks using DGS. The PURIA model situated the participants’ conceptualized roles in technology use (Beaudin & Bowers, 1997; Zbiek & Hollebrands, 2008). Findings showed that the preservice teachers only employed geometric construction types on low level geometric DGS tasks, which relied on technological step-by-step procedures students would follow in order to arrive at the same results. The preservice teachers transformed those low level tasks into high level tasks by preparing DGS tasks in advance in accordance with the black box approach and by encouraging students to explore the tasks by posing appropriate questions. However, as soon as they prepared high level DGS tasks with deductive proofs, low level procedure-based tasks followed in their lesson planning. The participants showed positive attitudes towards using DGS to prepare high level geometric tasks that differ from textbook-like procedural tasks. Major factors influencing preservice teachers’ preparation of high level tasks included teachers’ knowledge of mathematics, pedagogy, and technology, as well as ways of using curriculum resources and teachers’ abilities to set appropriate lesson goals. Findings of this investigation can provide guidelines for integrating DGS in designing high level geometric tasks for teacher educators, researchers, and textbook publishers.

Submitted by Verena Bastos (verena@uefs.br) on 2015-08-04T00:26:41Z No. of bitstreams: 1 DISSERTACAO_A PARTICIPA??O DE PROFESSORES DE MATEM?TICA_WEDESON(1).pdf: 981158 bytes, checksum: 46925da71f4e589bb5d90f6af24a99a1 (MD5) / Made available in DSpace on 2015-08-04T00:26:41Z (GMT). No. of bitstreams: 1 DISSERTACAO_A PARTICIPA??O DE PROFESSORES DE MATEM?TICA_WEDESON(1).pdf: 981158 bytes, checksum: 46925da71f4e589bb5d90f6af24a99a1 (MD5) Previous issue date: 2015-03-16 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior - CAPES / The aim of this work is to analyze how math teachers participate in the development of mathematical tasks that constitute an educational curriculum material in a collaborative work and to analyze mathematical tasks produced by these teachers in this context. Thus, the focus of this research is the participation of teachers and the production of mathematical tasks. Thereby, theoretical constructs presented by Jean Lave and Etienne Wenger were used to understand the participation of teachers and tasks markers presented by Jonei Barbosa used as reference for the analysis of mathematical tasks. The context of this research were the meetings of the Mathematics Educational Watch (OEM-Bahia) based at the Federal University of Bahia (UFBA) and formed by teachers (basic and college education) and students with Degree in Mathematics and Post-Graduate at the State University of Feira Santana and Federal University of Bahia. The methodological approach was qualitative, in which we used as procedures of data collection observation and interviews stimulated for analyzes relating to the participation of teachers, and document analysis of mathematical tasks that constitute an educational curriculum materials developed by teachers during the collaborative work. The results showed that the participation of teachers in the development of mathematical tasks occurs in three distinct ways: contemplating common goals of the group, sharing ways to develop investigative or exploratory mathematical questions and producing tasks in view of the practices involving other math teachers. Ways to participate relate to the joint undertaking established by the group for the preparation of tasks, with the list shared repertoire group members, which allowed evidence of changes in practices in which teachers participate and with what the group reifies from participation in social practice. In terms of the tasks that were developed through the participation of teachers in the OEM-Bahia we can see that the mathematical tasks performed in convergent, divergent and alternate forms, depending on the task markers, allowing an analysis of the freedom of choice of group during the design of tasks, depending on the practice that teachers involved and the context of the classroom. In terms of mathematical tasks markers, the analysis allowed us to expand the theoretical construct, indicating the possibility of tasks with alternative insulation because the relationships established during the implementation of tasks can be in charge of teachers and negotiations with the students. In addition, we propose the marker in relation to the procedures that are required from students during the implementation of mathematical tasks, called teaching focus. This analysis allowed us to realize that both tasks markers as the framework for analyzing mathematical tasks theoretically allow further analysis of mathematical tasks designed in different contexts and social practices. / O objetivo desta disserta??o ? analisar como professores de matem?tica participam da elabora??o de tarefas matem?ticas que constituem um material curricular educativo em um trabalho colaborativo e analisar as tarefas matem?ticas produzidas por esses professores nesse contexto. Assim, o foco desta investiga??o ? a participa??o dos professores e a produ??o das tarefas matem?ticas. Desta forma, os constructos te?ricos apresentados por Jean Lave e Etienne Wenger foram utilizados para compreender a participa??o dos professores e os marcadores de tarefas apresentados por Jonei Barbosa utilizados como refer?ncia para a an?lise das tarefas matem?ticas. O contexto desta pesquisa foram as reuni?es do Observat?rio da Educa??o Matem?tica (OEM-Bahia) sediado na Universidade Federal da Bahia (UFBA) formados por professores (educa??o b?sica e superior) e estudantes da Licenciatura em Matem?tica e P?s-Gradua??o da Universidade Estadual de Feira de Santana e da UFBA. A abordagem metodol?gica utilizada foi a qualitativa, na qual utilizamos como procedimentos de coleta de dados a observa??o e as entrevistas estimuladas para as an?lises referentes ? participa??o dos professores, e an?lise documental das tarefas matem?ticas que constituem um material curricular educativo elaboradas pelos professores durante o trabalho colaborativo. Os resultados apontaram que a participa??o dos professores na elabora??o de tarefas matem?ticas ocorre de tr?s formas distintas: contemplando objetivos comuns do grupo, compartilhando sobre modos de elaborar quest?es matem?ticas investigativas ou explorat?rias e produzindo tarefas na perspectiva das pr?ticas que participam outros professores de matem?tica. Essas formas de participar t?m rela??o com o empreendimento conjunto estabelecido pelo grupo para a elabora??o das tarefas, com o repert?rio compartilhado entre os membros do grupo que possibilitou ind?cios de mudan?as nas pr?ticas que os professores participam e com o que o grupo reifica a partir da participa??o na pr?tica social. Em termos das tarefas que foram elaboradas por meio da participa??o dos professores no OEM-Bahia podemos observar que as tarefas matem?ticas se apresentaram nas formas convergentes, divergentes e alternadas, a depender dos marcadores de tarefas, permitindo uma an?lise sobre a liberdade de escolha do grupo durante o delineamento das tarefas, a depender da pr?tica que os professores participam e do contexto das salas de aula. Em termos dos marcadores de tarefas matem?ticas, a an?lise nos permitiu expandir o constructo te?rico, apontando a possibilidade de tarefas com isolamento alternativo, pois as rela??es estabelecidas durante a implementa??o de tarefas podem ficar por conta dos professores e das negocia??es com os estudantes. Al?m disso, propomos o marcador com rela??o aos procedimentos que s?o requeridos aos estudantes durante a implementa??o de tarefas matem?ticas, a este denominamos foco de ensino. Essa an?lise possibilitou compreendermos que tanto os marcadores de tarefas quanto o quadro de an?lise de tarefas matem?ticas permitem aprofundar teoricamente a an?lise de tarefas matem?ticas elaboradas em diferentes contextos e pr?ticas sociais.

This study investigated how student teachers used their mathematical knowledge for teaching and pedagogical knowledge to design and modify mathematical tasks. It also examined the relationship between teacher knowledge and the cognitive demands of a task. The study relied heavily on the framework in Hill, Ball, and Shilling (2008), which describes the different domains of knowledge in mathematical knowledge for teaching, and the framework on the cognitive demands of mathematical tasks in Stein, Smith, Henningsen, and Silver (2000). Results of the study indicated that the student teachers used their common content knowledge when they lacked sufficient knowledge in other domains, especially specialized content knowledge, to perform a particular job of teaching. There was often a decrease in the cognitive demands of a task when it was modified by the student teachers. These drops were often associated with a lack of specialized content knowledge.

Students’ struggle with learning mathematics is often cast in a negative light. Mathematics educators and researchers, however, suggest that struggling to make sense of mathematics is a necessary component of learning mathematics with understanding. In order to investigate the possible connection between struggle and learning, this study examined students’ productive struggle as students worked on tasks of higher cognitive demand in middle school mathematics classrooms. Students’ productive struggle refers to students’ “effort to make sense of mathematics, to figure something out that is not immediately apparent” (Hiebert & Grouws, 2007, p. 287) as opposed to students’ effort made in despair or frustration. As an exploratory case study using embedded multiple cases, the study examined 186 episodes of student‐teacher interactions in order to identify the kinds and nature of student struggles that occurred in a naturalistic classroom setting as students engaged in mathematical tasks focused on proportional reasoning. The study identified the kinds of teacher responses used in the interaction with the students and the types of resolutions that occurred. The participants were 327 6th and 7th grade students and their six mathematics teachers from three middle schools located in mid‐size Texas cities. Findings from the study identified four basic types of student struggles: get started, carry out a process, give a mathematical explanation, and express misconception and errors. Four kinds of teacher responses to these struggles were identified as situated along a continuum: telling, directed guidance, probing guidance, and affordance. The outcomes of the student‐teacher interactions that resolved the students’ struggles were categorized as: productive, productive at a lower level, or unproductive. These categories were based on how the interactions maintained the cognitive level of the implemented task, addressed the externalized student struggle, and built on student thinking. Findings provide evidence that there are aspects of student‐teacher interactions that appear to be productive for student learning of mathematics. The struggle‐response framework developed in the study can be used to further examine the phenomenon of student struggle from initiation, interaction, to its resolution, and measure learning outcomes of students who experience struggle to make sense of mathematics. / text

The aim of this study is to examine how teachers explain mathematical tasks in the mathematical education in grade 2, and also which perception they have of explanations and their own teaching. The survey is supported by theories that focuses on explanations in the education. These are Vygotsky´s theory of the development of the scientific concepts, general and literary concepts, proximal zone and scaffolding. To achieve the aim, the following questions has been formulated: How do teachers explain mathematical tasks in the mathematical education in grade 2? How do teachers speak about explanations and their own teaching? In summary, the results of this study are that teachers explain mathematical tasks in the mathematical education in grade 2 by using so called mathematical, daily and visual explanations when explaining these tasks in their education. Beyond this, the results of this study are also that teachers speak about explanations as something clear and concrete. They also define explanations as something that promote for student development and learning, and this is something the teachers in this study want to achieve in their own education when explaining mathematical tasks.

Several studies show that students with a mother tongue other than Swedish are disadvantaged in the meeting with text rich mathematical tasks. This is especially so when the tasks have a context that is linked to Swedish culture and tradition that is unknown to the students. This study examines how multilingual middle school students respond to text-rich mathematical tasks. More specifically, the study has investigated multilingual students' difficulties and resources in meeting text-rich mathematical tasks.To answer the questions, qualitative interviews were conducted with a total of eight pupils who, on the basis of four text-rich textbook tasks, had to reason and reflect on the content, language and problems of the tasks. The theoretical approach to analyze the empirical material was social semiotics. Social semiotics is about how people communicate, create meaning and interact with the outside world through different social and cultural contexts. In these social and cultural contexts, humans creates so-called resources that are available to interpret, communicate and create meaning.The results of this study indicate that multilingual students encounter linguistic, cultural and contextual difficulties in the mathematical text-rich tasks. Furthermore, the results of this study also show how these multilingual students use different semiotic resources in their encounter with the text-rich mathematical task such as; dynamically thinking through two languages in parallel, pointing at words, pictures and using a mental “number-exchange”, where the students exchange the mathematical numbers, from Swedish to their native language to cognitively calculate the mathematical numbers.

The tasks with which students engage in their mathematics courses determine, for a large part, what students learn. Therefore, it is essential that teachers are able to design tasks that are worthwhile for developing mathematical understanding. Since practicing teachers seldom incorporate worthwhile mathematical tasks in their lessons, we would expect that they did not become proficient at designing worthwhile tasks while in their teacher education programs. This thesis describes a study that investigated what preservice secondary teachers attend to as they attempt to design worthwhile mathematical tasks. Three participants were selected from a course at a large private university where preservice teachers are taught and practice the skill of task design. This "Task Design" course was observed, and the three participants were interviewed to determine what they attend to while designing tasks. There were seven main characteristics that the main participants in the study attended to the most often and thought were the most important: sound and significant mathematics, reasoning, appropriateness, clarity, communication, engagement, and openness. How the participants attended to these characteristics is described. Some implications for teacher education, such as requiring preservice teachers to explain how their tasks embody certain characteristics, are given based on the results.

In this thesis, focused attention is given to the idea of task solvers as active participants in mathematical practices. The theoretical assumptions of the study, reported in this thesis, are inspired by socio-political concerns. The aim of the study is to investigate the underlying view of participation in mathematical practices, as understood in a nationwide teacher professional development programme, the Boost for Mathematics, in Sweden. To be more precise, the study is arranged to problematise ways of encouraging students as active participants. This aim is approached by means of the following research questions: (1) What messages do mathematical tasks in the Boost for Mathematics send about people as participants in mathematical practices? and (2) What is the role of multiple representations in these messages? An empirical study is reported. The data of the study, i.e., three collections of problems, are drawn from the Boost for Mathematics. Data processing is conducted by using a modified version of a pre-existing data processing framework, focusing on mathematical practices as socio-political practices. The empirical study uncovers an implicit view of task solvers in mathematical practices and especially a detachment between students, as potential task solvers, and the social contexts where mathematical ideas and concepts are embedded. This implicit view is challenged from the assumption that it is motivating for a student to conceive him/herself as someone who is ‘qualified’ to take part in mathematical practices.

The overall purpose of the study is partly to increase knowledge about the use of open mathematics by primary school teachers, and partly if the tasks are considered functional to use to include high-performing students in mathematics education. In order to achieve the purpose of the study, an internet survey was constructed. Through the survey, data were collected from 104 active primary school teachers who teach mathematics in Sweden. The results show that a large part of the teachers who participated in the study use open mathematical tasks in the teaching. In addition, more than half of the respondents considered that the tasks could be used to include high-performing students in the heterogeneous classroom. The study suggests that professional development among primary school teachers is needed to enable an improvement in the ability to work with open mathematical tasks and the inclusion of high-performing students. / Det övergripande syftet med studien är dels att öka kunskapen om lågstadielärares användning av öppna matematiska uppgifter, dels om sådana uppgifter anses funktionella att använda för att inkludera högpresterande elever i matematikundervisningen. För att uppnå syftet med studien konstruerades en internetenkät. Via enkäten samlades data in från 104 verksamma lågstadielärare som undervisar i matematik runt om i Sverige. Resultaten visar att en stor del av de lågstadielärare som deltog i studien använder öppna matematiska uppgifter i undervisningen. Dessutom ansåg drygt hälften av respondenterna att nämnda uppgifter kunde användas för att inkludera högpresterande elever i det heterogena klassrummet. Studien antyder att det behövs kompetensutveckling bland lågstadielärare för att möjliggöra en förbättring av förmågan att arbeta med öppna matematiska uppgifter samt inkludering av högpresterande elever.