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- A prática de umaprofessora de Matemática ao conduzir uma discussão coletiva sobre sequênciasPublication . Rodrigues, Cátia; Menezes, Luís; Da Ponte, João PedroMathematics learning with understanding can be improved by engaging students in whole-class discussions as they are called upon to present, justify and argue their strategies and the strategies of their colleagues and, later, to systematize the main ideas emerging from this discussion. This qualitative and interpretive study, is a case study of a grade 7 mathematics teacher, and aims to understand this practice in conducting the whole-class discussions, in connection with her didactic knowledge, when exploring a task involving sequences. The results show that the teacher, based on her didactic knowledge and preparation of the lesson, organizes the whole-class discussion in three moments (presentation; comparison, evaluation and filtering; and conclusion) in which she undertakes various teaching actions.
- Mathematical pedagogical content knowledge in Early Childhood Education: tales from the ‘great unknown’ in teacher education in PortugalPublication . Figueiredo, Maria Pacheco; Gomes, Helena Margarida dos Santos Vasconcelos; Rodrigues, CátiaThe study aims to explore the specificity of Mathematics pedagogical content knowledge in Early Childhood Education Pedagogy in Initial Teacher Education. It addresses the issue by characterizing student teachers’ perspectives and by analyzing student teachers’ knowledge mobilized in a situation of planning for teaching. The answers to a task developed by students in an Initial Teacher Education program are analyzed in terms of the mathematical knowledge and pedagogical options presented. The results contribute to the discussion in terms of (un)balance between teacher-initiated and child-led activities. The discussion deepens the importance to assert specific/particular ways of teaching in Early Childhood Education, contrasting with the more restricted view of only adult-led moments being teaching. Strong content knowledge and pedagogical content knowledge are valued because of their relevance both at the level of adults’ knowledge needed to support children initiatives and plan curricular/didactic activities and at the level of the knowledge children interact within their daily environment and routine
- A prática de uma professora de Matemática ao conduzir uma discussão coletiva sobre sequênciasPublication . Rodrigues, Cátia; Menezes, Luís; Da Ponte, João PedroMathematics learning with understanding can be improved by engaging students in whole-class discussions as they are called upon to present, justify and argue their strategies and the strategies of their colleagues and, later, to systematize the main ideas emerging from this discussion. This qualitative and interpretive study, is a case study of a grade 7 mathematics teacher, and aims to understand this practice in conducting the whole-class discussions, in connection with her didactic knowledge, when exploring a task involving sequences. The results show that the teacher, based on her didactic knowledge and preparation of the lesson, organizes the whole-class discussion in three moments (presentation; comparison, evaluation and filtering; and conclusion) in which she undertakes various teaching actions.
- Penso, comunico, resolvoPublication . Menezes, Luís; Ribeiro, António Augusto Gaspar; martins, ana patrícia; Rodrigues, Cátia; Gomes, Helena Margarida dos Santos Vasconcelos; Novo, SóniaEsta brochura destina-se a alunos do ensino básico (1.º a 9.º anos), tendo como objetivo fornecer informação sobre as capacidades transversais que o novo Programa de Matemática português (2007) veio destacar.
- Práticas de discussão matemática no ensino da ÁlgebraPublication . Rodrigues, Cátia; Menezes, Luís; Da Ponte, João PedroMathematical discussions contribute strongly to pupils’ learning, as they put into play a set of social interactions and processes of negotiation of mathematical meanings. This communication analyzes the discussion practices of a mathematics teacher of the 3rd cycle of basic education, in the context of collaborative work. The study is interpretive, qualitative and draws on interviews and participant observation of collaborative work sessions and of the teacher’s lessons, supported in field notes. The results show that the teacher focuses on supporting and extending actions in collective discussions.
- Mathematical pedagogical content knowledge in early childhood education: A study in initial teacher education in PortugalPublication . Figueiredo, Maria Pacheco; Gomes, Helena Margarida dos Santos Vasconcelos; Rodrigues, CátiaThe study aims to explore the specificity of mathematics Pedagogical Content Knowledge in Early Childhood Education Pedagogy. The pedagogy of ECE (Siraj-Blatchford, 2010) and the didactics of ECE (Pramling & Pramling-Samuelsson, 2011) suggest dimensions of knowledge that require strong content and PC knowledge of teachers. Recent studies about PCK of ECE teachers highlight similar specific dimensions: organization of educational environment and interactions with children (Lee, 2010, McCray, 2008, Rojas, 2008). The current framework for ECE Teacher Education in Portugal (since 2007) focuses both content knowledge and subject didactics. PCK has been labelled the 'great unknown' in ECE (Rojas, 2008) in traditions where the child's development is considered as the main knowledge base for ECE (Chen & McNamee, 2006, Cullen, 2005, Hedges & Cullen, 2005). We studied the perspectives of 27 initial teacher education students about knowledge for teaching and about ECE Pedagogy. We used one open-ended questionnaire and students' analysis of episodes focusing children's answers or discourse relevant for mathematics (about high numbers and square root). The questionnaire was anonymous and students’ permission to use the answers was obtained. In the questionnaire, interactions with children (62%) and organization of the educational environment (38%) are highlighted as the most important focus for the teacher. Students suggested tasks that were adult planned and oriented to further the situations presented in the episodes. Very few references to children's exploratory actions (Bonawitz et al., 2011) were made. The specificity of ECE (child initiated activities, e.g.) needs to be further developed in initial teacher education.
- Tarefas matemáticas no ensino da álgebraPublication . Rodrigues, Cátia; Menezes, Luís; Da Ponte, João PedroAs tarefas matemáticas desempenham um papel importante na aprendizagem dos alunos, ao contribuírem para o desenvolvimento do seu pensamento matemático. Nesta comunicação procuramos compreender de que forma o conhecimento mobilizado por uma professora na seleção e exploração de tarefas contribui para essa aprendizagem dos alunos, nomeadamente ao nível da justificação e generalização de raciocínios algébricos. A metodologia envolve a observação de aulas e de sessões de trabalho colaborativo com a professora, que são complementadas pela elaboração de notas de campo. Os resultados mostram que a professora mobiliza diversos aspetos do seu conhecimento didático na seleção das tarefas e na forma de as explorar em sala de aula, tanto no momento de planificação como na ação em aula. Em particular, na aula, acompanha os alunos e favorece os momentos de apresentação e discussão de ideias matemáticas, desafiando-os a justificarem e generalizarem ideias, aspetos importante do conhecimento da prática letiva.
- Avaliação em Matemática: Problemas e desafiosPublication . Menezes, Luís; Santos, Leonor; Gomes, Helena Margarida dos Santos Vasconcelos; Rodrigues, CátiaEste livro contém um conjunto de artigos sobre a temática da avaliação em Matemática, identificando problemas e desafios.
- Práticas discursivas de professores de Matemática na condução de discussões coletivasPublication . Rodrigues, Cátia; Da Ponte, João Pedro; Menezes, LuísMathematical discussions are a tool within the reach of the teacher to promote communication in the mathematics class, as students are asked to present their resolution strategies, to compare them with those of their peers, to justify and argue about the mobilized reasoning in these solutions of a task proposed by the teacher. When class discourse focuses on algebraic ideas, symbolization and generalization are enhanced, as students are encouraged to use increasingly formal language and broaden their thinking beyond the considered cases. This qualitative and interpretative study seeks to understand how three teachers of mathematics in the 3rd cycle of basic education manage and promote discourse during a collective discussion with mathematical tasks with a view to teaching Algebra. The results show that teachers, supported by the teaching actions of eliciting, supporting, informing, and challenging, direct the discourse through a cyclical process of requesting and discussing many ideas and of filtering, seeking to lead students to relate, generalize and justify algebraic ideas.
- Desafios e tensões de professores de matemática na dinamização de discussões em temas da ÁlgebraPublication . Rodrigues, Cátia; Menezes, Luís; Da Ponte, João PedroWhole class discussions have come to be seen in mathematics teaching as a tool available to teachers to promote their students’ learning, insofar as they can involve them in the presentation and justification of previously selected strategies for solving mathematical tasks. In these discussions, the teacher relies on his/her didactic knowledge of mathematics to carry out a set of teaching actions to involve the students in the classroom discourse. The orchestration of these actions generates tensions in teachers, especially when they try to reconcile the discussion of important mathematical ideas with students’ participation. In this research, part of a larger study, we try to understand the tensions that three mathematics teachers face in leading whole class discussions based on algebraic tasks and how they try to overcome them. This qualitative and interpretative study is based on case studies of three teachers who are part of a collaborative group. Data were collected by interview, participant observation of teachers’ classes and collaborative work sessions and documentary analysis. Data analysis is based on a recursive process that led to the construction of formal categories, supported by the theoretical framework. The results show that teachers face diverse tensions when they try to involve the students in sharing and justifying algebraic ideas, recognizing that collaboration and planning the discussions are important mechanisms to help them cope with these tensions.