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- O papel dos modelos no processo de construção dos conceitos: O caso da divisão dos números racionaisPublication . Pinto, Hélia; Ponte, João Pedro daNeste capítulo analisa-se o papel dos modelos na aprendizagem dos números racionais.
- Exploratory mathematics teaching and the development of students’ use of representations and reasoning processes: An illustration with rational numbersPublication . Ponte, João Pedro da; Quaresma, MarisaThis chapter presents a perspective about the exploratory approach as a possible way to enact inquiry based mathematics teaching. In this approach, the teacher, instead of beginning the class by presenting explanations and examples to the students, proposes them to work on tasks that may lead to the construction of new knowledge. We use as illustration the work of a grade 6 class of students solving tasks involving rational numbers. Our aim is to know how students use representations and reasoning processes, seeking to find out how they deal with different representations and how they formulate generalizations and justifications. We follow a qualitative and interpretative approach, with participant observation of a teaching experiment that included five lessons that were integrally videotaped and transcribed. We analyse episodes from the work of the students in two tasks, one involving a complex relationship between fractions and the other involving the use of fractions as operators. The results show that when solving a task that involves rational numbers given as fractions, the students mostly use the decimal representation, with which they feel rather comfortable. In another task, involving rational numbers as operators, most students use fractions, but some of them also use of decimal numbers and pictorial representations. In both cases, the students chose the representation that they considered best suit their needs. In their written work, the students justify their choices by presenting the computations done when solving a task, adding explanations in natural language. Just by themselves, they are able to use counterexamples to refute a statement, and, during whole class discussions, prompted by the teacher, they are able to make generalizations and justifications based on definitions.
- Different Levels of Sophistication in Solving and Expressing Mathematical Problems with Digital ToolsPublication . Jacinto, Hélia; Nobre, Sandra; Carreira, Susana; Amado, NéliaAll over the world, several organizations have nurtured the development of students’ problem-solving abilities by organizing competitions and tournaments of different kinds. This is the case of the Mathematical Competitions SUB12 and SUB14, promoted by the University of Algarve, addressing students from grades 5 to 8 (10–14 year olds) in the south of Portugal. To each proposed problem, participants are required to explain their problem-solving process and find ways to express their thinking. They may use any of the digital tools they have available and they find useful for solving a given problem. Our research has uncovered the aptitudes of young competitors in taking advantage of everyday digital tools and its representational expressiveness to give form and substance to their reasoning and strategies. Another emerging aspect is the apparent existence of different degrees of robustness of the solutions submitted, mainly in terms of the strategies that competitors develop, with a particular technological tool, to solve the problems. In this chapter, we are taking a selection of solutions submitted to two problems, in which competitors resort to GeoGebra, in one case, and to Excel, in the other. We offer a proposal for identifying levels of sophistication and robustness of technology-based solutions to the problems, according to the characteristics of the tool use and its connection to the conceptual models underlying students’ thinking on the problems.
- A Model of Mathematical Problem Solving with Technology: The Case of Marco Solving-and-Expressing Two Geometry ProblemsPublication . Carreira, Susana; Jacinto, HéliaResearch has long been using analytical tools to describe the processes students engage in when solving non-routine mathematical problems. In this chapter we describe and discuss our progress on devising and implementing an analytical tool that aims to account for the use of technological tools by combining a mathematical problem solving model with a digital problem solving framework. By means of the Mathematical Problem Solving with Technology model (MPST) we report the case of Marco using technologies for solving two problems from a beyond school competition. Results show that Marco’s choice of the tools is grounded on an explicit knowledge of their affordances and how they enhance his mathematical thinking, mainly by triggering visual approaches that support the development of conceptual models for solving-and-expressing the solutions to the problems.
- Theory and practice of lesson study in mathematics around the worldPublication . Huang, Rongjin; Takahashi, Akihiko; Ponte, João Pedro daLesson study, as powerful teacher professional development approach, originating in Asia, has spread globally.
- Echoes and influences of Realistic Mathematics Education in PortugalPublication . Ponte, João Pedro da; Brocardo, JoanaThis chapter traces the connections between Realistic Mathematics Education (RME) and Portuguese developments in mathematics education in terms of research studies and curriculum development. The basis for this work is a literature review of papers and other documents, with special attention to the period 2005-2015, and research studies organised by mathematical topic. Although there is no research group in Portugal that is perfectly aligned with RME principles and curriculum materials, noticeable influences may be seen in the frequent references made in some research groups to key RME ideas, notably the importance of students working from tasks in meaningful contexts, the role of representations and models to support students’ thinking, and the levels of students’ mathematical activity. This is most noticeable in conceptual frameworks for developmental research studies in the area of number and in the use of realistic contexts in task design, and it is also apparent in the official 2007 Portuguese curriculum document.
- Lesson study as a learning context in mathematics educationPublication . Ponte, João Pedro da; Wake, Geoffrey; Quaresma, MarisaThis chapter presents lesson study, a professional development process that originated in Japan, describing its main features with emphasis on how it may be regarded as a special form of teachers’ research on their own practice. We indicate some adaptations of lesson study to fit different purposes and pay attention to the participants’ experiences of collaboration, reflection and work in communities of practice. To illustrate these features, we present two case studies, one from Portugal and another from the UK.
- A colaboração profissional em estudos de aula sob a perspectiva de professores do ensino básicoPublication . Richit, Adriana; Ponte, João Pedro daDestacamos aspetos da colaboração docente em um processo de desenvolvimento profissional baseado em estudos de aula - abordagem baseada na prática docente e que assume natureza colaborativa e reflexiva.
- Desenvolvendo o raciocínio relacional nos alunos dos primeiros anosPublication . Ponte, João Pedro da; Cerca, Mónica RaquelNeste capítulo procuramos mostrar como se pode desenvolver o raciocínio relacional dos alunos do 3.º ano (alunos com 8-9 anos) ao longo de uma experiência de ensino, dando importância às relações de igualdade e desigualdade e à capacidade de generalização a partir de tarefas que envolvem quantidades desconhecidas.
- O conhecimento para ensinar Matemática na prática letiva de uma futura professora do 2º ciclo: O conceito de percentagemPublication . Ferreira, Nadia; Ponte, João Pedro daNeste capítulo caracterizamos a prática letiva de uma professora estagiária, Berta, evidenciando as suas ações e conhecimento no momento de planificação e concretização. Damos atenção à natureza do conhecimento para ensinar Matemática, com foco no conhecimento sobre os alunos e tarefas e nas ações e comunicação durante a exploração de uma tarefa. Para isso recolhemos e analisámos dados de entrevistas, planificações, reflexões e vídeos das aulas observadas. Analisamos as opções e dificuldades vividas perante situações previstas e imprevistas na sala de aula. O caso de Berta evidencia a importância de antecipar possíveis resoluções dos alunos, preparar questões para a fase de exploração da tarefa e identificar representações eficientes para realçar as ideias matemáticas a construir.
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