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In this chapter, the theoretical construct of guided reinvention is extended to include desirable pedagogical practices for teachers implementing RME sequences. First, we explain what a guided reinvention teaching approach looks like and how it evolved out of over 25 years of research. We then articulate the planning and teaching practices of guided reinvention teachers and describe how those practices move beyond what many call “inquiry approaches” to mathematics teaching. We end the chapter by offering a set of learning goals that professional developers might use when mentoring aspiring guided reinvention teachers.
... As tarefas contextualizadas desempenham um papel central na EMR (e.g., Gravemeijer & Doorman, 1999;Stephan, Underwood-Gregg, & Yackel, 2014) . O termo Educação Matemática Realista está relacionado com o verbo holandês zich realiseren que significa imaginar (Van den Heuvel-Panhuizen, 2003). ...
... Neste sentido, os recursos educacionais são construídos objetivando apoiar os alunos na reinvenção de ideias e conceitos em curtos períodos de tempo, através da sequenciação de tarefas, da seleção de materiais e da orientação do professor. As sequências de aprendizagem são construídas de modo a que os conhecimentos vão emergindo à medida que os alunos se empenham na resolução das tarefas propostas (Stephan et al., 2014). Este é um dos aspetos em que a abordagem da EMR se distingue, na medida em que a trajetória de aprendizagem se desenrola de tal modo que a matemática formal emerge da atividade matemática dos alunos (Gravemeijer & Doorman, 1999). ...
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Este trabalho resulta do envolvimento, no âmbito da disciplina de Matemática A, dos alunos duma turma do décimo segundo ano numa experiência de ensino com o objetivo de os apoiar na reinvenção guiada das fórmulas das operações combinatórias dos arranjos sem repetição e das combinações, num ambiente exploratório de ensino aprendizagem. Numa perspetiva de Educação Matemática Realista, procurámos analisar o modo como os alunos reinventaram estas duas fórmulas recorrendo ao Modelo de Pensamento Combinatório dos Alunos desenvolvido por Lockwood (2013). Na investigação realizada, os dados foram recolhidos através da observação participante, gravações em vídeo de aulas e recolha documental. Os resultados sugerem que o sucesso na reinvenção das fórmulas das operações combinatórias está relacionado com a familiarização do aluno com os processos de contagem que conduzem às operações algébricas expressas nas diferentes expressões, o que reflete as relações descritas entre as componentes do modelo de Lockwood. Os dados recolhidos indicam uma relação muito forte entre os processos de contagem e as fórmulas/expressões que traduzem a resposta aos problemas em causa. Verificámos que o insucesso na execução dos processos de contagem conduziu ao insucesso no estabelecimento de uma fórmula/expressão que represente o cardinal do conjunto de resultados, o que sugere a necessidade de uma reformulação das tarefas, de modo a apoiar os alunos na reinvenção guiada das fórmulas de um modo mais eficaz.
... 2 Theorising the pedagogy of Realistic Mathematics Education: Guided reinvention and the devolution of authority As a theory of learning, RME emphasises the role of emergent mathematics (Gravemeijer & Stephan, 2002): students move from models of their informal activity in realisable contexts towards more formal mathematics via a process of progressive mathematisation of models (Van den Heuvel-Panhuizen, 2003). This development is supported by "guided reinvention" (Stephan, Underwood-Gregg, & Yackel, 2014), requiring particular pedagogic practices on the part of the teacher, and corresponding student modes of participation. These include the following: social norms of communication (e.g., students listen to others' contributions, and ask questions when they disagree or do not understand; teacher avoids saying if strategies/ solutions are right or wrong); sociomathematical norms regarding the nature of mathematical statements (e.g., the class agrees on what counts as a mathematically different solution, cf. ...
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Realistic Mathematics Education (RME) relies on the pedagogy of guided reinvention, in which opportunities for learning are created through the teacher’s orchestration of whole-class mathematical discussion towards a specific goal. However, introducing an RME approach to students who are accustomed to traditional teaching requires a substantial shift in roles, particularly with respect to the devolution of authority from teacher to student. In this study, we worked with low-attaining students, implementing RME to improve understanding of fractions. The analysis highlights how the introduction of guided reinvention is supported by extended wait time and teacher neutrality, but also by teachers’ appropriation of student strategies as a basis for supporting shared authority in the joint construction of mathematical ideas. The article considers the relationship between guided reinvention, appropriation and student agency.
... Students could choose between two problem-solving tasks: one based on the most popular engineering problem from the first pilot study and one set in the most popular bio-medical context. Both tasks had the same structure based on the model of guided reinvention (Stephan, Underwood-Gregg, & Yackel, 2014). The ways the students engaged with the new materials were observed and ...
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This study shows that using authentic contexts for learning differential equations in a differentiation-by-interest setting can enhance students’ beliefs about the relevance of mathematics. The students in this study were studying advanced mathematics (wiskunde D) at upper secondary school in the Netherlands. These students are often not aware of the relevance of the mathematics they have to learn in school. More insights into the application of mathematics in other sciences can be beneficial for these students in terms of preparation for their future study and career. A course differentiating by student interest with new context-rich curriculum materials was developed in order to enhance students’ beliefs about the relevance of mathematics. The intervention aimed at teaching differential equations through guided small-group tasks in scientific, medical or economical contexts. The results show that students’ beliefs about the relevance of mathematics improved, and they appreciated experiencing how the mathematics was applied in real-life situations.
... A abordagem da Educação Matemática Realista, ancorada nas ideias de Freudenthal (1973), dá uma relevância muito particular às tarefas com contextos reais (Gravemeijer & Doorman, 1999;Stephan, Underwood-Gregg, & Yackel, 2014). Nestes contextos, para além das situações reais do quotidiano, inclui-se também o mundo da fantasia e tudo o que esteja ao alcance da imaginação dos alunos (Shannon, 2007). ...
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Formular problemas é uma atividade essencial no contexto da formação de professores que ensinam matemática. A construção de trilhos matemáticos permite conjugar a formulação de problemas com aspetos lúdicos da atividade matemática, promovendo uma afetividade positiva com a disciplina e dando azo ao desenvolvimento da criatividade matemática. Neste sentido, desafiámos dois grupos de futuros professores, em duas instituições de formação inicial diferentes, a desenhar trilhos matemáticos, focando a nossa análise na natureza das tarefas construídas e na criatividade matemática manifestada nas produções apresentadas. Os participantes revelaram algumas dificuldades na compreensão da noção de problema e, ao nível da capacidade de formulação de problemas, a experiência realizada permitiu perceber que prevalecem bastantes dificuldades, sobretudo nos futuros professores do 3.º ciclo do ensino básico e do ensino secundário, em comparação com os seus colegas dos 1.º e 2.º ciclos do ensino básico. Encontrámos algumas diferenças entre os grupos de futuros professores no que toca à flexibilidade, elemento essencial da criatividade matemática. A construção de trilhos matemáticos mostrou ser uma atividade promissora no âmbito da formação inicial de professores, possibilitando aprofundar conceitos e processos matemáticos e, ao mesmo tempo, aspetos centrais da didática da matemática, como o desenho de tarefas.
Chapter
In this chapter, we discuss the question of how we can encourage mathematics education to shift towards more inquiry-oriented practices in schools and what role textbooks and teachers play in such a reform. The stage is set by an exposition on the need for curriculum innovation in light of the demands of the twenty-first century. This points to a need to address goals in the area of critical thinking, problem solving, collaborating, and communicating. However, previous efforts to effectuate a change in mathematics education in that direction have not been very successful. This is illustrated by experiences in the Netherlands. In relation to this, the limitations of transforming education using textbooks and problems with up-scaling are discussed. To find ways to address these problems, an inventory is made of what can be learned from decades of experimenting with reform mathematics education while trying to achieve the very goals that are discerned as crucial for the twenty-first century. On the basis of this inventory, suggestions are made for shaping textbooks in such a manner that they may better support this kind of transformation. At the same time it is pointed out that the latter requires a complementary effort in teacher professionalization and a well-considered alignment of both efforts.
Chapter
Mathematics classroom instruction, often seen as a key contributing factor to students’ learning, has remained virtually unchanged for the past several decades. Efforts to improve the quality of mathematics education have led to multiple approaches and research that target different contributing factors but lack a systematic account of diverse approaches and practices for improving mathematics instruction. This book is thus designed to survey, synthesize, and extend current research on specific approaches and practices that are developed and used in different education systems for transforming mathematics instruction. In this introduction chapter, we highlight the background of this book project, its purposes, and what can be learned from reading this book.
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Small-group problem solving was used as a primary instructional strategy for all aspects of second-grade mathematics, including computation, for the entire school year. This gave rise to learning opportunities that do not typically occur in traditional classrooms, including those that arise from collaborative dialogue as well as from the resolution of conflicting points of view. The nature of these learning opportunities is elaborated and illustrated. The manner in which the teacher used paradigm cases as she initiated and guided discussion of obligations and expectations to make possible the mutual construction of classroom norms for cooperative learning is also illustrated. This and the use of cognitively based activities designed to be problematic for children at a variety of conceptual levels are the crucial features of a cooperative learning environment in the absence of extrinsic rewards.
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This paper sets forth a way of interpreting mathematics classrooms that aims to account for how students develop mathematical beliefs and values and, consequently, how they become intellectually autonomous in mathematics. To do so, we advance the notion of sociomathematical norms, that is, normative aspects of mathematical discussions that are specific to students' mathematical activity. The explication of sociomathematical norms extends our previous work on general classroom social norms that sustain inquiry-based discussion and argumentation. Episodes from a second-grade classroom where mathematics instruction generally followed an inquiry tradition are used to clarify the processes by which sociomathematical norms are interactively constituted and to illustrate how these norms regulate mathematical argumentation and influence learning opportunities for both the students and the teacher. In doing so, we both clarify how students develop a mathematical disposition and account for students' development of increasing intellectual autonomy in mathematics. In the process, the teacher's role as a representative of the mathematical community is elaborated.
Article
Increasing emphasis on “Algebra for all” (NCTM 1997a, 1997b) compels educators to identify and address fundamental ideas that build the foundations for algebraic thinking and reasoning. Identifying these foundational concepts and developing appropriate instructional approaches are the focuses of our work. One area in which students often experience difficulty is adding and subtracting algebraic expressions. Although students may be able to memorize a procedure, such as “distribute the negative” when subtracting algebraic expressions, they are often unable to make sense of this procedure. Our work suggests that part of students' difficulty in this area is that they do not conceptualize an algebraic expression as a composite unit . In the paragraphs below, we explain what is meant by composite units and how this construct helped frame our development of an instructional sequence to help students make sense of, and find meaning in, algebraic expressions and operations on algebraic expressions.
Chapter
In this chapter, the design of an instructional sequence dealing with flexible mental computation strategies for addition and subtraction up to one hundred, is taken as an instance for elaborating on the role of ‘emergent models’ as an RME design heuristic. It is explicated how the label ‘emergent’ refers both to the character of the process by which models emerge within RME, and to the process by which these models support the emergence of formal mathematical ways of knowing. According to the emergent-models design heuristic, the model first comes to the fore as a model of the students’ situated informal strategies. Then, over time the model gradually takes on a life of its own. The model becomes an entity in its own right and starts to serve as a model for more formal, yet personally meaningful, arithmetical reasoning. The analysis of the exemplary instructional sequence, is used to show that the transition from model-of to model-for involves the constitution of a new mathematical reality that can be denoted ‘formal’ in relation to the original starting points of the students. Furthermore, attention is given to the dynamical character of the emergent model; there is not one model, but the model is actually shaped as a cascade of inscriptions. The latter observation forms the starting point for a more detailed discussion of the role of these individual inscriptions in the learning process.