How teachers interpret and express fractions critically influences their teaching and their students’ fraction knowledge. Internationally, the mathematics education community has been studying ways to enhance pre-service elementary teachers’ rational number knowledge, particularly fractions. To address the challenge of augmenting pre-service teachers’ fraction knowledge warrants theoretical and empirical revisions to standardized practices for teaching fractions. This study investigates how reexamining fractions from a distinctive measuring perspective influences pre-service teachers’ reasoning about fractions. For four 75-minute sessions, 46 pre-service teachers enrolled in a teacher preparation program at a university in the United States revisited fractions from a measuring perspective. They engaged in tasks that focused on comparing continuous quantities and identifying relative magnitudes. The data for this study comprise their pre- and post-tests that assessed how they identify and represent fractions with discrete and continuous models. For each model, we analyzed participants’ reasoning by attending to their written strategies. Findings revealed three main strategies: partition, construction, and symbolic manipulation. In general, participants expressed more strategies on the post-test for all fraction models. Partitioning was the most frequent strategy on the pre- and post-tests. However, the frequencies of strategies changed after the intervention. For example, with all models, there was an increase in partitioning strategy and a decrease in symbolic manipulation strategy. The results highlight affordances of a measuring perspective to support participants to shift from procedural strategies such as symbolic manipulation to more conceptual strategies to identify and represent fractions.
Com o intuito de contribuir para o entendimento de como os professores podem desenvolver a compreensão da geometria, este artigo trata do desenvolvimento discursivo do raciocínio geométrico dos professores através de apropriação de instrumentos enquanto colaborando em um ambiente de geometria dinâmica (AGD) online. Utilizando a teoria da atividade mediada por instrumentos, analisamos o discurso e as ações AGD de um grupo de professores de matemática do ensino fundamental e médio que participaram de um curso de desenvolvimento profissional com duração de um semestre. Trabalhando em pequenos grupos, eles interagiram para resolver problemas geométricos. Nossos resultados mostram que na medida em que se apropriam dos artefatos AGD e transformam seus componentes em instrumentos, os professores desenvolvem o conhecimento e raciocínio geométricos em geometria dinâmica. Nosso estudo contribui para uma compreensão ampla de como os professores desenvolvem o conhecimento matemático para o ensino. Palavras-chave: Geometria dinâmica, Gênese instrumental, Conhecimento para o ensino de matemática, Colaboração.
We draw on the theory of instrumental genesis (RABARDEL; BEGUIN, 2005) and the notion of co-action (HEGEDUS; MORENO-ARMELLA, 2010) to understand how teachers’ instrumentation of dynamic geometry environment (DGE) and how this instrumentation shapes their geometric knowledge. In small groups, six middle and high school mathematics teachers engaged in solving open-ended geometric problems in an online dynamic geometry environment for 15 weeks. Our analysis of their interactions indicates that the co-action between the teachers and the environment helped them appropriate the dragging feature of DGE, which shaped their understanding of geometrical relations, particularly dependencies. Designing tasks that support teachers’ effective appropriation of DGEs requires special attention to the co-active nature of DGEs. This study provides insights into aspects of learners’ collaborative interaction with certain technologies.
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