What strategies could be used by teachers to connect learner preconceptions with mathematically acceptable ways of reflective symmetry?Ī number of strategies have been put in place to mitigate the problem of poor performances in Mathematics. What are the consistent preconceptions of reflective symmetry that are evident in learner responses? Hence here we are making such learner constructions the objects of our analyses, and we do so by raising the following research questions: Because these regularities were not connected with the wider mathematical community's ways of knowing the problem of explaining how students make such constructions seemed intractable as long as 'we' failed to make them objects of 'our' analyses (Cobb, Yackel & Wood, 1992). different children, with consistency far beyond chance, tend to notice the same correspondences? With specific reference to reflective symmetry, Hoyles and Healy (1997) posited that the very existence of these regularities suggested that learners had constructed a socially shared understanding of reflection. Out of the infinitude of correspondences that might be noticed between one event and another, how does it happen that. Berieter (1985:24) has previously pondered such regularities, which were usually not connected with any accepted mathematical definition, through the following question: In our analyses of learner responses we observed a rather disturbing consistency in the way in which they conceptualised reflective symmetry. In this article we analyse Grade 11 learners' responses to a benchmark task on reflective symmetry with the aim of understanding: (a) preconceptions that were common or typical in learners' responses (b) how both practice and previous research findings have explained such regularities and (c) some suggestions that Mathematics Education literature has put forward to mitigate similar challenges. Keywords: preconceptions, reflective symmetry, regularities, SOLO taxonomy, transformationsĬhanges to assessment have always been recognised as an important means of achieving curriculum change in practice and in setting standards (Sieborger & Macintosh, 2004), yet in the past assessment was rarely integrated with the development process. We suggest that if this gap is to be closed, learners need to construct these reflections physically so that they may think of reflections beyond motion. While this understanding is useful in some cases, it is not an essential aspect of mapping understanding, which is critical for application in function notations and other analytical geometry contexts. The results indicated that 85% of learner responses reflect a motion understanding of reflections, where learners considered geometric figures as physical motions on top of the plane. Our framework for analysing the responses was based on the taxonomy of structure of the observed learning outcome. A total of 235 Grade 11 learners, from 13 high schools that participate in the First Rand Foundation-funded Mathematics Education project in the Eastern Cape, responded to a task on reflective symmetry. Such a gap inhibits further understanding and application, and hence needed to be investigated. Literature suggests that the very existence of such regularities indicates a gap between what learners know and what they need to know. Consistencies far beyond chance: an analysis of learner preconceptions of reflective symmetryĮducation Department, FRF Mathematics Education Chair, Rhodes University, South Africa article reports on regularities observed in learners' preconceptions of reflective symmetry.
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