'The Area of a Triangle is 1800C’: an analysis of Learners' Idiosyncratic Geometry Responses through the Lenses of Vygotsky's Theory of Concept Formation
- Mhlolo, Michael K, Schäfer, Marc
- Authors: Mhlolo, Michael K , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141127 , vital:37946 , DOI: 10.1080/10288457.2013.826973
- Description: In this paper, we focus specifically on Vygotsky's theory of concept formation to gain initial insights into seemingly garbled and incoherent connections we observed in learners' responses to a geometry task. While acknowledging research that is supportive of the van Hiele model as being useful when analysing concept formation in geometry, in this paper we use empirical evidence to argue that today's geometry requires learners to reason with many tools which the van Hiele model does not seem to accommodate. Working with 470 written responses of 12–14-year-old learners to a geometry task, we then explored the potential of Vygotsky's theory of concept formation as an alternative framework. We conclude by discussing the paper's contribution both to theory and to classroom practice.
- Full Text:
- Date Issued: 2013
- Authors: Mhlolo, Michael K , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141127 , vital:37946 , DOI: 10.1080/10288457.2013.826973
- Description: In this paper, we focus specifically on Vygotsky's theory of concept formation to gain initial insights into seemingly garbled and incoherent connections we observed in learners' responses to a geometry task. While acknowledging research that is supportive of the van Hiele model as being useful when analysing concept formation in geometry, in this paper we use empirical evidence to argue that today's geometry requires learners to reason with many tools which the van Hiele model does not seem to accommodate. Working with 470 written responses of 12–14-year-old learners to a geometry task, we then explored the potential of Vygotsky's theory of concept formation as an alternative framework. We conclude by discussing the paper's contribution both to theory and to classroom practice.
- Full Text:
- Date Issued: 2013
Consistencies far beyond chance: an analysis of learner preconceptions of reflective symmetry
- Mhlolo, Michael K, Schäfer, Marc
- Authors: Mhlolo, Michael K , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141139 , vital:37947 , DOI: 10.15700/saje.v33n2a686
- Description: This article reports on regularities observed in learners’ preconceptions of reflective symmetry. Literature suggests that the very existence of such regularities indicates a gap between what learners know and what they need to know. Such a gap inhibits further understanding and application, and hence needed to be investigated. 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. Our framework for analysing the responses was based on the taxonomy of structure of the observed learning outcome. 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. 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. 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.
- Full Text:
- Date Issued: 2013
- Authors: Mhlolo, Michael K , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141139 , vital:37947 , DOI: 10.15700/saje.v33n2a686
- Description: This article reports on regularities observed in learners’ preconceptions of reflective symmetry. Literature suggests that the very existence of such regularities indicates a gap between what learners know and what they need to know. Such a gap inhibits further understanding and application, and hence needed to be investigated. 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. Our framework for analysing the responses was based on the taxonomy of structure of the observed learning outcome. 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. 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. 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.
- Full Text:
- Date Issued: 2013
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