'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:
- 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.
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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:
- 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.
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Dynamic geometry software as a dynamic tool for spatial exploration:
- Holzl, Reinhard, Schäfer, Marc
- Authors: Holzl, Reinhard , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140871 , vital:37925 , https://hdl.handle.net/10520/EJC146065
- Description: Dynamic Geometry Software (DGS) such as GeoGebra, Geometer's Sketchpad and Cabri Geometry offer a wealth of opportunities for an exploratory style of teaching and learning Mathematics, particularly in the exploration of space and shape. The new Curriculum and Assessment Policy Statement (CAPS) foregrounds the use of spatial skills and properties of shapes and objects "to identify, pose and solve problems creatively and critically" (South Africa. DBE, 2011, p. 9). Although the South African Mathematics curriculum no longer places importance on traditional Euclidean construction by means of straightedge and compass, DGS can nonetheless be used to engage with fundamental ideas relating to geometric shapes, symmetry and transformations. This was the impetus behind encouraging the use of GeoGebra in four township schools in the Grahamstown Education District whose Mathematics teachers participate in an ongoing in-service research and development programme hosted by Rhodes University.
- Full Text:
- Authors: Holzl, Reinhard , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140871 , vital:37925 , https://hdl.handle.net/10520/EJC146065
- Description: Dynamic Geometry Software (DGS) such as GeoGebra, Geometer's Sketchpad and Cabri Geometry offer a wealth of opportunities for an exploratory style of teaching and learning Mathematics, particularly in the exploration of space and shape. The new Curriculum and Assessment Policy Statement (CAPS) foregrounds the use of spatial skills and properties of shapes and objects "to identify, pose and solve problems creatively and critically" (South Africa. DBE, 2011, p. 9). Although the South African Mathematics curriculum no longer places importance on traditional Euclidean construction by means of straightedge and compass, DGS can nonetheless be used to engage with fundamental ideas relating to geometric shapes, symmetry and transformations. This was the impetus behind encouraging the use of GeoGebra in four township schools in the Grahamstown Education District whose Mathematics teachers participate in an ongoing in-service research and development programme hosted by Rhodes University.
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The nature of geometry instruction and observed learning-outcomes opportunities in Nigerian and South African high schools:
- Atebe, Humphrey U, Schäfer, Marc
- Authors: Atebe, Humphrey U , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140919 , vital:37929 , DOI: 10.1080/10288457.2011.10740712
- Description: The purpose of this qualitative case study involving six secondary school teachers was to obtain insight into how geometry is taught in selected Nigerian and South African high schools. It also aimed, by making use of the van Hiele model of geometry instruction, to elucidate what possible learning opportunities observed instructional methods could offer learners in the subject. The sample comprised three mathematics teachers from Nigeria and three mathematics teachers from South Africa, all of whom were selected using purposive sampling techniques. Instructional activities in six geometry classrooms were recorded on videotape. The van Hiele learning phases provided the framework for data analysis. The findings of this study indicate that observed teaching methods in geometry classrooms in the participating schools offer learners scant opportunity to learn geometry.
- Full Text:
- Authors: Atebe, Humphrey U , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140919 , vital:37929 , DOI: 10.1080/10288457.2011.10740712
- Description: The purpose of this qualitative case study involving six secondary school teachers was to obtain insight into how geometry is taught in selected Nigerian and South African high schools. It also aimed, by making use of the van Hiele model of geometry instruction, to elucidate what possible learning opportunities observed instructional methods could offer learners in the subject. The sample comprised three mathematics teachers from Nigeria and three mathematics teachers from South Africa, all of whom were selected using purposive sampling techniques. Instructional activities in six geometry classrooms were recorded on videotape. The van Hiele learning phases provided the framework for data analysis. The findings of this study indicate that observed teaching methods in geometry classrooms in the participating schools offer learners scant opportunity to learn geometry.
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“As soon as the four sides are all equal, then the angles must be 90° each”: children's misconceptions in geometry
- Atebe, Humphrey U, Schäfer, Marc
- Authors: Atebe, Humphrey U , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140983 , vital:37934 , DOI: 10.1080/10288457.2008.10740634
- Description: This study describes Nigerian and South African students' conceptual understanding in high school geometry based on the van Hiele model of geometric thinking levels. The study further highlights students' misconceptions in school geometry. Concepts of triangles and quadrilaterals were investigated among 36 mathematics learners drawn from grades 10 through 12 who participated in this study. The tasks included identifying and naming shapes, sorting of shapes, stating the properties of shapes, defining of shapes and establishing class inclusions of shapes. The results indicated that many Nigerian and South African high school learners in Grade 10, 11 and 12 hold a number of misconceptions about geometric concepts of triangles and quadrilaterals.
- Full Text:
- Authors: Atebe, Humphrey U , Schäfer, Marc
- Date: 2013
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140983 , vital:37934 , DOI: 10.1080/10288457.2008.10740634
- Description: This study describes Nigerian and South African students' conceptual understanding in high school geometry based on the van Hiele model of geometric thinking levels. The study further highlights students' misconceptions in school geometry. Concepts of triangles and quadrilaterals were investigated among 36 mathematics learners drawn from grades 10 through 12 who participated in this study. The tasks included identifying and naming shapes, sorting of shapes, stating the properties of shapes, defining of shapes and establishing class inclusions of shapes. The results indicated that many Nigerian and South African high school learners in Grade 10, 11 and 12 hold a number of misconceptions about geometric concepts of triangles and quadrilaterals.
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