Potential Gaps during the Transition from the Embodied through Symbolic to Formal Worlds of Reflective Symmetry:
- Mhlolo, Michael K, Schäfer, Marc
- Authors: Mhlolo, Michael K , Schäfer, Marc
- Date: 2014
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141117 , vital:37945 , https://doi.org/10.1080/10288457.2014.925269
- Description: Even though reflective symmetry is heavily embedded in the everyday, learners continue to experience challenges when they mathematize concepts from this informal/everyday context. In this article we argue that symmetry exists in nature, it can also be symbolized algebraically and it can be abstracted into the world of axioms and theorems. We problematize this multiple nature of symmetry which on one hand is supportive and on the other acts as a contributory factor to learners' gaps in knowledge. Tall's three worlds of mathematics helped us to show the transition of symmetry from the embodied through symbolic to the formal world and the inherent gaps attributed to the shifts in thinking thereof. We then used this same framework to analyse learners' responses to a reflective symmetry task. The results show that many learner responses could be explained explicitly by the lack of flexibility in the applicability of experiences in the embodied world of reflective symmetry.
- Full Text:
- Date Issued: 2014
- Authors: Mhlolo, Michael K , Schäfer, Marc
- Date: 2014
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141117 , vital:37945 , https://doi.org/10.1080/10288457.2014.925269
- Description: Even though reflective symmetry is heavily embedded in the everyday, learners continue to experience challenges when they mathematize concepts from this informal/everyday context. In this article we argue that symmetry exists in nature, it can also be symbolized algebraically and it can be abstracted into the world of axioms and theorems. We problematize this multiple nature of symmetry which on one hand is supportive and on the other acts as a contributory factor to learners' gaps in knowledge. Tall's three worlds of mathematics helped us to show the transition of symmetry from the embodied through symbolic to the formal world and the inherent gaps attributed to the shifts in thinking thereof. We then used this same framework to analyse learners' responses to a reflective symmetry task. The results show that many learner responses could be explained explicitly by the lack of flexibility in the applicability of experiences in the embodied world of reflective symmetry.
- Full Text:
- Date Issued: 2014
'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
The nature and quality of the mathematical connections teachers make:
- Mhlolo, Michael K, Schäfer, Marc, Venkat, Hamsa
- Authors: Mhlolo, Michael K , Schäfer, Marc , Venkat, Hamsa
- Date: 2012
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140893 , vital:37927 , https://0-hdl.handle.net.wam.seals.ac.za/10520/EJC120887
- Description: Current reforms in mathematics education emphasise the need for pedagogy because it offers learners opportunities to develop their proficiency with complex high-level cognitive processes. One has always associated the ability to make mathematical connections, together with the teacher's role in teaching them, with deep mathematical understanding. This article examines the nature and quality of the mathematical connections that the teachers' representations of those connections enabled or constrained. The researchers made video recordings of four Grade 11 teachers as they taught a series of five lessons on algebra-related topics. The results showed that the teachers' representations of mathematical connections were either faulty or superficial in most cases. It compromised the learners' opportunities for making meaningful mathematical connections. The researchers concluded by suggesting that helping teachers to build their representation repertoires could increase the effectiveness of their instructional practices.
- Full Text:
- Date Issued: 2012
- Authors: Mhlolo, Michael K , Schäfer, Marc , Venkat, Hamsa
- Date: 2012
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140893 , vital:37927 , https://0-hdl.handle.net.wam.seals.ac.za/10520/EJC120887
- Description: Current reforms in mathematics education emphasise the need for pedagogy because it offers learners opportunities to develop their proficiency with complex high-level cognitive processes. One has always associated the ability to make mathematical connections, together with the teacher's role in teaching them, with deep mathematical understanding. This article examines the nature and quality of the mathematical connections that the teachers' representations of those connections enabled or constrained. The researchers made video recordings of four Grade 11 teachers as they taught a series of five lessons on algebra-related topics. The results showed that the teachers' representations of mathematical connections were either faulty or superficial in most cases. It compromised the learners' opportunities for making meaningful mathematical connections. The researchers concluded by suggesting that helping teachers to build their representation repertoires could increase the effectiveness of their instructional practices.
- Full Text:
- Date Issued: 2012
Towards empowering learners in a democratic mathematics classroom: to what extent are teachers' listening orientations conducive to and respectful of learners' thinking?
- Mhlolo, Michael K, Schäfer, Marc
- Authors: Mhlolo, Michael K , Schäfer, Marc
- Date: 2012
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140882 , vital:37926 , https://0-hdl.handle.net.wam.seals.ac.za/10520/EJC129235
- Description: In an effort to make education accessible, to 'heal the divisions of the past and establish a society based on democratic values', the South African Department of Education claims that a series of mathematics reforms that has so far been introduced is underpinned by the principles of 'social justice, fundamental human rights and inclusivity'. Critics however argue that the system has remained 'undemocratic' in that those groups of learners who were supposed to be 'healed' continue to underperform and hence be disempowered. In this study, we conceptualised a democratic and mathematically empowering classroom as one that is consistent with the principle of inclusivity and in which a hermeneutic listening orientation towards teaching promotes such a democratic and mathematically empowering learning environment. We then worked with three different orientations teachers might have towards listening in the mathematics classroom: evaluative, interpretive and hermeneutic. We then used these orientations to analyse 20 video-recorded lessons with a specific focus on learners' unexpected contributions and how teachers listened and responded to such contributions. The results were consistent with the literature, which shows that teachers tend to dismiss learners' ways of thinking by imposing their own formalised constructions.
- Full Text:
- Date Issued: 2012
- Authors: Mhlolo, Michael K , Schäfer, Marc
- Date: 2012
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140882 , vital:37926 , https://0-hdl.handle.net.wam.seals.ac.za/10520/EJC129235
- Description: In an effort to make education accessible, to 'heal the divisions of the past and establish a society based on democratic values', the South African Department of Education claims that a series of mathematics reforms that has so far been introduced is underpinned by the principles of 'social justice, fundamental human rights and inclusivity'. Critics however argue that the system has remained 'undemocratic' in that those groups of learners who were supposed to be 'healed' continue to underperform and hence be disempowered. In this study, we conceptualised a democratic and mathematically empowering classroom as one that is consistent with the principle of inclusivity and in which a hermeneutic listening orientation towards teaching promotes such a democratic and mathematically empowering learning environment. We then worked with three different orientations teachers might have towards listening in the mathematics classroom: evaluative, interpretive and hermeneutic. We then used these orientations to analyse 20 video-recorded lessons with a specific focus on learners' unexpected contributions and how teachers listened and responded to such contributions. The results were consistent with the literature, which shows that teachers tend to dismiss learners' ways of thinking by imposing their own formalised constructions.
- Full Text:
- Date Issued: 2012
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