A Case Study of Two Selected Teachers as they Integrated Dynamic Geometry Software as a Visualisation Tool in Teaching Geometry:
- Authors: Mavani, Deepak , Mavani, Beena , Schäfer, Marc
- Date: 2018
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141076 , vital:37942 , https://doi.org/10.1080/18117295.2018.1522716
- Description: This paper reports on an aspect of a larger research study conceptualised within a teacher development project in Mthatha, Eastern Cape Province. The project was initiated with the objective to develop appropriate skills to use dynamic geometry software (DGS) effectively and strategically as a teaching and learning tool for mathematics. The study reported in this paper aims specifically to ascertain how selected mathematics teachers integrated co-developed technologically aided visualisation tools in the observed lessons. The case study involved two teachers from different schools. The data sources were the classroom observations followed by stimulated reflective interviews with the teachers. The data were analysed to study the use of DGS visualisation tools in relation to Kilpatrick’s framework of teaching proficiency.
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- Date Issued: 2018
Enactivism as a powerful theoretical framework for research and tool to reflect on my own role as a supervisor:
- Authors: Schäfer, Marc
- Date: 2016
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141102 , vital:37944 , DOI: 10.1080/18117295.2016.1243938
- Description: Enactivism, as an interesting and useful theoretical underpinning is gaining traction in Mathematics Education research. It forms the central theme of this paper whose aim is two-fold: first to describe and engage with how elements of enactivism informed a PhD study, both on a theoretical and analytical level, and second to reflect on the enacted role of the supervisor of this study. Despite the inevitable embodied relationship between the supervisor and the supervised PhD project, it is not often written about. This paper thus attempts to address this. The PhD study in question used constructs of enactivism such as autonomy, sense-making, emergence, embodiment and experience to design a pre-service mathematics education programme and then explored the growth of student teachers’ mathematical identity and disposition in their development of becoming mathematics teachers. The PhD supervision process was framed by the enactivist notion that learning and the construction of meaning and knowledge is co-created by the lecturer, the student and the particular context.
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- Date Issued: 2016
Teacher code switching consistency and precision in a multilingual mathematics classroom:
- Authors: Chikiwa, Clemence , Schäfer, Marc
- Date: 2016
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141087 , vital:37943 , https://doi.org/10.1080/18117295.2016.1228823
- Description: This paper reports on a study that investigated teacher code switching consistency and precision in multilingual secondary school mathematics classrooms in South Africa. Data was obtained through interviewing and observing five lessons of each of three mathematics teachers purposively selected from three township schools in the Eastern Cape Province. Elements of Gumperz and Mercer’s work on lesson categories and Dowling’s Domains of Mathematical Practice were used to analyse data. Results showed that code switching frequency in general was inconsistent across different lessons for the same teacher.
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- Date Issued: 2016
Potential Gaps during the Transition from the Embodied through Symbolic to Formal Worlds of Reflective Symmetry:
- 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.
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- 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
- 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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- Date Issued: 2013
Consistencies far beyond chance: an analysis of learner preconceptions of reflective symmetry
- 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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- Date Issued: 2013
Dynamic geometry software as a dynamic tool for spatial exploration:
- 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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- Date Issued: 2013
The nature of geometry instruction and observed learning-outcomes opportunities in Nigerian and South African high schools:
- 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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- Date Issued: 2013
“As soon as the four sides are all equal, then the angles must be 90° each”: children's misconceptions in geometry
- 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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- Date Issued: 2013
Teacher education in post-apartheid South Africa: navigating a way through competing state and global imperatives for change
- Authors: Schäfer, Marc , Wilmot, P Dianne
- Date: 2012
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140905 , vital:37928 , DOI: 10.1007/s11125-012-9220-3
- Description: This article focuses on teacher education in post-apartheid South Africa. It argues that the restructuring and reorganization of teacher education is at the nexus of the axes of tension created by national and global imperatives for change. Along with the dismantling of apartheid and the transition to a free and democratic state in 1994 came the urgent need for social reconstruction, democratization, redress, social justice, and equity. At the same time, and as part of a global context, the country needed global competitiveness, human capital development, global skills, international standards, and accountability. These competing modernist discourses have informed the design and orientation of the National Qualifications Framework and national curriculum that took place in parallel with, and simultaneous to, the restructuring and reform of teacher education. This article reviews literature pertinent to understanding the post-apartheid transformation in South African education in general and teacher education in particular. It concludes that policy makers have managed to navigate a way through the axis of tension created by opposed orientations to transformation. A more equitable and improved system of teacher education has been achieved but critical issues of teacher quality and quantity have emerged which urgently need resolution.
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- Date Issued: 2012
The nature and quality of the mathematical connections teachers make:
- 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.
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- 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?
- 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.
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- Date Issued: 2012
Beyond teaching language: towards terminological primacy in learners' geometric conceptualisation
- Authors: Atebe, Humphrey U , Schäfer, Marc
- Date: 2010
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140960 , vital:37932 , https://0-hdl.handle.net.wam.seals.ac.za/10520/EJC20922
- Description: This paper reports on a specific aspect of a broader geometry conceptualisation study that sought to explore and explicate learners' knowledge of basic geometric terminology in selected Nigerian and South African high schools. It is framed by the notion that students' acquisition of the correct terminology in school geometry is important for their success in the subject. The original study further aimed to determine the relationship that might exist between a learner's ability in verbal geometry terminology tasks and his/her ability in visual geometry terminology tasks. A total of 144 learners (72 each from South Africa and Nigeria) were selected for the study, using both the stratified and the fish-bowl sampling techniques. A questionnaire consisting of a sixty-item multiple-choice objective test provided the data for the study. An overall percentage mean score of 44,17% obtained in the test indicated that learners in this study had only a limited knowledge of basic geometric terminology. The Nigerian subsample in the study had a weaker understanding of basic geometric terminology than their South African counterparts. Importantly, there were high positive correlations between participants' ability in verbal geometry terminology tasks and their ability in visual geometry terminology tasks. These results are consistent with those of several earlier studies, and provide a reasonably firm basis for certain recommendations to be made.
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- Date Issued: 2010
Visual technology for the autonomous learning of mathematics:
- Authors: Linneweber-Lammerskitten, Helmut , Schäfer, Marc , Samson, Duncan
- Date: 2010
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/140949 , vital:37931 , https://0-hdl.handle.net.wam.seals.ac.za/10520/EJC20930
- Description: This paper describes a collaborative research and development project between the University of Applied Sciences Northwestern Switzerland and Rhodes University in South Africa. The project seeks to establish, disseminate and research the efficacy and use of short video clips designed specifically for the autonomous learning of mathematics. Specific to the South African context is our interest in capitalising on the ubiquity of cellphone technology and the autonomous affordances offered by mobile learning. This paper engages with a number of theoretical and pedagogical issues relating to the design, production and use of these video clips. Although the focus is specific to the contexts of South Africa and Switzerland, the discussion is of broad applicability.
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- Date Issued: 2010
An analysis of the influence of question design on learners' approaches to number pattern generalisation tasks:
- Authors: Samson, Duncan , Schäfer, Marc
- Date: 2007
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141026 , vital:37938 , https://0-hdl.handle.net.wam.seals.ac.za/10520/EJC20890
- Description: This paper reports on a study of the extent to which question design affects the solution strategies adopted by children when solving linear number pattern generalisation tasks presented in pictorial and numeric contexts. The research tool comprised a series of 22 pencil-and-paper exercises based on linear generalisation tasks set in both numeric and two-dimensional pictorial contexts. The responses to these linear generalisation questions were classified by means of stage descriptors as well as stage modifiers. The method or strategy adopted was analysed and classified into one of seven categories. In addition, a meta-analysis focused on the formula derived for the nth term in conjunction with its justification. The results of this study strongly support the notion that question design can play a critical role in influencing learners' choice of strategy and level of attainment when solving pattern generalisation tasks. An understanding of the importance of appropriate question design has direct pedagogical application within the context of the mathematics classroom.
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- Date Issued: 2007
Does 'African mathematics' facilitate access to mathematics?: towards an ongoing critical analysis of ethnomathematics in a South African context
- Authors: Horsthemke, Kai , Schäfer, Marc
- Date: 2007
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141004 , vital:37936 , https://0-hdl.handle.net.wam.seals.ac.za/10520/EJC20885
- Description: Mosibudi Mangena, the Minister of Science and Technology, said in an address to the Annual Congress of the South African Mathematical Society at the University of the Potchefstroom, November 2, 2004: "There is one thing we need to address before anything else. We need to increase the number of young people, particularly blacks and women, who are able to successfully complete the first course in Mathematics at our universities." How is this to be achieved? A popular trend involves a call for the introduction and incorporation of so-called ethnomathematics, and more particularly 'African mathematics', into secondary and tertiary curricula. Although acknowledging the obvious benefits of so-called ethnomathematics, this paper critically analyses three aspects of ethnomathematics that have been neglected in past critiques. Our focus is not on the relationship as such between ethnomathematics and mathematics education. Our critique involves (1) epistemological and logical misgivings, (2) a new look at practices and skills, (3) concerns about embracing 'African mathematics' as valid and valuable - just because it is African.
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- Date Issued: 2007
Teacher education for Mathematical Literacy: a modelling approach
- Authors: Brown, Bruce J L , Schäfer, Marc
- Date: 2006
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141040 , vital:37939 , DOI: 10.4102/pythagoras.v0i64.98
- Description: This paper reports on a study of the extent to which question design affects the solution strategies adopted by children when solving linear number pattern generalisation tasks presented in pictorial and numeric contexts. The research tool comprised a series of 22 pencil-and-paper exercises based on linear generalisation tasks set in both numeric and two-dimensional pictorial contexts. The responses to these linear generalisation questions were classified by means of stage descriptors as well as stage modifiers. The method or strategy adopted was analysed and classified into one of seven categories. In addition, a meta-analysis focused on the formula derived for the nth term in conjunction with its justification. The results of this study strongly support the notion that question design can play a critical role in influencing learners' choice of strategy and level of attainment when solving pattern generalisation tasks. An understanding of the importance of appropriate question design has direct pedagogical application within the context of the mathematics classroom.
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- Date Issued: 2006
World view theory and the conceptualisation of space in mathematics education:
- Authors: Schäfer, Marc
- Date: 2004
- Language: English
- Type: text , article
- Identifier: http://hdl.handle.net/10962/141052 , vital:37940 , DOI: 10.4102/pythagoras.v0i59.127
- Description: The cornerstone of current education trends that recognise prior knowledge as fundamental to the learning process, is the notion that beliefs and experiences that learners bring to the classroom influence their learning experiences in the classroom (Cobern, Gibson and Underwood, 1999).
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- Date Issued: 2004
Designing and making a difference: an exploration of technology education for rural school teachers
- Authors: Schäfer, Marc
- Date: 2000
- Subjects: Technology -- Study and teaching -- South Africa -- Eastern Cape Technology -- Study and teaching -- South Africa Rural schools -- South Africa -- Eastern Cape Education, Rural -- South Africa Winterberg schools trust
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1491 , http://hdl.handle.net/10962/d1003372
- Description: This qualitative study focused on a Technology Education programme for farm school teachers in the Eastern Cape province of South Africa. These teachers were faced with the challenge of incorporating Technology Education into their curriculum. The study was conducted within the context of an investigation into the conceptual nature of technology and an exploration of the theoretical underpinnings of Technology Education within both the international and South African context. Technology Education is being introduced into the South African curriculum against a background of educational transformation and the building of capacity to solve real life problems. This calls for a curriculum that will empower learners to be innovative, creative and skilled problem solvers. The introduction and incorporation of Technoiogy Education into the school curriculum poses a formidable challenge to farm schools in particular. Their unique history of neglect and legacy of underqualified teachers has made the introduction of any curriculum innovation process very difficult and challenging. This study analyses how an introductory Technology Education programme for farm school teachers in the Winterberg area of the Eastern Cape impacted~ on the teachers' professional and personal lives. It shows the importance of developing teclpological skills in conjunction with life skills in cO.ntributing to the empowerment, both in the work place and in the wider context, of rural school teachers. It highlights the need for supportive in-service education programmes and strengthens the argument for an integrative and mulitidisciplinary approach to the introduction of Technology Education in farm schools. Data was collected by means of questionnaires, interviews and photographs.
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- Date Issued: 2000