An analysis of how the use of geoboards as visualisation tools can be utilised in the teaching of quadrilaterals
- Authors: Matengu, Given Kahale
- Date: 2019
- Subjects: Manipulatives (Education) , Information visualization , Visualization , Mathematics -- Study and teaching , Mathematics -- Study and teaching -- Activity programs , Geometry -- Study and teaching
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/96724 , vital:31312
- Description: The relationship between visualisation processes and using manipulatives in the teaching and learning of mathematics is apparent and yet not so vocal in the literature. This could be because of the common mistaken understanding that because manipulatives are visual in nature, then visualisation processes should be obvious. Literature warns that just because something is visual therefore it is transparent, is incorrect. This study argues that the effective use of manipulatives in the teaching of mathematics helps learners to effectively understand mathematical concepts. Research on the teaching and learning of mathematics suggests that physical manipulation experiences, especially of concrete materials concerning shapes, is an important process in learning at all ages. One such teaching tool, the Geoboard, a physical manipulative that employs visualisation processes when correctly used, is explored in this study. The aim of this interpretive case study was to investigate and analyse the use of Geoboards as a visualisation tool in the teaching of the properties of quadrilaterals. The study focused on visualisation processes and the use of Geoboards through a teaching framework that was informed by the Van Hiele phases of teaching geometry. The study was conducted in the Opuwo circuit of the Kunene region, Namibia, and it involved three selected Grade 7 mathematics teachers, each from a different primary school. It was underpinned by a constructivist theory using the Van Hiele phases of teaching geometry and framed within visualisation processes. The study employed the use of qualitative data collection techniques such as observations and interviews. The analysis of the findings of this study revealed that Geoboards were very useful in demonstrating the visual representations of the properties of quadrilaterals in a cheap and yet novel way in the selected teachers’ classes. Moreover, the use of Geoboards by the selected teachers effectively fostered visualisation processes such as concrete pictorial imagery, dynamic imagery, perceptual apprehension, sequential apprehension, discursive apprehension and operative apprehension. It was also revealed that Geoboards enabled the selected teachers to structure and teach their lessons in a well-planned manner according to the Van Hiele phases, although it was difficult for them to adhere strictly to the hierarchy of the phases.
- Full Text:
- Authors: Matengu, Given Kahale
- Date: 2019
- Subjects: Manipulatives (Education) , Information visualization , Visualization , Mathematics -- Study and teaching , Mathematics -- Study and teaching -- Activity programs , Geometry -- Study and teaching
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/96724 , vital:31312
- Description: The relationship between visualisation processes and using manipulatives in the teaching and learning of mathematics is apparent and yet not so vocal in the literature. This could be because of the common mistaken understanding that because manipulatives are visual in nature, then visualisation processes should be obvious. Literature warns that just because something is visual therefore it is transparent, is incorrect. This study argues that the effective use of manipulatives in the teaching of mathematics helps learners to effectively understand mathematical concepts. Research on the teaching and learning of mathematics suggests that physical manipulation experiences, especially of concrete materials concerning shapes, is an important process in learning at all ages. One such teaching tool, the Geoboard, a physical manipulative that employs visualisation processes when correctly used, is explored in this study. The aim of this interpretive case study was to investigate and analyse the use of Geoboards as a visualisation tool in the teaching of the properties of quadrilaterals. The study focused on visualisation processes and the use of Geoboards through a teaching framework that was informed by the Van Hiele phases of teaching geometry. The study was conducted in the Opuwo circuit of the Kunene region, Namibia, and it involved three selected Grade 7 mathematics teachers, each from a different primary school. It was underpinned by a constructivist theory using the Van Hiele phases of teaching geometry and framed within visualisation processes. The study employed the use of qualitative data collection techniques such as observations and interviews. The analysis of the findings of this study revealed that Geoboards were very useful in demonstrating the visual representations of the properties of quadrilaterals in a cheap and yet novel way in the selected teachers’ classes. Moreover, the use of Geoboards by the selected teachers effectively fostered visualisation processes such as concrete pictorial imagery, dynamic imagery, perceptual apprehension, sequential apprehension, discursive apprehension and operative apprehension. It was also revealed that Geoboards enabled the selected teachers to structure and teach their lessons in a well-planned manner according to the Van Hiele phases, although it was difficult for them to adhere strictly to the hierarchy of the phases.
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Generating shared interpretive resources in the mathematics classroom: using philosophy of mathematics to teach mathematics better
- Authors: De Lange, Laura
- Date: 2017
- Subjects: Mathematics -- Study and teaching , Mathematics -- Philosophy
- Language: English
- Type: Thesis , Masters , MA
- Identifier: http://hdl.handle.net/10962/4293 , vital:20645
- Description: Every student has a unique mathematical lived experience: a unique amalgamation of ideas about mathematics, exposure to mathematical concepts and feelings about mathematics. A student's unique set of circumstances means that not every explanatory account of mathematics will cohere with her previous experiences. For an explanation to have explanatory potential, it must provide an account which coheres with the other beliefs a student has about mathematics. If an explanation has no such coherence, it will not be recognisable as an explanation of the phenomenon of mathematics for the student. Our explanatory accounts of mathematics and mathematical knowledge are our philosophies of mathematics. Different philosophies of mathematics will better explain different sets of mathematical lived experiences. In this thesis I will argue that students should be exposed to a multiplicity of philosophies of mathematics so that they can endorse the philosophy of mathematics which has the most explanatory potential for their particular set of mathematical lived experiences. I argue that this will improve student understanding of mathematics. The claims inherent in any given philosophy of mathematics, when combined with other stereotypes or prejudices, can work to unjustly exclude members of subordinated groups, such as poor, black or female students, from mathematical participation. If we want to avoid reinforcing and reinscribing prejudicial claims about people in the mathematics classroom, we need to be aware of how a certain philosophy of mathematics can exclude certain students. In this thesis I will be defending the idea that, as mathematics educators, we should diversify the way we see mathematics so that we decrease this exclusion from mathematics. In order to diversify the way in which we see mathematics so as to decrease unjust exclusion, members of subordinated groups should be encouraged to share their mathematical experiences in a space sensitive to the power dynamics present in the mathematics classroom. These accounts can then be combined with existing philosophies of mathematics to create new ways of making sense of mathematics which do not unjustly exclude members of subordinated groups.
- Full Text:
- Authors: De Lange, Laura
- Date: 2017
- Subjects: Mathematics -- Study and teaching , Mathematics -- Philosophy
- Language: English
- Type: Thesis , Masters , MA
- Identifier: http://hdl.handle.net/10962/4293 , vital:20645
- Description: Every student has a unique mathematical lived experience: a unique amalgamation of ideas about mathematics, exposure to mathematical concepts and feelings about mathematics. A student's unique set of circumstances means that not every explanatory account of mathematics will cohere with her previous experiences. For an explanation to have explanatory potential, it must provide an account which coheres with the other beliefs a student has about mathematics. If an explanation has no such coherence, it will not be recognisable as an explanation of the phenomenon of mathematics for the student. Our explanatory accounts of mathematics and mathematical knowledge are our philosophies of mathematics. Different philosophies of mathematics will better explain different sets of mathematical lived experiences. In this thesis I will argue that students should be exposed to a multiplicity of philosophies of mathematics so that they can endorse the philosophy of mathematics which has the most explanatory potential for their particular set of mathematical lived experiences. I argue that this will improve student understanding of mathematics. The claims inherent in any given philosophy of mathematics, when combined with other stereotypes or prejudices, can work to unjustly exclude members of subordinated groups, such as poor, black or female students, from mathematical participation. If we want to avoid reinforcing and reinscribing prejudicial claims about people in the mathematics classroom, we need to be aware of how a certain philosophy of mathematics can exclude certain students. In this thesis I will be defending the idea that, as mathematics educators, we should diversify the way we see mathematics so that we decrease this exclusion from mathematics. In order to diversify the way in which we see mathematics so as to decrease unjust exclusion, members of subordinated groups should be encouraged to share their mathematical experiences in a space sensitive to the power dynamics present in the mathematics classroom. These accounts can then be combined with existing philosophies of mathematics to create new ways of making sense of mathematics which do not unjustly exclude members of subordinated groups.
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An investigation of teachers' mathematical task selection in the Zambia context
- Authors: Kangwa, Evaristo
- Date: 2013
- Subjects: Mathematics -- Study and teaching
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1384 , http://hdl.handle.net/10962/d1001512
- Description: This research sought to investigate the sources and type of tasks used in the teaching of trigonometry in Zambia’s secondary schools, and to investigate the criteria used and decisions made by teachers in their selection and implementation of tasks. The study was conducted in three different school types located in high cost, medium cost and low cost respectively. One participant was chosen from each of the different categories of schools. The research was located within an interpretive paradigm. Data were collected through semi-structured interviews, lesson observations and document analysis which include: lesson plans for five consecutive days, pupils’ activity books and three textbooks predominantly used by the teachers. Document analysis was informed by the task analysis guide and essential themes which were used to tease out teachers’ task practice with regard to criteria used and decisions made in the selection and implementation of tasks. Essential themes that were qualitatively established were validated and explicated by the qualitative analysis. The findings of the study indicate that teachers picked tasks from prescribed textbooks. The study further suggests that teachers selected a mix of low and high level tasks, procedures without connections and procedures with connections tasks to be specific. There were no memorisations and doing mathematics tasks. Their choice of tasks was based on the purpose for which the task was intended. Some tasks were selected for the purpose of practicing the procedures and skills, other tasks for the promotion conceptual development. Most of high level tasks decline to low level tasks during implementation. The findings also indicate that teachers selected and implemented a variety of tasks and concepts. Furthermore, teachers presented tasks in various forms of representations and in a variety of ways. However, the results of this study could not be generalized because of the small sample involved. The results presented reflect the views and task practices of the target group. A possibility for future study would be to consider a large population, drawn across the country.
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- Authors: Kangwa, Evaristo
- Date: 2013
- Subjects: Mathematics -- Study and teaching
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1384 , http://hdl.handle.net/10962/d1001512
- Description: This research sought to investigate the sources and type of tasks used in the teaching of trigonometry in Zambia’s secondary schools, and to investigate the criteria used and decisions made by teachers in their selection and implementation of tasks. The study was conducted in three different school types located in high cost, medium cost and low cost respectively. One participant was chosen from each of the different categories of schools. The research was located within an interpretive paradigm. Data were collected through semi-structured interviews, lesson observations and document analysis which include: lesson plans for five consecutive days, pupils’ activity books and three textbooks predominantly used by the teachers. Document analysis was informed by the task analysis guide and essential themes which were used to tease out teachers’ task practice with regard to criteria used and decisions made in the selection and implementation of tasks. Essential themes that were qualitatively established were validated and explicated by the qualitative analysis. The findings of the study indicate that teachers picked tasks from prescribed textbooks. The study further suggests that teachers selected a mix of low and high level tasks, procedures without connections and procedures with connections tasks to be specific. There were no memorisations and doing mathematics tasks. Their choice of tasks was based on the purpose for which the task was intended. Some tasks were selected for the purpose of practicing the procedures and skills, other tasks for the promotion conceptual development. Most of high level tasks decline to low level tasks during implementation. The findings also indicate that teachers selected and implemented a variety of tasks and concepts. Furthermore, teachers presented tasks in various forms of representations and in a variety of ways. However, the results of this study could not be generalized because of the small sample involved. The results presented reflect the views and task practices of the target group. A possibility for future study would be to consider a large population, drawn across the country.
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Case study : using visual representations to enhance conceptual knowledge of division in mathematics
- Authors: Joel, Linea Beautty
- Date: 2013
- Subjects: Mathematics -- Study and teaching , Visual learning , Division -- Study and teaching
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1992 , http://hdl.handle.net/10962/d1013356
- Description: Literature emphasizes how important it is that procedural and conceptual knowledge of mathematics should be learned in integration. Yet, generally, the learning and teaching in mathematics classrooms relies heavily on isolated procedures. This study aims to improve teaching and learning of partitive and quotitive division, moving away from isolated procedural knowledge to that of procedures with their underlying concepts through the use of manipulatives, visual representation and questioning. Learning and teaching lessons were designed to teach partitive and quotitive division both procedurally and conceptually. The study explored the roles these manipulatives, visual representations and questioning played toward the conceptual learning of partitive and quotitive division. It was found that manipulatives and iconic visualization enhanced learning, and this could be achieved through scaffolding using a questioning approach. It was concluded that manipulatives and iconic visualization need to be properly planned and used, and integrated with questioning to achieve success in the learning of procedural and conceptual knowledge.
- Full Text:
Case study : using visual representations to enhance conceptual knowledge of division in mathematics
- Authors: Joel, Linea Beautty
- Date: 2013
- Subjects: Mathematics -- Study and teaching , Visual learning , Division -- Study and teaching
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1992 , http://hdl.handle.net/10962/d1013356
- Description: Literature emphasizes how important it is that procedural and conceptual knowledge of mathematics should be learned in integration. Yet, generally, the learning and teaching in mathematics classrooms relies heavily on isolated procedures. This study aims to improve teaching and learning of partitive and quotitive division, moving away from isolated procedural knowledge to that of procedures with their underlying concepts through the use of manipulatives, visual representation and questioning. Learning and teaching lessons were designed to teach partitive and quotitive division both procedurally and conceptually. The study explored the roles these manipulatives, visual representations and questioning played toward the conceptual learning of partitive and quotitive division. It was found that manipulatives and iconic visualization enhanced learning, and this could be achieved through scaffolding using a questioning approach. It was concluded that manipulatives and iconic visualization need to be properly planned and used, and integrated with questioning to achieve success in the learning of procedural and conceptual knowledge.
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Project 1 - Student teachers' exploration of beadwork : cultural heritage as a resource for mathematical concepts
- Authors: Dabula, Nomonde Patience
- Date: 2001
- Subjects: Ethnomathematics , Beadwork -- South Africa -- Eastern Cape , Beadwork , Mathematics -- Study and teaching , Culture -- Study and teaching
- Language: English
- Type: Thesis , Masters , MA
- Identifier: vital:1415 , http://hdl.handle.net/10962/d1003292
- Description: This portfolio consists of three research projects that predominantly lie within the socio-cultural strand. The first project is a qualitative ethnomathematical study that links students' knowledge of mathematics to their cultural heritage. The study was conducted with a group of final year student teachers at a College of Education in Umtata, Eastern Cape. These students visited a city museum where mathematics concepts were identified from beadwork artifacts. Mathematics concepts that were identified consisted of symmetry, tessellation and number patterns. Students' views about the nature of mathematics shifted radically after their own explorations. Initially students did not perceive mathematics as relating to socio-cultural practices. But now, they have reviewed their position and see mathematics as inextricably interwoven in everyday activities and as such, a product of all cultures. They also pride themselves of their own cultural heritage to have mathematical connections. A more positive attitude towards studying mathematics in this approach was noticed. Data was collected by means of interviews, reflective journal entries and photographs. The second project is a survey with a group of practising teachers who have already implemented Curriculum 2005, and a group which is about to implement it in 2001. The study sought teachers' understanding of connections between mathematics and socio-cultural issues. The new mathematics curriculum in South Africa calls for teachers to grapple well with these issues. About a third of the articulated specific outcomes specifically relate to socio-cultural issues. If teachers' understanding of these issues is poor, implementation of the new curriculum will remain a mere dream. The findings of the survey revealed that the majority of teachers could not identify the culture related specific outcomes in the new mathematics curriculum. Complicated language used in the OBE policy documents was found to inhibit meaning to these teachers. Although, all teachers showed a positive attitude towards the inclusion of socio-cultural issues in the mathematics classroom, the implementation of these outcomes was found to be very problematic. In this survey data was collected by means of questionnaires. The third project is a literature review on the need to popularise mathematics to students in particular, and to the broader public in general. The 21 st century places great technological demands. Mathematics underpins most thinking behind technological development. The role played by mathematics in advancing other fields is largely hidden to the majority of people. There is, therefore, a need to bring forth the vital role that mathematics plays in these fields. The number of students participating in mathematics is decreasing. Mathematics, as a field, is experiencing competition from other science fields. There is a need to bring some incentives to attract more students into this field and retain those mathematicians already involved. Also important, is the need to change the negative image that the public often holds about mathematics. Many people are mathematically illiterate and do not see mathematics as an everyday activity that relates to their needs. There is, therefore, a need to change the face of mathematics.
- Full Text:
- Authors: Dabula, Nomonde Patience
- Date: 2001
- Subjects: Ethnomathematics , Beadwork -- South Africa -- Eastern Cape , Beadwork , Mathematics -- Study and teaching , Culture -- Study and teaching
- Language: English
- Type: Thesis , Masters , MA
- Identifier: vital:1415 , http://hdl.handle.net/10962/d1003292
- Description: This portfolio consists of three research projects that predominantly lie within the socio-cultural strand. The first project is a qualitative ethnomathematical study that links students' knowledge of mathematics to their cultural heritage. The study was conducted with a group of final year student teachers at a College of Education in Umtata, Eastern Cape. These students visited a city museum where mathematics concepts were identified from beadwork artifacts. Mathematics concepts that were identified consisted of symmetry, tessellation and number patterns. Students' views about the nature of mathematics shifted radically after their own explorations. Initially students did not perceive mathematics as relating to socio-cultural practices. But now, they have reviewed their position and see mathematics as inextricably interwoven in everyday activities and as such, a product of all cultures. They also pride themselves of their own cultural heritage to have mathematical connections. A more positive attitude towards studying mathematics in this approach was noticed. Data was collected by means of interviews, reflective journal entries and photographs. The second project is a survey with a group of practising teachers who have already implemented Curriculum 2005, and a group which is about to implement it in 2001. The study sought teachers' understanding of connections between mathematics and socio-cultural issues. The new mathematics curriculum in South Africa calls for teachers to grapple well with these issues. About a third of the articulated specific outcomes specifically relate to socio-cultural issues. If teachers' understanding of these issues is poor, implementation of the new curriculum will remain a mere dream. The findings of the survey revealed that the majority of teachers could not identify the culture related specific outcomes in the new mathematics curriculum. Complicated language used in the OBE policy documents was found to inhibit meaning to these teachers. Although, all teachers showed a positive attitude towards the inclusion of socio-cultural issues in the mathematics classroom, the implementation of these outcomes was found to be very problematic. In this survey data was collected by means of questionnaires. The third project is a literature review on the need to popularise mathematics to students in particular, and to the broader public in general. The 21 st century places great technological demands. Mathematics underpins most thinking behind technological development. The role played by mathematics in advancing other fields is largely hidden to the majority of people. There is, therefore, a need to bring forth the vital role that mathematics plays in these fields. The number of students participating in mathematics is decreasing. Mathematics, as a field, is experiencing competition from other science fields. There is a need to bring some incentives to attract more students into this field and retain those mathematicians already involved. Also important, is the need to change the negative image that the public often holds about mathematics. Many people are mathematically illiterate and do not see mathematics as an everyday activity that relates to their needs. There is, therefore, a need to change the face of mathematics.
- Full Text:
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