The use of dynamic software to potentially enhance conceptual understanding and a productive disposition in the visual learning of algebra: an interventionist case study
- Authors: Junius, Daniel Franscius
- Date: 2023-10-13
- Subjects: Mathematics education , Algebra Study and teaching (Secondary) Namibia , Educational technology , Visual learning , GeoGebra , High school students Attitudes
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/431542 , vital:72784 , DOI 10.21504/10962/431542
- Description: Over the decades, the didactics and practice of teaching mathematics has offered many unique challenges and opportunities for exploration and understanding. The introduction and development of technology into mathematics is one of the occurrences which has also contributed to a new discourse in teaching mathematics – in this case the teaching of algebra. Algebra is still seen as a gatekeeper and remains as one of the key reasons for a negative disposition amongst learners towards learning the subject. Across the globe psychologists, philosophers and educators continue to engage in debates and research projects in search of answers and solutions for the improvement of algebra teaching and an improvement in dispositions towards learning algebra. This thesis reports on a research project that focused on the use of dynamic software to enhance the conceptual understanding and productive dispositions of selected learners through the visual learning of abstract algebraic concepts. The research was executed as an interventionist case study. A case study methodological strategy was adopted with two groups of 30 Grade 9 learners. One group was a Grade 9 mathematics class of a school in Windhoek who scored above average in algebra but showed a very low disposition score, while the other group was made up of learners from a community project who scored high on the disposition scale but achieved below average results in algebra. The analytical framework of the case study is structured around a combination of complementary algebraic topics presented through visual learning, with GeoGebra as a medium of instruction. With the focus on visualisation and the use of technology the study investigated and attempted to understand how participants processed and internalised algebraic concepts to make sense of abstract algebraic concepts and eventually gain sustained conceptual understanding. The study was framed by the theoretical theories of constructivism and the Dual Coding Theory. For the collection of data, a mixed methods approach was adopted following three cycles. Three algebraic topics were taught with GeoGebra applets yielding both qualitative and quantitative data, through the observation of participants, screen captures and reflective interviews, using instruments designed specifically for the study and collecting quantitative achievement test results. The study, a journey that both the participants and the researcher embarked upon, revealed that the use of technology enhanced conceptual understanding for both groups and both groups showed a positive change in disposition towards learning algebra. The intervention with GeoGebra consistently and progressively improved in terms of conceptual understanding and dispositions towards learning algebra significant improvements in results were achieved. The findings showed that this approach to teaching algebra yielded positive results and gave new insights into visual teaching with technology. New opportunities for further research were created. , Thesis (PhD) -- Faculty of Education, Secondary and Post-School Education, 2023
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- Authors: Junius, Daniel Franscius
- Date: 2023-10-13
- Subjects: Mathematics education , Algebra Study and teaching (Secondary) Namibia , Educational technology , Visual learning , GeoGebra , High school students Attitudes
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/431542 , vital:72784 , DOI 10.21504/10962/431542
- Description: Over the decades, the didactics and practice of teaching mathematics has offered many unique challenges and opportunities for exploration and understanding. The introduction and development of technology into mathematics is one of the occurrences which has also contributed to a new discourse in teaching mathematics – in this case the teaching of algebra. Algebra is still seen as a gatekeeper and remains as one of the key reasons for a negative disposition amongst learners towards learning the subject. Across the globe psychologists, philosophers and educators continue to engage in debates and research projects in search of answers and solutions for the improvement of algebra teaching and an improvement in dispositions towards learning algebra. This thesis reports on a research project that focused on the use of dynamic software to enhance the conceptual understanding and productive dispositions of selected learners through the visual learning of abstract algebraic concepts. The research was executed as an interventionist case study. A case study methodological strategy was adopted with two groups of 30 Grade 9 learners. One group was a Grade 9 mathematics class of a school in Windhoek who scored above average in algebra but showed a very low disposition score, while the other group was made up of learners from a community project who scored high on the disposition scale but achieved below average results in algebra. The analytical framework of the case study is structured around a combination of complementary algebraic topics presented through visual learning, with GeoGebra as a medium of instruction. With the focus on visualisation and the use of technology the study investigated and attempted to understand how participants processed and internalised algebraic concepts to make sense of abstract algebraic concepts and eventually gain sustained conceptual understanding. The study was framed by the theoretical theories of constructivism and the Dual Coding Theory. For the collection of data, a mixed methods approach was adopted following three cycles. Three algebraic topics were taught with GeoGebra applets yielding both qualitative and quantitative data, through the observation of participants, screen captures and reflective interviews, using instruments designed specifically for the study and collecting quantitative achievement test results. The study, a journey that both the participants and the researcher embarked upon, revealed that the use of technology enhanced conceptual understanding for both groups and both groups showed a positive change in disposition towards learning algebra. The intervention with GeoGebra consistently and progressively improved in terms of conceptual understanding and dispositions towards learning algebra significant improvements in results were achieved. The findings showed that this approach to teaching algebra yielded positive results and gave new insights into visual teaching with technology. New opportunities for further research were created. , Thesis (PhD) -- Faculty of Education, Secondary and Post-School Education, 2023
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Visual representations of linear algebraic expressions: a case study in a Grade 9 after-school mathematics club
- Authors: Herbert, Sindisiwe
- Date: 2023-10-13
- Subjects: Visual learning , Algebra Study and teaching (Secondary) South Africa , Algebras, Linear , After-school programs , Kinaesthetic learning
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/424019 , vital:72116
- Description: Visualisation is commonly used as a tool in introducing algebra through visual or kinaesthetic sequences designed to prompt learners' development of a general rule for moving from a term's position to its output value. Fluency in both the concepts and the conventions of elementary algebra are essential to learners, as algebra forms the language in which many advanced mathematical ideas are encoded. Moreover, algebraic fluency is often associated with an ability to think abstractly about arithmetic processes. In many classrooms, however, research has shown that learners often focus on fluency in algebraic conventions rather than concepts, learning how to manipulate expressions without understanding the algorithms they are taught to follow. This trend can be linked to several causes, including teacher-centred mathematics classrooms in which learners are – whether implicitly or explicitly – encouraged to copy formulae and methods in order to ‘get it right in a test’ without necessarily grasping the underlying logical relationships. This case study, therefore, aimed to determine whether there was value in using visual, kinaesthetic models to broaden and deepen learners' use of algebra. To that end, in the context of an extra-curricular mathematics club that aimed to decentre the teachers and demand innovative ideas of the participants, six pairs of Grade 9 learners were tasked with creating visual representations of a linear algebraic expression using coloured building cubes. The responses to this task over the course of five assignments were many and varied and almost universally displayed a sustained internal logic that the learners were able to explain and develop. Most pairs began with a visual list of terms arranged in sets of towers, pyramids or, in one case, a spiral. At the end of the study, all but one of the pairs had settled on a Visual Expression, in which various colours were used to represent elements of the algebra such as the values of the coefficient, the variable and the constant term. The participants' representations grew in complexity over the course of the study and the conformity of the final responses showed that the club was a collaborative space in which learners shared ideas. However, the structure of the Visual Expressions and their own confessions of nerves about ‘getting it wrong’ in the interviews suggest that the participants were stuck in a mindset that led them to seek out and idealise the representation closest to the original algebra, even though that representation revealed little about the structural relationship underlying the expression. , Thesis (MEd) -- Faculty of Education, Secondary and Post-School Education, 2023
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- Authors: Herbert, Sindisiwe
- Date: 2023-10-13
- Subjects: Visual learning , Algebra Study and teaching (Secondary) South Africa , Algebras, Linear , After-school programs , Kinaesthetic learning
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/424019 , vital:72116
- Description: Visualisation is commonly used as a tool in introducing algebra through visual or kinaesthetic sequences designed to prompt learners' development of a general rule for moving from a term's position to its output value. Fluency in both the concepts and the conventions of elementary algebra are essential to learners, as algebra forms the language in which many advanced mathematical ideas are encoded. Moreover, algebraic fluency is often associated with an ability to think abstractly about arithmetic processes. In many classrooms, however, research has shown that learners often focus on fluency in algebraic conventions rather than concepts, learning how to manipulate expressions without understanding the algorithms they are taught to follow. This trend can be linked to several causes, including teacher-centred mathematics classrooms in which learners are – whether implicitly or explicitly – encouraged to copy formulae and methods in order to ‘get it right in a test’ without necessarily grasping the underlying logical relationships. This case study, therefore, aimed to determine whether there was value in using visual, kinaesthetic models to broaden and deepen learners' use of algebra. To that end, in the context of an extra-curricular mathematics club that aimed to decentre the teachers and demand innovative ideas of the participants, six pairs of Grade 9 learners were tasked with creating visual representations of a linear algebraic expression using coloured building cubes. The responses to this task over the course of five assignments were many and varied and almost universally displayed a sustained internal logic that the learners were able to explain and develop. Most pairs began with a visual list of terms arranged in sets of towers, pyramids or, in one case, a spiral. At the end of the study, all but one of the pairs had settled on a Visual Expression, in which various colours were used to represent elements of the algebra such as the values of the coefficient, the variable and the constant term. The participants' representations grew in complexity over the course of the study and the conformity of the final responses showed that the club was a collaborative space in which learners shared ideas. However, the structure of the Visual Expressions and their own confessions of nerves about ‘getting it wrong’ in the interviews suggest that the participants were stuck in a mindset that led them to seek out and idealise the representation closest to the original algebra, even though that representation revealed little about the structural relationship underlying the expression. , Thesis (MEd) -- Faculty of Education, Secondary and Post-School Education, 2023
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An investigation of how a visual teaching approach can possibly address issues of mathematics anxiety at a selected school in the Oshikoto region of Namibia
- Ngonga, Daniel Nghifikepunye
- Authors: Ngonga, Daniel Nghifikepunye
- Date: 2022-04-08
- Subjects: Visual learning Namibia , Math anxiety Namibia , Mathematics Study and teaching Namibia , After-school programs Namibia , Mathematics Study and teaching Activity programs
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/290637 , vital:56770
- Description: This Namibian case study aimed to explore a visual teaching approach (VTA) used by three selected teachers to address issues of mathematics anxiety (MA). The three teachers took part in an intervention program that was looking at how a VTA could be grown in the context of an after-school club (ASC) at my school. The selected teachers were the senior primary teachers at my school. The focus of the research was on how they taught mathematics using visuals after participating in an intervention programme. Their VTA made use of manipulatives, visuals, and concrete materials. The learners of the participating teachers completed a big MA pre-test, small MA tests, and a big MA post-test to determine their levels of MA as the teaching programme unfolded. The study hoped to create awareness amongst teachers and education researchers about the significant use of a VTA in the teaching and learning of mathematics to address issues of MA among the learners. It aimed to answer three research questions. One was on teachers’ use of a VTA in the context of an ASC; the second one was on comparisons of learners’ MA big pre and post-tests to detect any change of MA, and the last was on the enabling and constraining factors encountered when using a VTA. The main argument was that a VTA can encourage learners to be more confident and less anxious about doing mathematics. This study was framed by a constructivist perspective and its design and methodology were underpinned by an interpretive paradigm. This mixed-method research study employed video-recorded observations and stimulated recall interviews, learners’ MA test results, and the teachers’ focus group interviews as the means of collecting data. To generate rich data and support validity, four lessons per selected teacher were observed and video recorded; 54 learners completed the MA tests of 16 questions, and three teachers answered seven questions each in the focus group interview (FGI) after the stimulus recall interviews (SRI) which were done immediately after the lesson presentations. The study found that the participating teachers incorporated a variety of visuals into their lessons to make the mathematics fun, inspiring, visible, hands-on, and activity-oriented. They engaged the learners and also found that the use of visuals motivated learners and reduced their MA. , Thesis (MED) -- Faculty of Education, Education, 2022
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- Authors: Ngonga, Daniel Nghifikepunye
- Date: 2022-04-08
- Subjects: Visual learning Namibia , Math anxiety Namibia , Mathematics Study and teaching Namibia , After-school programs Namibia , Mathematics Study and teaching Activity programs
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/290637 , vital:56770
- Description: This Namibian case study aimed to explore a visual teaching approach (VTA) used by three selected teachers to address issues of mathematics anxiety (MA). The three teachers took part in an intervention program that was looking at how a VTA could be grown in the context of an after-school club (ASC) at my school. The selected teachers were the senior primary teachers at my school. The focus of the research was on how they taught mathematics using visuals after participating in an intervention programme. Their VTA made use of manipulatives, visuals, and concrete materials. The learners of the participating teachers completed a big MA pre-test, small MA tests, and a big MA post-test to determine their levels of MA as the teaching programme unfolded. The study hoped to create awareness amongst teachers and education researchers about the significant use of a VTA in the teaching and learning of mathematics to address issues of MA among the learners. It aimed to answer three research questions. One was on teachers’ use of a VTA in the context of an ASC; the second one was on comparisons of learners’ MA big pre and post-tests to detect any change of MA, and the last was on the enabling and constraining factors encountered when using a VTA. The main argument was that a VTA can encourage learners to be more confident and less anxious about doing mathematics. This study was framed by a constructivist perspective and its design and methodology were underpinned by an interpretive paradigm. This mixed-method research study employed video-recorded observations and stimulated recall interviews, learners’ MA test results, and the teachers’ focus group interviews as the means of collecting data. To generate rich data and support validity, four lessons per selected teacher were observed and video recorded; 54 learners completed the MA tests of 16 questions, and three teachers answered seven questions each in the focus group interview (FGI) after the stimulus recall interviews (SRI) which were done immediately after the lesson presentations. The study found that the participating teachers incorporated a variety of visuals into their lessons to make the mathematics fun, inspiring, visible, hands-on, and activity-oriented. They engaged the learners and also found that the use of visuals motivated learners and reduced their MA. , Thesis (MED) -- Faculty of Education, Education, 2022
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The use of smartphones and visualisation processes for conceptual understanding of mensuration: a case study of the Mathcitymap Project in Namibia
- Authors: Shimakeleni, Liina
- Date: 2022-04-08
- Subjects: Mathematics Study and teaching (Secondary) Namibia Oshana , Smartphones , Visual learning , Measurement , Concept learning , MathCityMap (MCM) project
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/290649 , vital:56771
- Description: The aim of this study was to investigate and analyse the potential use of smartphones as visualisation tools by learners to enhance conceptual understanding through mathematics trails developed using the MathCityMap (MCM) project. This research study is part of the VIPROmaths project which seeks to research the effective use of visualisation processes in mathematics classrooms in South Africa, Namibia, Zambia, Switzerland and Germany. This study adopted a case of twelve purposively selected Grade 9 learners from a school in the Eheke circuit of the Oshana region, Namibia. This study was framed within a social constructivist perspective and sought to investigate visualisation processes as well as conceptual understanding of learners as they conceptualised the MCM tasks in new, outdoor and collaborative learning situations. The MCM app was installed on selected learners’ smartphones to access and to walk the MCM trails located in various places in the schoolyard. Three MCM trails based on three key themes of mensuration (perimeter, area and volume) were created. Each consisted of four tasks that were sourced and developed in line with the Grade 9 Namibian mathematics syllabus. This study is oriented in an interpretive paradigm and employed video-recorded observations and focus group interviews as qualitative data collection methods. Data collected were analysed first using the themes developed from Ho’s (2010) work on visualisation processes and Kilpatrick, Swafford, and Findell’s (2001) conceptual understanding. During analysis, some themes emerged from the data and were considered. Findings from this study revealed that smartphones afforded learners ample opportunities to enhance the visualisation processes that they went through as they learned the concepts of measurement. In addition to this, some learners were initially pessimistic regarding the use of smartphones for learning purposes. This study recommends that resources such as MCM be effectively be used in formal school settings. The learning of measurement can be advanced in outdoor settings where learners have physical and spatial access to the learning content. Smartphone technology can be used as an additional tool to integrate Information and Communication Technology (ICT) in learning mathematics within the Namibian context. , Thesis (MED) -- Faculty of Education, Education, 2022
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- Authors: Shimakeleni, Liina
- Date: 2022-04-08
- Subjects: Mathematics Study and teaching (Secondary) Namibia Oshana , Smartphones , Visual learning , Measurement , Concept learning , MathCityMap (MCM) project
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/290649 , vital:56771
- Description: The aim of this study was to investigate and analyse the potential use of smartphones as visualisation tools by learners to enhance conceptual understanding through mathematics trails developed using the MathCityMap (MCM) project. This research study is part of the VIPROmaths project which seeks to research the effective use of visualisation processes in mathematics classrooms in South Africa, Namibia, Zambia, Switzerland and Germany. This study adopted a case of twelve purposively selected Grade 9 learners from a school in the Eheke circuit of the Oshana region, Namibia. This study was framed within a social constructivist perspective and sought to investigate visualisation processes as well as conceptual understanding of learners as they conceptualised the MCM tasks in new, outdoor and collaborative learning situations. The MCM app was installed on selected learners’ smartphones to access and to walk the MCM trails located in various places in the schoolyard. Three MCM trails based on three key themes of mensuration (perimeter, area and volume) were created. Each consisted of four tasks that were sourced and developed in line with the Grade 9 Namibian mathematics syllabus. This study is oriented in an interpretive paradigm and employed video-recorded observations and focus group interviews as qualitative data collection methods. Data collected were analysed first using the themes developed from Ho’s (2010) work on visualisation processes and Kilpatrick, Swafford, and Findell’s (2001) conceptual understanding. During analysis, some themes emerged from the data and were considered. Findings from this study revealed that smartphones afforded learners ample opportunities to enhance the visualisation processes that they went through as they learned the concepts of measurement. In addition to this, some learners were initially pessimistic regarding the use of smartphones for learning purposes. This study recommends that resources such as MCM be effectively be used in formal school settings. The learning of measurement can be advanced in outdoor settings where learners have physical and spatial access to the learning content. Smartphone technology can be used as an additional tool to integrate Information and Communication Technology (ICT) in learning mathematics within the Namibian context. , Thesis (MED) -- Faculty of Education, Education, 2022
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An analysis of selected Grade 8 mathematics teachers’ use of gestures as visualisation tools to support mathematical meaning-making
- Authors: Haipinge, David Tuhafeni
- Date: 2021-10-29
- Subjects: Gesture , Gesture in mathematics education , Visual learning , Visualization , Mathematics Study and teaching (Secondary) Namibia
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/191969 , vital:45184
- Description: The purpose of this Namibian case study was to investigate how different types of gestures are used to support the construction of mathematical meaning making in teaching and learning. Gestures of three selected Grade 8 mathematics teachers were observed and analysed. This study was intended to answer the following research question: how do selected Grade 8 mathematics teachers use gestures as visualisation tools to support mathematical meaning making? The study was framed by an enactivist perspective and the research was oriented in the interpretive paradigm. Data were collected through video-recorded observations of three selected teachers and through stimulus recall interviews. In order to generate rich data and support validity, five lessons per selected teacher were video recorded. The study found that the participating teachers incorporated a variety of mathematical gestures into their lessons in order to support and provide mathematical meaning. Further, this study found that gestures facilitated meaning making in mathematics. The findings in the study suggest a need for mathematics teachers to be trained in using gestures appropriately to communicate mathematically in their lessons. In addition, this study discovered a new type of gesture – the overlapping gesture in addition to McNeill's (1992) types, namely: pointing gestures, metaphor gestures, beating gestures and iconic gestures. This case study also showed that the more experienced the teachers are, the more mathematical gestures they produce during their mathematics lessons. , Thesis (MEd) -- Faculty of Education, Education, 2021
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- Authors: Haipinge, David Tuhafeni
- Date: 2021-10-29
- Subjects: Gesture , Gesture in mathematics education , Visual learning , Visualization , Mathematics Study and teaching (Secondary) Namibia
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/191969 , vital:45184
- Description: The purpose of this Namibian case study was to investigate how different types of gestures are used to support the construction of mathematical meaning making in teaching and learning. Gestures of three selected Grade 8 mathematics teachers were observed and analysed. This study was intended to answer the following research question: how do selected Grade 8 mathematics teachers use gestures as visualisation tools to support mathematical meaning making? The study was framed by an enactivist perspective and the research was oriented in the interpretive paradigm. Data were collected through video-recorded observations of three selected teachers and through stimulus recall interviews. In order to generate rich data and support validity, five lessons per selected teacher were video recorded. The study found that the participating teachers incorporated a variety of mathematical gestures into their lessons in order to support and provide mathematical meaning. Further, this study found that gestures facilitated meaning making in mathematics. The findings in the study suggest a need for mathematics teachers to be trained in using gestures appropriately to communicate mathematically in their lessons. In addition, this study discovered a new type of gesture – the overlapping gesture in addition to McNeill's (1992) types, namely: pointing gestures, metaphor gestures, beating gestures and iconic gestures. This case study also showed that the more experienced the teachers are, the more mathematical gestures they produce during their mathematics lessons. , Thesis (MEd) -- Faculty of Education, Education, 2021
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An evaluation of the impact of the Rhodes University Master of Education in Mathematics Education Programme in Namibia from the perspectives of its graduates
- Authors: David, Johannes
- Date: 2021-10-29
- Subjects: Master of education degree Namibia , Rhodes University. Department of Education , Mathematics Study and teaching , Critical thinking , Master of education degree Evaluation , Career development Namibia , Educational change Namibia , College students Attitudes
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/191942 , vital:45181
- Description: Rhodes University (RU) has been offering a MEdME (Masters in Education, Mathematics Education) degree in Namibia for over 12 years without evaluating its success and impact. This study aimed to evaluate the MEdME programme’s impact on the graduates and on mathematics education in Namibia from the perspectives of its graduates. The study also evaluated the factors that enabled or limited the impact of the programme from the perspective of RU staff members. A three-phased research study was implemented to achieve this. Phase I was a MEdME theses audit to analyze what each graduate researched which then guided the sampling process. Phase II was a distribution of a questionnaire to willing MEdME graduates to elicit their views about the programme's impact. In Phase III, selected graduates were interviewed to follow up on the questionnaire. The RU academic staff members were also interviewed about the programme's design and rationale, as well as their perspectives on its impact on graduates' professional lives. The study found that graduates believed that the programme made them more critical reflexive practitioners (CRPs) who: are now self-introspective, can reflect in context and are competent researchers. This is a sign of the programme attaining its objectives of developing and growing CRPs and graduates, who have advanced in their fields, including their ability to do research. The programme also created opportunities for graduates to study further and/or publish papers. The programme further opened up opportunities for graduates’ promotion into positions of influence in the Ministry of Education where they can positively impact on mathematics education transformation in Namibia (METN). The graduates consequently noticed improvements in some areas but they also noticed room for improvement in other areas of mathematics education in Namibia. This study therefore suggests that more mathematics teachers study further, improve their research capacity and contribute more to METN. The study also suggests a national campaign to increase mathematics teachers' content knowledge. Furthermore, the programme should reintroduce coursework to address students’ subject content knowledge. RU is also advised to explore offering a Professional MEd in Namibia for students who want to focus more on improving their mathematics content knowledge. It is also proposed that Namibia establish a research bank where researchers can upload their theses for wider dissemination. , Thesis (MEd) -- Faculty of Education, Education, 2021
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- Authors: David, Johannes
- Date: 2021-10-29
- Subjects: Master of education degree Namibia , Rhodes University. Department of Education , Mathematics Study and teaching , Critical thinking , Master of education degree Evaluation , Career development Namibia , Educational change Namibia , College students Attitudes
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/191942 , vital:45181
- Description: Rhodes University (RU) has been offering a MEdME (Masters in Education, Mathematics Education) degree in Namibia for over 12 years without evaluating its success and impact. This study aimed to evaluate the MEdME programme’s impact on the graduates and on mathematics education in Namibia from the perspectives of its graduates. The study also evaluated the factors that enabled or limited the impact of the programme from the perspective of RU staff members. A three-phased research study was implemented to achieve this. Phase I was a MEdME theses audit to analyze what each graduate researched which then guided the sampling process. Phase II was a distribution of a questionnaire to willing MEdME graduates to elicit their views about the programme's impact. In Phase III, selected graduates were interviewed to follow up on the questionnaire. The RU academic staff members were also interviewed about the programme's design and rationale, as well as their perspectives on its impact on graduates' professional lives. The study found that graduates believed that the programme made them more critical reflexive practitioners (CRPs) who: are now self-introspective, can reflect in context and are competent researchers. This is a sign of the programme attaining its objectives of developing and growing CRPs and graduates, who have advanced in their fields, including their ability to do research. The programme also created opportunities for graduates to study further and/or publish papers. The programme further opened up opportunities for graduates’ promotion into positions of influence in the Ministry of Education where they can positively impact on mathematics education transformation in Namibia (METN). The graduates consequently noticed improvements in some areas but they also noticed room for improvement in other areas of mathematics education in Namibia. This study therefore suggests that more mathematics teachers study further, improve their research capacity and contribute more to METN. The study also suggests a national campaign to increase mathematics teachers' content knowledge. Furthermore, the programme should reintroduce coursework to address students’ subject content knowledge. RU is also advised to explore offering a Professional MEd in Namibia for students who want to focus more on improving their mathematics content knowledge. It is also proposed that Namibia establish a research bank where researchers can upload their theses for wider dissemination. , Thesis (MEd) -- Faculty of Education, Education, 2021
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Exploring visual probability teaching strategies for enhancing mathematical thinking in grade 11 classrooms
- Nghidinwa, Lavinia Tangi-Jehova
- Authors: Nghidinwa, Lavinia Tangi-Jehova
- Date: 2021-10-29
- Subjects: Mathematics Study and teaching (Secondary) Namibia , Probabilities , Visualization , Learning models (Stochastic processes) , VIPROMaths project
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/192002 , vital:45187
- Description: This Namibian case study aimed to explore the use of visualisation tools associated with different teaching strategies in the teaching of probability concepts in Grade 11 by selected teachers, to promote mathematical thinking. This research project is an integral component of the VIPROMaths project whose goal is to research the effective use of visualisation strategies in the mathematics classroom in the Southern African region. As a mathematics teacher, I have observed that mathematics teaching practices in our classrooms have relatively little connection with actual mathematics and as a result, teaching misses opportunities to promote mathematical thinking. This qualitative case study is underpinned by an interpretive paradigm and it is informed by the dual coding theory. Data was collected through survey questionnaires, reflective journals, field notes, observation schedules and stimulus-recall interviews. Firstly, I piloted my study by conducting a survey with the Grade 10-12 mathematics teachers in the Khomas region. The aim of this survey was to understand and explore how teachers in the Khomas region taught probability prior to the intervention programme. The data was analysed quantitatively using descriptive statistics such as tables and bar graphs. The findings from the survey necessitated the need for an intervention programme with some teachers in the region, focused on the use of visual tools to promote mathematical thinking. Lastly, three schools were selected from which three Grade 11 mathematics teachers were chosen to take part in an intervention programme. The goal was to observe how these three teachers use visual probability teaching strategies to enhance mathematical thinking after participating in an intervention programme. Lesson observations showed that all observed teachers used visual models to generate images and used models to develop a probability idea as well as to create platforms for classroom discussions. Interviews revealed that teachers’ views towards probability have shifted from that of being the centre of knowledge to that of a facilitator. As a result, teachers used different models to build on learners’ prior knowledge, to assess whether they grasped the probability concept and extend their teaching to real-life situations. This study concluded that the teachers need to consider using mathematical models for creating a platform for discussion to ensure that their verbal explanations are in line with the visuals incorporated. Coupled with that, the teachers’ correct use of visual probability teaching strategies has the potential of enhancing learners’ mathematical thinking. Therefore, teachers need to teach the learners how to create visuals for enhancing maximise understanding of probability concepts in mathematics. Furthermore, it is hoped that the findings will be useful to mathematics teachers, scholars and educators to improve the teaching of probability. , Thesis (MEd) -- Faculty of Education, Education, 2021
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- Authors: Nghidinwa, Lavinia Tangi-Jehova
- Date: 2021-10-29
- Subjects: Mathematics Study and teaching (Secondary) Namibia , Probabilities , Visualization , Learning models (Stochastic processes) , VIPROMaths project
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/192002 , vital:45187
- Description: This Namibian case study aimed to explore the use of visualisation tools associated with different teaching strategies in the teaching of probability concepts in Grade 11 by selected teachers, to promote mathematical thinking. This research project is an integral component of the VIPROMaths project whose goal is to research the effective use of visualisation strategies in the mathematics classroom in the Southern African region. As a mathematics teacher, I have observed that mathematics teaching practices in our classrooms have relatively little connection with actual mathematics and as a result, teaching misses opportunities to promote mathematical thinking. This qualitative case study is underpinned by an interpretive paradigm and it is informed by the dual coding theory. Data was collected through survey questionnaires, reflective journals, field notes, observation schedules and stimulus-recall interviews. Firstly, I piloted my study by conducting a survey with the Grade 10-12 mathematics teachers in the Khomas region. The aim of this survey was to understand and explore how teachers in the Khomas region taught probability prior to the intervention programme. The data was analysed quantitatively using descriptive statistics such as tables and bar graphs. The findings from the survey necessitated the need for an intervention programme with some teachers in the region, focused on the use of visual tools to promote mathematical thinking. Lastly, three schools were selected from which three Grade 11 mathematics teachers were chosen to take part in an intervention programme. The goal was to observe how these three teachers use visual probability teaching strategies to enhance mathematical thinking after participating in an intervention programme. Lesson observations showed that all observed teachers used visual models to generate images and used models to develop a probability idea as well as to create platforms for classroom discussions. Interviews revealed that teachers’ views towards probability have shifted from that of being the centre of knowledge to that of a facilitator. As a result, teachers used different models to build on learners’ prior knowledge, to assess whether they grasped the probability concept and extend their teaching to real-life situations. This study concluded that the teachers need to consider using mathematical models for creating a platform for discussion to ensure that their verbal explanations are in line with the visuals incorporated. Coupled with that, the teachers’ correct use of visual probability teaching strategies has the potential of enhancing learners’ mathematical thinking. Therefore, teachers need to teach the learners how to create visuals for enhancing maximise understanding of probability concepts in mathematics. Furthermore, it is hoped that the findings will be useful to mathematics teachers, scholars and educators to improve the teaching of probability. , Thesis (MEd) -- Faculty of Education, Education, 2021
- Full Text:
Re-imagining the use of the abacus as a visualization tool to develop number sense in Grade 3 learners
- Authors: Elifas, Taimi Ndinelago
- Date: 2021-10-29
- Subjects: Abacus , Mathematics Study and teaching (Elementary) Namibia , Visual learning , Teaching Aids and devices , Constructivism (Education) , Effective teaching Namibia
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/191953 , vital:45182
- Description: The poor performance of learners in mathematics has long been a matter of concern in Namibia. After more than three decades after independence, the country's education system is still seeking ways of addressing the high rate of poor performance in mathematics. Research previously conducted pointed out the lack of number sense in learners, as one of the contributing factors to the poor performance in mathematics. This research study is a contribution towards using the abacus as a visualization tool to develop number sense in the Foundation Phase. The study was premised on the thought – supported by literature – that effective use of concrete equipment and manipulatives, of which the abacus is part of, does affect the teaching and learning of number sense. The study further examines the use of the abacus as a visualization tool to develop number sense by the four selected Grade 3 teachers. The study argues that the effective use of the abacus develops number sense in learners. This study was framed as a multiple case study that was grounded within an interpretive paradigm and informed by the constructivist learning theory. The qualitative data of this study were collected using questionnaires, observation, reflective journals, and semi-structured interviews. The data were then analyzed using thematic analysis and an analytical tool developed from relevant literature. A survey was conducted using 50 Foundation Phase teachers in the //Kharas region, while the intervention programme consisted of four purposively selected teachers from the four primary schools in the Kalahari circuit in the Keetmanshoop district. The findings of this study revealed that the majority of teachers were not aware of abacus use in teaching mathematics before the intervention programme. It was also revealed that the few teachers that use the abacus as a visualization tool to develop number sense, employ it as a counting tool for explaining certain concepts and as well as for teaching simple arithmetic. In this research study, the selected teachers use the abacus to link the abstract mathematics content to a concrete way of doing mathematics. In the absence of the abacus in classrooms, various manipulatives are used to develop learners’ number sense. Mathematical games, verbalizing mathematics concepts, and drawing pictures to visualize abstract concepts among others are used by the selected teachers to enhance the development of number sense in the Foundation Phase. The use of the abacus by the selected teachers effectively fostered the visualization process and the conceptual understanding of number sense in learners. Through the abacus, teachers led their learners into visualizing number sense concepts such as subitizing, computing, performing mental mathematics and physical representation of numbers in different ways. The abacus was used by teachers to enhance listening skills, improve concentration and strengthen the memory of learners. On the other hand, the study also revealed that despite the various benefits of the abacus, it is time consuming. The lack of abacus use in previous grades has a huge impact on the use of it and the development of learners’ number sense in the grades being studied in this research study. Teachers are recommended to make their own abacuses, encourage learners to make their own abacuses from the readily available materials, and to allow the learners to realize that the mathematics they are doing in classrooms is around them. The use of re-imagining, re-envisioning, re-conceptualizing and re-examining of so-called ‘old teaching tools’ such as the abacus, needs to be encouraged through in-service and pre-service teacher training programmes. The study concludes that the use of traditional algorithm methods do not promote conceptual understanding and visual strategies for Foundation Phase learners and should be discouraged. It is hoped that this study will contribute towards improving the practices of mathematics teachers primarily in the Keetmanshoop district, //Kharas Region and in the rest our Namibian schools. , Thesis (MEd) -- Faculty of Education, Education, 2021
- Full Text:
- Authors: Elifas, Taimi Ndinelago
- Date: 2021-10-29
- Subjects: Abacus , Mathematics Study and teaching (Elementary) Namibia , Visual learning , Teaching Aids and devices , Constructivism (Education) , Effective teaching Namibia
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/191953 , vital:45182
- Description: The poor performance of learners in mathematics has long been a matter of concern in Namibia. After more than three decades after independence, the country's education system is still seeking ways of addressing the high rate of poor performance in mathematics. Research previously conducted pointed out the lack of number sense in learners, as one of the contributing factors to the poor performance in mathematics. This research study is a contribution towards using the abacus as a visualization tool to develop number sense in the Foundation Phase. The study was premised on the thought – supported by literature – that effective use of concrete equipment and manipulatives, of which the abacus is part of, does affect the teaching and learning of number sense. The study further examines the use of the abacus as a visualization tool to develop number sense by the four selected Grade 3 teachers. The study argues that the effective use of the abacus develops number sense in learners. This study was framed as a multiple case study that was grounded within an interpretive paradigm and informed by the constructivist learning theory. The qualitative data of this study were collected using questionnaires, observation, reflective journals, and semi-structured interviews. The data were then analyzed using thematic analysis and an analytical tool developed from relevant literature. A survey was conducted using 50 Foundation Phase teachers in the //Kharas region, while the intervention programme consisted of four purposively selected teachers from the four primary schools in the Kalahari circuit in the Keetmanshoop district. The findings of this study revealed that the majority of teachers were not aware of abacus use in teaching mathematics before the intervention programme. It was also revealed that the few teachers that use the abacus as a visualization tool to develop number sense, employ it as a counting tool for explaining certain concepts and as well as for teaching simple arithmetic. In this research study, the selected teachers use the abacus to link the abstract mathematics content to a concrete way of doing mathematics. In the absence of the abacus in classrooms, various manipulatives are used to develop learners’ number sense. Mathematical games, verbalizing mathematics concepts, and drawing pictures to visualize abstract concepts among others are used by the selected teachers to enhance the development of number sense in the Foundation Phase. The use of the abacus by the selected teachers effectively fostered the visualization process and the conceptual understanding of number sense in learners. Through the abacus, teachers led their learners into visualizing number sense concepts such as subitizing, computing, performing mental mathematics and physical representation of numbers in different ways. The abacus was used by teachers to enhance listening skills, improve concentration and strengthen the memory of learners. On the other hand, the study also revealed that despite the various benefits of the abacus, it is time consuming. The lack of abacus use in previous grades has a huge impact on the use of it and the development of learners’ number sense in the grades being studied in this research study. Teachers are recommended to make their own abacuses, encourage learners to make their own abacuses from the readily available materials, and to allow the learners to realize that the mathematics they are doing in classrooms is around them. The use of re-imagining, re-envisioning, re-conceptualizing and re-examining of so-called ‘old teaching tools’ such as the abacus, needs to be encouraged through in-service and pre-service teacher training programmes. The study concludes that the use of traditional algorithm methods do not promote conceptual understanding and visual strategies for Foundation Phase learners and should be discouraged. It is hoped that this study will contribute towards improving the practices of mathematics teachers primarily in the Keetmanshoop district, //Kharas Region and in the rest our Namibian schools. , Thesis (MEd) -- Faculty of Education, Education, 2021
- Full Text:
How selected Grade 7 participants develop conceptual understanding in solving algebraic problems as a result of participating in a screencast intervention
- Authors: Wienekus, George Renier
- Date: 2021-04
- Subjects: Algebra -- Study and teaching -- South Africa , Mathematics -- Study and teaching (Elementary) -- South Africa , Problem solving in children , Algebra -- Ability testing , Algebra -- Computer-assisted instruction
- Language: English
- Type: thesis , text , Masters , MEd
- Identifier: http://hdl.handle.net/10962/176833 , vital:42763
- Description: This research project is an interventionist case study, oriented in the interpretive paradigm, which aims to investigate how selected Grade 7 participants develop conceptual understanding in solving algebraic problems as a result of participating in screencast interventions. The aim of my screencast intervention programme, which lies at the heart of this study, is to develop practices, inter alia, of how such devices and software may be “used to develop conceptual rather than procedural or decorative knowledge” (Larkin & Calder, 2015:1) in solving linear equations. The planned intervention was delivered in the form of a series of screencasts: these take the form of audio-video lessons with an emphasis on the visual impact, and were recorded using an application called Explain Everything. The screencast interventions were delivered via Google Classroom and included animations supported by such conceptual explanations of early algebra as are relevant to Grade 7 students, and in line with the South African Curriculum and Assessment Policy Statements - Department of Education, 2011. The fundamental components of an early algebraic equation that would be relevant to a Grade 7 student were considered and used to develop an analytic framework. This was based on a taxonomy designed according to four identified “clusters” in order to analyse the workings of the purposefully selected Grade 7 participants who were video recorded and questioned in a talk-aloud interview while they completed a post-intervention pencil-and-paper test. What emerges from this research project is that there is a significant need for specific and concentrated technology-based techniques, such as the interventions undertaken here, and that exploration and development in the field could benefit the delivery of a pedagogy for algebra. The pedagogical methods implemented and studied in the form of screencasts proved to be successful and were well received by the learners particularly in relation to the conceptualisation of “symbol sense” and transformation in early algebra. The structure and design of the screencast interventions were important in supporting the acquisition of these concepts and were demonstrated to be worthwhile tools for an epistemological application in a classroom or teaching context. , Thesis (MEd) -- Rhodes University, Faculty of Education, Education, 2021
- Full Text:
- Authors: Wienekus, George Renier
- Date: 2021-04
- Subjects: Algebra -- Study and teaching -- South Africa , Mathematics -- Study and teaching (Elementary) -- South Africa , Problem solving in children , Algebra -- Ability testing , Algebra -- Computer-assisted instruction
- Language: English
- Type: thesis , text , Masters , MEd
- Identifier: http://hdl.handle.net/10962/176833 , vital:42763
- Description: This research project is an interventionist case study, oriented in the interpretive paradigm, which aims to investigate how selected Grade 7 participants develop conceptual understanding in solving algebraic problems as a result of participating in screencast interventions. The aim of my screencast intervention programme, which lies at the heart of this study, is to develop practices, inter alia, of how such devices and software may be “used to develop conceptual rather than procedural or decorative knowledge” (Larkin & Calder, 2015:1) in solving linear equations. The planned intervention was delivered in the form of a series of screencasts: these take the form of audio-video lessons with an emphasis on the visual impact, and were recorded using an application called Explain Everything. The screencast interventions were delivered via Google Classroom and included animations supported by such conceptual explanations of early algebra as are relevant to Grade 7 students, and in line with the South African Curriculum and Assessment Policy Statements - Department of Education, 2011. The fundamental components of an early algebraic equation that would be relevant to a Grade 7 student were considered and used to develop an analytic framework. This was based on a taxonomy designed according to four identified “clusters” in order to analyse the workings of the purposefully selected Grade 7 participants who were video recorded and questioned in a talk-aloud interview while they completed a post-intervention pencil-and-paper test. What emerges from this research project is that there is a significant need for specific and concentrated technology-based techniques, such as the interventions undertaken here, and that exploration and development in the field could benefit the delivery of a pedagogy for algebra. The pedagogical methods implemented and studied in the form of screencasts proved to be successful and were well received by the learners particularly in relation to the conceptualisation of “symbol sense” and transformation in early algebra. The structure and design of the screencast interventions were important in supporting the acquisition of these concepts and were demonstrated to be worthwhile tools for an epistemological application in a classroom or teaching context. , Thesis (MEd) -- Rhodes University, Faculty of Education, Education, 2021
- Full Text:
A critical analysis of selected teachers’ perceptions and experiences of the role that visualisation processes play in their Van Hiele level 1 teaching to migrate their learners to the next Van Hiele level
- Authors: Munichinga, Ben Muyambango
- Date: 2019
- Subjects: Hiele, Pierre M. van. Structure and insight , Visualization , Mathematics -- Study and teaching (Secondary) -- Namibia , Geometry -- Study and teaching (Secondary) -- Namibia , Mathematics -- Study and teaching (Secondary) -- Activity programs -- Namibia
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/96735 , vital:31313
- Description: Learning is a process that involves building on prior knowledge, enriching and exchanging existing understanding where learners’ knowledge base is scaffolded in the construction of knowledge. Research on the teaching and learning of geometry in mathematics suggests that physical manipulation experiences, especially of shapes, is an important process in learning at all ages. The focus of the study was the migration of Grade 8 learners from one Van Hiele level to the next as a result of teachers incorporating visualisation processes and Van Hiele phases of instructions in their teaching. The study underpinned by the social constructivist’s theory, therefore aimed at teachers developing visual materials and using Van Hiele’s phases of instruction to teach two dimensional figures in Geometry. The study was carried out in Namibia, Zambezi region in Bukalo circuit. It involved four schools, with 93 learners and three teacher participants. The research is an interpretive case study of a planned intervention programme, which took a four weeks to complete. Participating learners wrote a Van Hiele Geometric test prior and post the intervention programme to determine their geometric level of thought. Participating teachers all received training on visualisation in mathematics and the Van Hiele theory before the intervention. During the intervention, teacher planned and each taught three lessons on two-dimensional figures. Qualitative data was collected from classroom observation, stimulus recall interviews and focus group interviews. Quantitative data came from the pre and post-test of learners. This study found that on average, Grade 8 learners who participated in the study were operating at levels lower than expected of pupils at their stage of schooling. This study also found that, visualisation processes and the Van Hiele phases are effective when used in geometry lessons to migrate learners from lower Van Hiele levels to higher. For teachers in the same circuit, partnership and planning of difficult topics on an agreed regular basis is recommended. When planning lessons teachers are encouraged to take advantage of the Van Hiele phases of instructions. This study thus recommends the incorporation of visualisation strategies of teaching geometry in particular at primary and lower secondary levels. Mathematics teachers are further encouraged to design visual materials such as Geoboards to use for every topic in geometry. Such visual materials should be carefully developed and evaluated to ensure that their use in the classroom is effectively linked to concepts under discussion in a given lesson.
- Full Text:
- Authors: Munichinga, Ben Muyambango
- Date: 2019
- Subjects: Hiele, Pierre M. van. Structure and insight , Visualization , Mathematics -- Study and teaching (Secondary) -- Namibia , Geometry -- Study and teaching (Secondary) -- Namibia , Mathematics -- Study and teaching (Secondary) -- Activity programs -- Namibia
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/96735 , vital:31313
- Description: Learning is a process that involves building on prior knowledge, enriching and exchanging existing understanding where learners’ knowledge base is scaffolded in the construction of knowledge. Research on the teaching and learning of geometry in mathematics suggests that physical manipulation experiences, especially of shapes, is an important process in learning at all ages. The focus of the study was the migration of Grade 8 learners from one Van Hiele level to the next as a result of teachers incorporating visualisation processes and Van Hiele phases of instructions in their teaching. The study underpinned by the social constructivist’s theory, therefore aimed at teachers developing visual materials and using Van Hiele’s phases of instruction to teach two dimensional figures in Geometry. The study was carried out in Namibia, Zambezi region in Bukalo circuit. It involved four schools, with 93 learners and three teacher participants. The research is an interpretive case study of a planned intervention programme, which took a four weeks to complete. Participating learners wrote a Van Hiele Geometric test prior and post the intervention programme to determine their geometric level of thought. Participating teachers all received training on visualisation in mathematics and the Van Hiele theory before the intervention. During the intervention, teacher planned and each taught three lessons on two-dimensional figures. Qualitative data was collected from classroom observation, stimulus recall interviews and focus group interviews. Quantitative data came from the pre and post-test of learners. This study found that on average, Grade 8 learners who participated in the study were operating at levels lower than expected of pupils at their stage of schooling. This study also found that, visualisation processes and the Van Hiele phases are effective when used in geometry lessons to migrate learners from lower Van Hiele levels to higher. For teachers in the same circuit, partnership and planning of difficult topics on an agreed regular basis is recommended. When planning lessons teachers are encouraged to take advantage of the Van Hiele phases of instructions. This study thus recommends the incorporation of visualisation strategies of teaching geometry in particular at primary and lower secondary levels. Mathematics teachers are further encouraged to design visual materials such as Geoboards to use for every topic in geometry. Such visual materials should be carefully developed and evaluated to ensure that their use in the classroom is effectively linked to concepts under discussion in a given lesson.
- Full Text:
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.
- Full Text:
An analysis of the roles and functions of teachers’ gestures as visualisation tools in the teaching of mathematics at the junior primary phase (Grades 0 –3)
- Authors: Namakalu, Dietlinde Nelao
- Date: 2019
- Subjects: Gesture , Gesture in mathematics education , Visual learning , Visualization , Education, Elementary -- Namibia
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/96713 , vital:31311
- Description: There is ample evidence in the literature that gestures are important hand and bodily actions that play vital roles during interactions between learners and teachers in the mathematics classroom. These gestures enable both the teacher and the learner to visualise mathematical concepts and ideas. The aim of this Namibian interpretive case study, located in enactivist theory, was to analyse the types of gestures that three selected teachers at the junior primary phase (Grades 0–3) used in their teaching of mathematics. The study also interrogated the roles that these gestures played in the teaching process of the three teachers. As I intended to analyse the types of gestures the selected teachers used and understand their views and perceptions of the roles and functions their gestures played as visualisation tools in the teaching of mathematics, I observed ten lessons of each teacher and video recorded them. I then interviewed them (one-on-one stimulus-recall interviews) and interpreted the gestures they used and utterances they made about using these gestures. The types of gestures that the participating teachers used are classified according to McNeill‘s (1992) framework, namely pointing (deictic) gestures, iconic (illustrators) gestures, metaphoric gestures, beat (motor) gestures and symbolic (emblems) gestures. The study found that the participating teachers incorporated a variety of gestures into their lessons and used them strategically. According to the teachers, gestures made the lessons interesting and encouraged active participation of the learners in the lessons. The teachers also revealed that gestures assist in explaining mathematical concepts and thus affected the learners‘ understanding positively. This study strongly suggests that the usefulness and efficacy of employing gestures as visualisation tools in mathematics education should be strategically harnessed.
- Full Text:
- Authors: Namakalu, Dietlinde Nelao
- Date: 2019
- Subjects: Gesture , Gesture in mathematics education , Visual learning , Visualization , Education, Elementary -- Namibia
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/96713 , vital:31311
- Description: There is ample evidence in the literature that gestures are important hand and bodily actions that play vital roles during interactions between learners and teachers in the mathematics classroom. These gestures enable both the teacher and the learner to visualise mathematical concepts and ideas. The aim of this Namibian interpretive case study, located in enactivist theory, was to analyse the types of gestures that three selected teachers at the junior primary phase (Grades 0–3) used in their teaching of mathematics. The study also interrogated the roles that these gestures played in the teaching process of the three teachers. As I intended to analyse the types of gestures the selected teachers used and understand their views and perceptions of the roles and functions their gestures played as visualisation tools in the teaching of mathematics, I observed ten lessons of each teacher and video recorded them. I then interviewed them (one-on-one stimulus-recall interviews) and interpreted the gestures they used and utterances they made about using these gestures. The types of gestures that the participating teachers used are classified according to McNeill‘s (1992) framework, namely pointing (deictic) gestures, iconic (illustrators) gestures, metaphoric gestures, beat (motor) gestures and symbolic (emblems) gestures. The study found that the participating teachers incorporated a variety of gestures into their lessons and used them strategically. According to the teachers, gestures made the lessons interesting and encouraged active participation of the learners in the lessons. The teachers also revealed that gestures assist in explaining mathematical concepts and thus affected the learners‘ understanding positively. This study strongly suggests that the usefulness and efficacy of employing gestures as visualisation tools in mathematics education should be strategically harnessed.
- Full Text:
Investigating how the use of visual models can enhance the teaching of common fractions for conceptual understanding to Grade 8 learners
- Authors: Katenda, Aune Kashikuka
- Date: 2019
- Subjects: Fractions -- Study and teaching (Secondary)-- Namibia , Mathematics -- Study and teaching (Secondary)-- Namibia , Information visualization , Visual learning -- Case studies
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/96746 , vital:31314
- Description: The intention of this study was to explore how selected mathematics teachers used visual models to improve the teaching of common fractions for conceptual understanding to Grade 8 learners as a result of an intervention programme. This research study is part of the VIPROmaths project which seeks to research the effective use of visualisation processes in the mathematics classroom in South Africa, Namibia, Zambia, Switzerland and Germany. This study which adopted a case study of teachers in Khomas Region, Namibia, is informed by constructivist learning theory. The study is situated within the interpretive paradigm and a multi-phase mixed method research approach was used. It focussed on analysing the use of visual models when teaching fractions namely: area model, number line model and a set model. The data were collected through survey questionnaires, observation and recall interview. The survey was conducted with the forty three mathematics teachers, from twenty secondary schools in Khomas region. The survey gave an overview of the nature and the use of visual models in schools. Three teachers purposively selected from the survey participated in the intervention program and were observed while teaching and interviewed after their teaching. Data were qualitatively and quantitatively analysed. The findings of this study reveal that visualising fractions is one of the methods that can improve both teaching and learning by providing concrete evidence of otherwise abstract ideas and concepts. The teachers highlighted that models themselves guide learners through to the answer, as compared to working out solutions using symbols only. They further indicated that visual models improve learners’ motivation, enhances understanding of fractions and encourages full participation of learners in the lesson. The study also found that use of visual models encouraged participation and it also boosted learners thinking capability. Teachers in this study preferred to use the area model as they found this model easier and more user-friendly in comparison with the number line and the set models. Teachers did not use the set model because of its complexity. This study concludes that the use of visual models can help enhance the conceptual teaching and understanding of common fractions. It is hoped that the study contributes towards improving the quality teaching and learning of fractions in Namibia. Furthermore, it informs the teacher-training institutions in Namibia to integrate the use of visualisation in their training programmes to promote conceptual understanding of mathematics.
- Full Text:
- Authors: Katenda, Aune Kashikuka
- Date: 2019
- Subjects: Fractions -- Study and teaching (Secondary)-- Namibia , Mathematics -- Study and teaching (Secondary)-- Namibia , Information visualization , Visual learning -- Case studies
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/96746 , vital:31314
- Description: The intention of this study was to explore how selected mathematics teachers used visual models to improve the teaching of common fractions for conceptual understanding to Grade 8 learners as a result of an intervention programme. This research study is part of the VIPROmaths project which seeks to research the effective use of visualisation processes in the mathematics classroom in South Africa, Namibia, Zambia, Switzerland and Germany. This study which adopted a case study of teachers in Khomas Region, Namibia, is informed by constructivist learning theory. The study is situated within the interpretive paradigm and a multi-phase mixed method research approach was used. It focussed on analysing the use of visual models when teaching fractions namely: area model, number line model and a set model. The data were collected through survey questionnaires, observation and recall interview. The survey was conducted with the forty three mathematics teachers, from twenty secondary schools in Khomas region. The survey gave an overview of the nature and the use of visual models in schools. Three teachers purposively selected from the survey participated in the intervention program and were observed while teaching and interviewed after their teaching. Data were qualitatively and quantitatively analysed. The findings of this study reveal that visualising fractions is one of the methods that can improve both teaching and learning by providing concrete evidence of otherwise abstract ideas and concepts. The teachers highlighted that models themselves guide learners through to the answer, as compared to working out solutions using symbols only. They further indicated that visual models improve learners’ motivation, enhances understanding of fractions and encourages full participation of learners in the lesson. The study also found that use of visual models encouraged participation and it also boosted learners thinking capability. Teachers in this study preferred to use the area model as they found this model easier and more user-friendly in comparison with the number line and the set models. Teachers did not use the set model because of its complexity. This study concludes that the use of visual models can help enhance the conceptual teaching and understanding of common fractions. It is hoped that the study contributes towards improving the quality teaching and learning of fractions in Namibia. Furthermore, it informs the teacher-training institutions in Namibia to integrate the use of visualisation in their training programmes to promote conceptual understanding of mathematics.
- Full Text:
An investigation into teaching mathematics using a visualisation approach to recontextualise indigenous knowledge
- Authors: Tshithigona, Gaus
- Date: 2018
- Subjects: Ethnoscience Namibia , Mathematics Study and teaching (Elementary) Namibia , Visual learning , Visualization , Interdisciplinary approach to knowledge , Teacher effectiveness Namibia , Traditional ecological knowledge Namibia
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/62461 , vital:28195
- Description: It can be argued that the Namibian curriculum is largely influenced by a Western epistemology. However, many studies suggest that learners make powerful meanings of mathematical concepts they are learning if they visualise these and experience them in relation to what they already know from their own cultural backgrounds. According to the Namibian National Curriculum for Basic Education (NNCBE, 2010), it is not only important for learners to acquire mathematical knowledge and skills, but also to develop and grow their identities, cultures and values as individuals. The aim of this study was to explore and investigate how selected mathematics teachers employ visualisation as a teaching approach to re-contextualise indigenous knowledge (IK). The study set out to explore how conceptual understanding is enhanced by participation in an intervention programme. The use of visualisation is considered an important mediating and pedagogical tool in the mathematics classroom to enhance the mathematics learning of learners. The research is informed by a socio-cultural theory of learning and is located within an interpretive paradigm. The study was conducted at four schools in the Oshana region of Namibia and involved four mathematics teachers who were purposefully selected due to their willingness to use visualisation-IK approaches in their teaching, based on their responses to the survey. The methodologies used are qualitative and quantitative case study. To gather data, document analysis, a survey, lesson observations and focus group interviews were used. The study found that most teachers in the Oshana region have an understanding of the effective use of visualisation-IK approaches. However, the study revealed that visualisation- IK approaches are mostly used in grades 4 to 7 mathematics classrooms. It also discovered that most of the visualisation approaches that teachers employed aligned well with the curriculum and promoted conceptual understanding in the teaching of mathematics. The results of the study showed that teachers have experience of the enabling and constraining factors in using a visualisation-IK approach to teaching mathematics.
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- Authors: Tshithigona, Gaus
- Date: 2018
- Subjects: Ethnoscience Namibia , Mathematics Study and teaching (Elementary) Namibia , Visual learning , Visualization , Interdisciplinary approach to knowledge , Teacher effectiveness Namibia , Traditional ecological knowledge Namibia
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/62461 , vital:28195
- Description: It can be argued that the Namibian curriculum is largely influenced by a Western epistemology. However, many studies suggest that learners make powerful meanings of mathematical concepts they are learning if they visualise these and experience them in relation to what they already know from their own cultural backgrounds. According to the Namibian National Curriculum for Basic Education (NNCBE, 2010), it is not only important for learners to acquire mathematical knowledge and skills, but also to develop and grow their identities, cultures and values as individuals. The aim of this study was to explore and investigate how selected mathematics teachers employ visualisation as a teaching approach to re-contextualise indigenous knowledge (IK). The study set out to explore how conceptual understanding is enhanced by participation in an intervention programme. The use of visualisation is considered an important mediating and pedagogical tool in the mathematics classroom to enhance the mathematics learning of learners. The research is informed by a socio-cultural theory of learning and is located within an interpretive paradigm. The study was conducted at four schools in the Oshana region of Namibia and involved four mathematics teachers who were purposefully selected due to their willingness to use visualisation-IK approaches in their teaching, based on their responses to the survey. The methodologies used are qualitative and quantitative case study. To gather data, document analysis, a survey, lesson observations and focus group interviews were used. The study found that most teachers in the Oshana region have an understanding of the effective use of visualisation-IK approaches. However, the study revealed that visualisation- IK approaches are mostly used in grades 4 to 7 mathematics classrooms. It also discovered that most of the visualisation approaches that teachers employed aligned well with the curriculum and promoted conceptual understanding in the teaching of mathematics. The results of the study showed that teachers have experience of the enabling and constraining factors in using a visualisation-IK approach to teaching mathematics.
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An analysis of how geogebra can be used as a visualisation tool by selected teachers to develop conceptual understanding of the properties of geometric shapes in grade 9 learners: a case study in Namibia
- Authors: Mwiikeni, Eramus
- Date: 2017
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/6989 , vital:21207
- Description: According to Rosken & Rolka (2006), learning mathematics through visualisations can be a powerful tool to explore mathematical problems and give meaning to mathematical concepts and relationships between them. “Visualisation can reduce the complexity of mathematical problems when dealing with a multitude of information” (p.458). This case study focused on using GeoGebra as a visualisation tool to teach angle properties in Grade 9 geometry. This study set out to analyse how GeoGebra visualisations can be used by selected teachers to teach for conceptual understanding. The research is based on a constructivist view of learning and is oriented within an interpretive paradigm. The methodology used is a qualitative case study. The study was conducted in one school and involved 3 mathematics teachers who were purposefully selected because they showed willingness to use technology in their teaching. I used classroom observations and interviews to collect the data. The study identified a number of factors from the participants that related to using GeoGebra in teaching for conceptual understanding. These include the effective use of dynamic visuals to build on prior knowledge, using multiple representations through image generation and image transformation to make connections and using visuals to justify mathematics ideas. The results from this study indicated that GeoGebra can indeed be used effectively as a teaching tool to teach for conceptual understanding in mathematics.
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- Authors: Mwiikeni, Eramus
- Date: 2017
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/6989 , vital:21207
- Description: According to Rosken & Rolka (2006), learning mathematics through visualisations can be a powerful tool to explore mathematical problems and give meaning to mathematical concepts and relationships between them. “Visualisation can reduce the complexity of mathematical problems when dealing with a multitude of information” (p.458). This case study focused on using GeoGebra as a visualisation tool to teach angle properties in Grade 9 geometry. This study set out to analyse how GeoGebra visualisations can be used by selected teachers to teach for conceptual understanding. The research is based on a constructivist view of learning and is oriented within an interpretive paradigm. The methodology used is a qualitative case study. The study was conducted in one school and involved 3 mathematics teachers who were purposefully selected because they showed willingness to use technology in their teaching. I used classroom observations and interviews to collect the data. The study identified a number of factors from the participants that related to using GeoGebra in teaching for conceptual understanding. These include the effective use of dynamic visuals to build on prior knowledge, using multiple representations through image generation and image transformation to make connections and using visuals to justify mathematics ideas. The results from this study indicated that GeoGebra can indeed be used effectively as a teaching tool to teach for conceptual understanding in mathematics.
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An analysis of selected grade 11 learners’ interactions with geometry tasks using visualization processes: a case study in Namibia
- Authors: Kabuku, Brian S
- Date: 2017
- Subjects: Mathematics -- Study and teaching -- Activity programs , Geometry -- Study and teaching (Secondary) -- Namibia , Geometry -- Study and teaching (Secondary) -- Namibia -- Cast studies , Visualization
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/5949 , vital:20997
- Description: This case study was conducted at a secondary school where I teach, situated in the semi-rural setting of Bukalo village in Namibia, and sought to gain insights into the nature and role of visualisation processes employed when selected grade 11 learners interacted with selected geometry problems. According to Mariotti and Pensci (1994), visualisation takes place when "thinking is spontaneously accompanied and supported by images”, and helps students to understand the problem at hand. Visualisation is regarded as "making the unseen visible and imagery as the power to imagine the possible and the impossible” (Mason 1992). The study is located within an interpretive research paradigm in order to obtain in-depth understanding of the participants’ visualisation processes. Within this paradigm, both quantitative and qualitative approaches were adopted. The eight Grade 11 participants engaged with 12 items of the Geometry Visualisation Tasks (GVT) worksheets. Data was collected using video-recorded learners’ interactions with the GVT, observations, stimulated recall interviews and post-GVT interviews with the learners. During the data analysis stage, I used inductive analysis to determine patterns evident in learners ‘thinking processes’. My analytical framework consisted of indicators that were used to identify and classify visualisation processes for each task of the GVT for each participant. I adapted this framework from Ho (2010) and Ho, Ramful and Lowrie’s (2015) clarification of the representations. The findings from this study revealed that the use of visualisations facilitated meaningful learning when learners made use of these to develop and scaffold their conceptual understanding. The findings revealed that most learners used visualisation processes fairly to very accurately when solving geometry problems. They used visualisation processes by using sketches and diagrams that transformed a mathematical problem pictorially, connected their thinking to previous knowledge and experience, clarified the algebraic task and assisted them to understand the spatial relationships within each task.
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- Authors: Kabuku, Brian S
- Date: 2017
- Subjects: Mathematics -- Study and teaching -- Activity programs , Geometry -- Study and teaching (Secondary) -- Namibia , Geometry -- Study and teaching (Secondary) -- Namibia -- Cast studies , Visualization
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/5949 , vital:20997
- Description: This case study was conducted at a secondary school where I teach, situated in the semi-rural setting of Bukalo village in Namibia, and sought to gain insights into the nature and role of visualisation processes employed when selected grade 11 learners interacted with selected geometry problems. According to Mariotti and Pensci (1994), visualisation takes place when "thinking is spontaneously accompanied and supported by images”, and helps students to understand the problem at hand. Visualisation is regarded as "making the unseen visible and imagery as the power to imagine the possible and the impossible” (Mason 1992). The study is located within an interpretive research paradigm in order to obtain in-depth understanding of the participants’ visualisation processes. Within this paradigm, both quantitative and qualitative approaches were adopted. The eight Grade 11 participants engaged with 12 items of the Geometry Visualisation Tasks (GVT) worksheets. Data was collected using video-recorded learners’ interactions with the GVT, observations, stimulated recall interviews and post-GVT interviews with the learners. During the data analysis stage, I used inductive analysis to determine patterns evident in learners ‘thinking processes’. My analytical framework consisted of indicators that were used to identify and classify visualisation processes for each task of the GVT for each participant. I adapted this framework from Ho (2010) and Ho, Ramful and Lowrie’s (2015) clarification of the representations. The findings from this study revealed that the use of visualisations facilitated meaningful learning when learners made use of these to develop and scaffold their conceptual understanding. The findings revealed that most learners used visualisation processes fairly to very accurately when solving geometry problems. They used visualisation processes by using sketches and diagrams that transformed a mathematical problem pictorially, connected their thinking to previous knowledge and experience, clarified the algebraic task and assisted them to understand the spatial relationships within each task.
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Observing and evaluating creative mathematical reasoning through selected VITALmaths video clips and collaborative argumentation
- Authors: Kellen, Matthew Earl
- Date: 2017
- Subjects: Mathematics Study and teaching (Secondary) South Africa Grahamstown , Mathematics Study and teaching (Secondary) Audio-visual aids , Reasoning , Mathematical ability
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/6107 , vital:21032
- Description: Creative mathematical reasoning is a definition that the NCS policies allude to when they indicate the necessity for students to, “identify and solve problems and make decisions using critical and creative thinking.”(NCS, 2011: 9). Silver (1997) and Lithner (2008) focus on creativity of reasoning in terms of the flexibility, fluency and novelty in which one approaches a mathematical problem. Learners who can creatively select appropriate strategies that are mathematically founded, and justify their answers use creative mathematical reasoning. This research uses Visual Technology for the Autonomous Learning of Mathematics (VITALmaths) video clips that pose mathematics problems to stimulate articulated reasoning among small multi-age, multi-ability Grade 9 peer groups. Using VITALmaths clips that pose visual and open-ended task, set the stage for collaborative argumentation between peers. This study observes creative mathematical reasoning in two ways: Firstly by observing the interaction between peers in the process of arriving at an answer, and secondly by examining the end product of the peer group’s justification of their solution. (Ball & Bass, 2003) Six grade 8 and 9 learners from no-fee public schools in the township of Grahamstown, South Africa were selected for this case study. Participants were a mixed ability, mixed gendered, sample group from an after-school programme which focused on creating a space for autonomous learning. The six participants were split into two groups and audio and video recorded as they solved selected VITALmaths tasks and presented their evidence and solutions to the tasks. Audio and video recordings and written work were used to translate, transcribe, and code participant interactions according to a framework adapted from Krummheuer (2007) and Lithner (2008) and Silver (1997) and Toulmin (1954). This constituted the analysis of the process of creative mathematical reasoning. Group presentations of evidence and solutions to the VITALmaths tasks, were used in conjunction with an evaluation framework adapted from Lithner (2008) and Campos (2010). This was the product analysis of creative mathematical reasoning. This research found that there was significant evidence of creative mathematical reasoning in the process and product evaluation of group interactions and solutions. Process analysis showed that participants were very active, engaged, and creative in their participation, but struggled to integrate and implement ideas cohesively. Product analysis similarly showed that depth and concentration of strategies implemented are key to correct and exhaustive mathematically grounded solutions.
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- Authors: Kellen, Matthew Earl
- Date: 2017
- Subjects: Mathematics Study and teaching (Secondary) South Africa Grahamstown , Mathematics Study and teaching (Secondary) Audio-visual aids , Reasoning , Mathematical ability
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/6107 , vital:21032
- Description: Creative mathematical reasoning is a definition that the NCS policies allude to when they indicate the necessity for students to, “identify and solve problems and make decisions using critical and creative thinking.”(NCS, 2011: 9). Silver (1997) and Lithner (2008) focus on creativity of reasoning in terms of the flexibility, fluency and novelty in which one approaches a mathematical problem. Learners who can creatively select appropriate strategies that are mathematically founded, and justify their answers use creative mathematical reasoning. This research uses Visual Technology for the Autonomous Learning of Mathematics (VITALmaths) video clips that pose mathematics problems to stimulate articulated reasoning among small multi-age, multi-ability Grade 9 peer groups. Using VITALmaths clips that pose visual and open-ended task, set the stage for collaborative argumentation between peers. This study observes creative mathematical reasoning in two ways: Firstly by observing the interaction between peers in the process of arriving at an answer, and secondly by examining the end product of the peer group’s justification of their solution. (Ball & Bass, 2003) Six grade 8 and 9 learners from no-fee public schools in the township of Grahamstown, South Africa were selected for this case study. Participants were a mixed ability, mixed gendered, sample group from an after-school programme which focused on creating a space for autonomous learning. The six participants were split into two groups and audio and video recorded as they solved selected VITALmaths tasks and presented their evidence and solutions to the tasks. Audio and video recordings and written work were used to translate, transcribe, and code participant interactions according to a framework adapted from Krummheuer (2007) and Lithner (2008) and Silver (1997) and Toulmin (1954). This constituted the analysis of the process of creative mathematical reasoning. Group presentations of evidence and solutions to the VITALmaths tasks, were used in conjunction with an evaluation framework adapted from Lithner (2008) and Campos (2010). This was the product analysis of creative mathematical reasoning. This research found that there was significant evidence of creative mathematical reasoning in the process and product evaluation of group interactions and solutions. Process analysis showed that participants were very active, engaged, and creative in their participation, but struggled to integrate and implement ideas cohesively. Product analysis similarly showed that depth and concentration of strategies implemented are key to correct and exhaustive mathematically grounded solutions.
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An ethnographic study of beginner mathematics teachers’ classroom practices in the first three years of their employment: Shaping of a Professional Identity
- Narayanan, Ajayagosh Ettappiriparambil
- Authors: Narayanan, Ajayagosh Ettappiriparambil
- Date: 2016
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:2069 , http://hdl.handle.net/10962/d1020967
- Description: The main theme in this study examines how beginner mathematics teachers (BTs) shape their professional identity in their first three years of classroom practices in Lesotho. This study, which focuses particularly on BTs’ second and third year of employment, gathers data with an understanding that the notion of professional identity is multi-faceted. Professional identity embraces a host of other identities such as personal identity, teacher identity, mathematics identity and community of practice identity. This study is framed by social theories of learning. Learning occurs by active participation and practice. BTs’ peripheral participation assists them in making sense of the activities (situated learning) in which they are engaged, in the classrooms. The sense making processes eventually shape their professional identity. In line with situated meanings that BTs form, the key notion (professional identity) is further categorised into personal identity, teacher identity, mathematics identity and community of practice identity. These identities integrate to become the professional identity of a beginner mathematics teacher. Using a narrative ethnographic approach as the research method, I have made use of extensive classroom observations and interviews to gather data. In this study, six volunteer participant BTs were originally selected. These teachers were from two districts, Berea and Maseru in Lesotho. After being observed in the classrooms, these teachers were interviewed. In the third year of the study, one participant withdrew from the study. I used vertical (descriptive) analysis to narrate their classroom practice followed by horizontal analysis to understand how they shape their professional identity. The analytical model enables the researcher to analyse the data in order to establish how the BTs’ actions, their reflexive stories and their journey in becoming a mathematics teacher shape their professional identity. The recurring themes that emerged from the horizontal analysis are the ways BTs approach the classroom practice which is dominated by teacher-centred learning. This involves demonstrating an example and then students following this model to practice more examples. In this sense, their approach is the same though these BTs started understanding how their classroom approaches can bring changes in the learning of mathematics. I analysed the utterances from the BTs’ classroom activities by separating these into mathematizing and subjectifying. The subjectifying utterances were further analysed to understand how these created meaning. These, in my view, are also central features of a teacher’s practice that need interpretation in order to understand the shaping of a professional identity. The key finding is that their narrative helped them to understand how they shape their professional identity. The study highlights the importance of listening to BTs’ stories of how they become mathematics teachers. Their narratives can be the benchmark for stake-holders, policy makers and potential researchers as the study on BTs’ professional identity is relatively new in Lesotho.
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- Authors: Narayanan, Ajayagosh Ettappiriparambil
- Date: 2016
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:2069 , http://hdl.handle.net/10962/d1020967
- Description: The main theme in this study examines how beginner mathematics teachers (BTs) shape their professional identity in their first three years of classroom practices in Lesotho. This study, which focuses particularly on BTs’ second and third year of employment, gathers data with an understanding that the notion of professional identity is multi-faceted. Professional identity embraces a host of other identities such as personal identity, teacher identity, mathematics identity and community of practice identity. This study is framed by social theories of learning. Learning occurs by active participation and practice. BTs’ peripheral participation assists them in making sense of the activities (situated learning) in which they are engaged, in the classrooms. The sense making processes eventually shape their professional identity. In line with situated meanings that BTs form, the key notion (professional identity) is further categorised into personal identity, teacher identity, mathematics identity and community of practice identity. These identities integrate to become the professional identity of a beginner mathematics teacher. Using a narrative ethnographic approach as the research method, I have made use of extensive classroom observations and interviews to gather data. In this study, six volunteer participant BTs were originally selected. These teachers were from two districts, Berea and Maseru in Lesotho. After being observed in the classrooms, these teachers were interviewed. In the third year of the study, one participant withdrew from the study. I used vertical (descriptive) analysis to narrate their classroom practice followed by horizontal analysis to understand how they shape their professional identity. The analytical model enables the researcher to analyse the data in order to establish how the BTs’ actions, their reflexive stories and their journey in becoming a mathematics teacher shape their professional identity. The recurring themes that emerged from the horizontal analysis are the ways BTs approach the classroom practice which is dominated by teacher-centred learning. This involves demonstrating an example and then students following this model to practice more examples. In this sense, their approach is the same though these BTs started understanding how their classroom approaches can bring changes in the learning of mathematics. I analysed the utterances from the BTs’ classroom activities by separating these into mathematizing and subjectifying. The subjectifying utterances were further analysed to understand how these created meaning. These, in my view, are also central features of a teacher’s practice that need interpretation in order to understand the shaping of a professional identity. The key finding is that their narrative helped them to understand how they shape their professional identity. The study highlights the importance of listening to BTs’ stories of how they become mathematics teachers. Their narratives can be the benchmark for stake-holders, policy makers and potential researchers as the study on BTs’ professional identity is relatively new in Lesotho.
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An exploration of how consistently and precisely mathematics teachers code-switch in multilingual classrooms
- Authors: Chikiwa, Clemence
- Date: 2016
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:2076 , http://hdl.handle.net/10962/d1021304
- Description: Many education research studies conducted in and outside South Africa encourage teachers to take advantage of the presence of multilingualism in their classrooms and to use it to the advancement of students’ conceptual learning. This study adopts the notion that code switching is a potential resource that teachers can use when teaching multilingual mathematics classes. The aim of this study is to determine how precisely and consistently selected teachers of multilingual mathematics classes code switched during teaching of trigonometry and geometry at secondary school. This study is informed by socio-cultural theory in general and Vygotsky’s work in particular. It focussed specifically on the critical role that language plays in the teaching and cognitive development of mathematics. My study situated within an interpretivist paradigm, used a case study research design and a mixed method research approach. Data were obtained through document collection, observing and interviewing three Grade 11 Mathematics teachers purposively selected from three secondary schools in Grahamstown and King Williamstown education districts of the Eastern Cape Province, South Africa. Data were quantitatively and qualitatively analysed. Findings from this study revealed that the frequency of code switching was not consistent across teachers, topics and lessons. Teachers taught predominantly in the public domain exposing students to compromised mathematical content through their code switching practices. Borrowing code switching was prevalently employed consistently across the participating teachers. Very little transparent code switching, from mainly those mathematical terms commonly used in the foundation and the intermediate phases, was evident in teacher language. No Grade 11 trigonometry and geometry terms in isiXhosa were transparently and consistently code switched. The data suggested that while precision was observed in some cases, it was not consistent. Inconsistencies were caused by lack of planning for code switching, lack of teaching materials in indigenous languages, selective code switching, and ‘safe mode’ code switching strategies which affected teachers’ pedagogical practices. Overall results in this study illustrate that the lack of planning for code switching and the lack of explicit policies and clear-cut official positions on code switching for teaching has contributed to inconsistent and imprecise code switching by the participating teachers. This study concludes that the development of supporting mechanisms, identifying and documenting best practices to encourage transparent, meaningful and beneficial code switching is urgently required to aid and promote conceptual understanding of strongly bounded sub-registers of secondary school mathematics such as trigonometry and geometry. It is anticipated that this study will contribute significantly to the ongoing debate on language use in education and to the institution of best practices for judicious, consistent and precise use of students’ home language during the teaching of mathematics in South Africa.
- Full Text:
- Authors: Chikiwa, Clemence
- Date: 2016
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:2076 , http://hdl.handle.net/10962/d1021304
- Description: Many education research studies conducted in and outside South Africa encourage teachers to take advantage of the presence of multilingualism in their classrooms and to use it to the advancement of students’ conceptual learning. This study adopts the notion that code switching is a potential resource that teachers can use when teaching multilingual mathematics classes. The aim of this study is to determine how precisely and consistently selected teachers of multilingual mathematics classes code switched during teaching of trigonometry and geometry at secondary school. This study is informed by socio-cultural theory in general and Vygotsky’s work in particular. It focussed specifically on the critical role that language plays in the teaching and cognitive development of mathematics. My study situated within an interpretivist paradigm, used a case study research design and a mixed method research approach. Data were obtained through document collection, observing and interviewing three Grade 11 Mathematics teachers purposively selected from three secondary schools in Grahamstown and King Williamstown education districts of the Eastern Cape Province, South Africa. Data were quantitatively and qualitatively analysed. Findings from this study revealed that the frequency of code switching was not consistent across teachers, topics and lessons. Teachers taught predominantly in the public domain exposing students to compromised mathematical content through their code switching practices. Borrowing code switching was prevalently employed consistently across the participating teachers. Very little transparent code switching, from mainly those mathematical terms commonly used in the foundation and the intermediate phases, was evident in teacher language. No Grade 11 trigonometry and geometry terms in isiXhosa were transparently and consistently code switched. The data suggested that while precision was observed in some cases, it was not consistent. Inconsistencies were caused by lack of planning for code switching, lack of teaching materials in indigenous languages, selective code switching, and ‘safe mode’ code switching strategies which affected teachers’ pedagogical practices. Overall results in this study illustrate that the lack of planning for code switching and the lack of explicit policies and clear-cut official positions on code switching for teaching has contributed to inconsistent and imprecise code switching by the participating teachers. This study concludes that the development of supporting mechanisms, identifying and documenting best practices to encourage transparent, meaningful and beneficial code switching is urgently required to aid and promote conceptual understanding of strongly bounded sub-registers of secondary school mathematics such as trigonometry and geometry. It is anticipated that this study will contribute significantly to the ongoing debate on language use in education and to the institution of best practices for judicious, consistent and precise use of students’ home language during the teaching of mathematics in South Africa.
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A case study of a pre-service mathematics education course to grow and develop proficient teaching in mathematics in the intermediate phase
- Authors: Lee, Amanda Jane
- Date: 2014
- Subjects: Mathematics -- Study and teaching -- South Africa , Mathematics teachers -- Training of -- South Africa , Mathematics -- Study and teaching (Elementary) -- South Africa
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:2004 , http://hdl.handle.net/10962/d1015664
- Description: This research study investigated the ways in which a mathematics module, informed by an enactivist philosophy, enabled pre-service teachers to unpack the reality of their teaching practice in terms of proficient teaching. Given the generally poor mathematics results in South Africa it is not enough for teachers to be merely proficient in Mathematics. They also need to be in a position to explain important mathematical concepts to children in a manner that will encourage and develop an understanding of the basic mathematical concepts. It was my intention with this study to determine whether a mathematics education module, that embraced the underlying themes of enactivism as part of its teaching pedagogy, could have the potential to develop and increase the skills of pre-service teachers’ teaching for proficiency in Mathematics. The mathematics module was underpinned by five themes of enactivism namely: autonomy, embodiment, emergence, sense-making and experience and was designed to supplement the pre-service teachers’ basic skills in Mathematics in the Intermediate Phase. This mathematics module was offered to fourth year pre-service teachers completing a B.Ed. in the Foundation Phase at a private institute specialising in the training of teachers. The theoretical framework was informed by enactivism and how the themes of enactivism could be used as a vehicle to develop teaching proficiency. The study was qualitative in nature and situated within an interpretivist paradigm. The specific perspectives of interpretivism that were used were hermeneutics, phenomenology and reflexivity. The research design was a case study that contained elements of action research and encompassed three phases of data collection. The first phase focused on the pre-service teachers’ approach to teaching Mathematics and what this brought forth in terms of the reality of their teaching practice and the problems they encountered. The second phase undertook to determine what growth and development of teaching proficiency in Mathematics had emerged over the research period. The final phase was undertaken after the pre-service teachers had graduated and were employed as full time teachers in the Intermediate Phase. The analytical framework and lens through which the data was analysed was that of Kilpatrick, Swafford and Findell’s (2001) strands of mathematical proficiency. The argument that I present is that the themes of enactivism did contribute to the growth of the pre-service teachers’ teaching for mathematical proficiency. The themes of embodiment and experience were major contributions in revealing that this was a reality for the pre-service teachers from a practical perspective and was what they would be able to take away with them. However the theme of emergence stood out as the principle that generated the most awareness and growth and which, in turn, affected the participants’ autonomy.
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- Authors: Lee, Amanda Jane
- Date: 2014
- Subjects: Mathematics -- Study and teaching -- South Africa , Mathematics teachers -- Training of -- South Africa , Mathematics -- Study and teaching (Elementary) -- South Africa
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:2004 , http://hdl.handle.net/10962/d1015664
- Description: This research study investigated the ways in which a mathematics module, informed by an enactivist philosophy, enabled pre-service teachers to unpack the reality of their teaching practice in terms of proficient teaching. Given the generally poor mathematics results in South Africa it is not enough for teachers to be merely proficient in Mathematics. They also need to be in a position to explain important mathematical concepts to children in a manner that will encourage and develop an understanding of the basic mathematical concepts. It was my intention with this study to determine whether a mathematics education module, that embraced the underlying themes of enactivism as part of its teaching pedagogy, could have the potential to develop and increase the skills of pre-service teachers’ teaching for proficiency in Mathematics. The mathematics module was underpinned by five themes of enactivism namely: autonomy, embodiment, emergence, sense-making and experience and was designed to supplement the pre-service teachers’ basic skills in Mathematics in the Intermediate Phase. This mathematics module was offered to fourth year pre-service teachers completing a B.Ed. in the Foundation Phase at a private institute specialising in the training of teachers. The theoretical framework was informed by enactivism and how the themes of enactivism could be used as a vehicle to develop teaching proficiency. The study was qualitative in nature and situated within an interpretivist paradigm. The specific perspectives of interpretivism that were used were hermeneutics, phenomenology and reflexivity. The research design was a case study that contained elements of action research and encompassed three phases of data collection. The first phase focused on the pre-service teachers’ approach to teaching Mathematics and what this brought forth in terms of the reality of their teaching practice and the problems they encountered. The second phase undertook to determine what growth and development of teaching proficiency in Mathematics had emerged over the research period. The final phase was undertaken after the pre-service teachers had graduated and were employed as full time teachers in the Intermediate Phase. The analytical framework and lens through which the data was analysed was that of Kilpatrick, Swafford and Findell’s (2001) strands of mathematical proficiency. The argument that I present is that the themes of enactivism did contribute to the growth of the pre-service teachers’ teaching for mathematical proficiency. The themes of embodiment and experience were major contributions in revealing that this was a reality for the pre-service teachers from a practical perspective and was what they would be able to take away with them. However the theme of emergence stood out as the principle that generated the most awareness and growth and which, in turn, affected the participants’ autonomy.
- Full Text: