Examining mathematical reasoning through enacted visualisation
- Authors: Dongwi, Beata Lididimikeni
- Date: 2019
- Subjects: Visualization , Mathematics -- Study and teaching -- Namibia , Mathematics -- Study and teaching -- Audio-visual aids , Geometry -- Study and teaching , Reasoning , Mathematical ability
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/68192 , vital:29217
- Description: This study sets out to analyse the co-emergence of visualisation and reasoning processes when selected learners engaged in solving word problems. The study argues that visualisation processes and mathematical reasoning processes are closely interlinked in the process of engaging in any mathematical activity. This qualitative research project adopted a case study methodology embedded within a broader interpretative orientation. The research participants were a cohort of 17 mixedgender and mixed-ability Grade 11 learners from a private school in southern Namibia. Data was collected in three phases and comprised of one-on-one task-based interviews in the first phase, focus group task-based interviews in the second, and semi-structured reflective interviews in the third. The analytical framework was informed by elements of enactivism and consisted of a hybrid of observable visualisation and mathematical reasoning indicators. The study was framed by an enactivist perspective that served as a linking mediator to bring visualisation and reasoning processes together, and as a lens through which the coemergence of these processes was observed and analysed. The key enactivist concepts of structural coupling and co-emergence were the two mediating ideas that enabled me to discuss the links between visualisation and reasoning that emerged whilst my participants solved the set word problems. The study argues that the visualisation processes enacted by the participants when solving these problems are inseparable from the reasoning processes that the participants brought to bear; that is, they co-emerged.
- Full Text:
- Date Issued: 2019
- Authors: Dongwi, Beata Lididimikeni
- Date: 2019
- Subjects: Visualization , Mathematics -- Study and teaching -- Namibia , Mathematics -- Study and teaching -- Audio-visual aids , Geometry -- Study and teaching , Reasoning , Mathematical ability
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/68192 , vital:29217
- Description: This study sets out to analyse the co-emergence of visualisation and reasoning processes when selected learners engaged in solving word problems. The study argues that visualisation processes and mathematical reasoning processes are closely interlinked in the process of engaging in any mathematical activity. This qualitative research project adopted a case study methodology embedded within a broader interpretative orientation. The research participants were a cohort of 17 mixedgender and mixed-ability Grade 11 learners from a private school in southern Namibia. Data was collected in three phases and comprised of one-on-one task-based interviews in the first phase, focus group task-based interviews in the second, and semi-structured reflective interviews in the third. The analytical framework was informed by elements of enactivism and consisted of a hybrid of observable visualisation and mathematical reasoning indicators. The study was framed by an enactivist perspective that served as a linking mediator to bring visualisation and reasoning processes together, and as a lens through which the coemergence of these processes was observed and analysed. The key enactivist concepts of structural coupling and co-emergence were the two mediating ideas that enabled me to discuss the links between visualisation and reasoning that emerged whilst my participants solved the set word problems. The study argues that the visualisation processes enacted by the participants when solving these problems are inseparable from the reasoning processes that the participants brought to bear; that is, they co-emerged.
- Full Text:
- Date Issued: 2019
An analysis of how visualisation processes can be used by teachers participating in an intervention programme to teach for conceptual understanding of geometry
- Authors: Muhembo, Gottfried Mbundu
- Date: 2018
- Subjects: Geometry -- Study and teaching (Secondary) -- Namibia -- Kavango East , Visualization , Mathematics teachers -- Namibia -- Kavango East , Effective teaching -- Namibia -- Kavango East , Mathematics -- Study and teaching -- Activity programs
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/62439 , vital:28190
- Description: Visualisation in general and visualisation processes in particular have received much attention in the mathematics education research literature. Literature suggests that the appropriate use of visualisation helps learners to develop their conceptual understanding and skills of geometry as it allows them to visually interpret and understand fundamental mathematical and geometrical concepts. It is claimed that visual tools play an important role in communicating mathematical ideas through diagrams, gestures, images, sketches or drawings. Learning mathematics through visualisation can be a powerful tool to explore mathematical problems and give meaning to mathematical concepts and relationships between them. This interpretive case study focused on how selected teachers taught concepts in geometry through visualisation processes for conceptual understanding as a result of an intervention programme. The study was conducted at four high schools by four mathematics teachers in the Kavango East Region in Northern Namibia. The participants were involved in a three-week intervention programme and afterwards taught three lessons each on the topic of geometry. The data collection method of this research was: focus group and stimulus recall interviews, classroom observations and recorded videos. This research is located in constructivism. I used vertical and horizontal analysis strategies to analyse the data. My analytical instrument consisted of an observation schedule which I used in each lesson to identify how each of the visualisation processes was evident in each of the observed lessons. This study revealed that the participant teachers used visualisation processes in most of their lessons and these processes were used accurately in line with the requirements of the grade 8 mathematics syllabi. The visualisation processes were used through designed visual materials, posters and through the use of geometrical objects such as chalkboard ruler, protractor and compass. The results from this study also confirmed that visualisation processes can be a powerful instructional tool for enhancing learners’ conceptual understanding of geometry.
- Full Text:
- Date Issued: 2018
- Authors: Muhembo, Gottfried Mbundu
- Date: 2018
- Subjects: Geometry -- Study and teaching (Secondary) -- Namibia -- Kavango East , Visualization , Mathematics teachers -- Namibia -- Kavango East , Effective teaching -- Namibia -- Kavango East , Mathematics -- Study and teaching -- Activity programs
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/62439 , vital:28190
- Description: Visualisation in general and visualisation processes in particular have received much attention in the mathematics education research literature. Literature suggests that the appropriate use of visualisation helps learners to develop their conceptual understanding and skills of geometry as it allows them to visually interpret and understand fundamental mathematical and geometrical concepts. It is claimed that visual tools play an important role in communicating mathematical ideas through diagrams, gestures, images, sketches or drawings. Learning mathematics through visualisation can be a powerful tool to explore mathematical problems and give meaning to mathematical concepts and relationships between them. This interpretive case study focused on how selected teachers taught concepts in geometry through visualisation processes for conceptual understanding as a result of an intervention programme. The study was conducted at four high schools by four mathematics teachers in the Kavango East Region in Northern Namibia. The participants were involved in a three-week intervention programme and afterwards taught three lessons each on the topic of geometry. The data collection method of this research was: focus group and stimulus recall interviews, classroom observations and recorded videos. This research is located in constructivism. I used vertical and horizontal analysis strategies to analyse the data. My analytical instrument consisted of an observation schedule which I used in each lesson to identify how each of the visualisation processes was evident in each of the observed lessons. This study revealed that the participant teachers used visualisation processes in most of their lessons and these processes were used accurately in line with the requirements of the grade 8 mathematics syllabi. The visualisation processes were used through designed visual materials, posters and through the use of geometrical objects such as chalkboard ruler, protractor and compass. The results from this study also confirmed that visualisation processes can be a powerful instructional tool for enhancing learners’ conceptual understanding of geometry.
- Full Text:
- Date Issued: 2018
An analysis of visualization processes used by selected Grade 11 and 12 learners when solving algebraic problems: a Namibian case study
- Authors: Josef, Joseane
- Date: 2017
- Subjects: Visualization , Mathematics -- Study and teaching -- Namibia -- Case studies , Mathematics -- Study and teaching (Secondary) -- Activity programs , Algebra -- Study and teaching (Secondary) -- Namibia -- Case studies
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/7843 , vital:21309
- Description: Visualisation is gaining visibility in mathematics education research. It is a powerful tool for solving different types of problems in many areas of mathematics, including Algebra - the mathematical domain of this study. The aim of this case study was to analyse the visualisation processes that selected senior secondary school learners used to solve a set of ten algebraic problems.The research was conducted at a secondary school in the Erongo region of Namibia. This is the school where the researcher is teaching. The sample consisted of six selected learners, three from Grade 11 and three from Grade 12. The learners were purposefully selected to participate in this study based on their mathematics performance in class and their willingness to participate. The participants were video recorded as they worked through the ten items of an Algebraic Visualisation Tasks (AVT) worksheet. They were also interviewed about the visualisation processes they employed when solving each of the tasks. The AVT, the videos and the interviews were analysed with the aid of an adapted visualisation template. The findings of the study show that learners used visualisations in all their problem-solving processes. These visualisations were used for different purposes such as starting points, for illustrative purposes, as organizational tools and as simplification tools. Visualisations as starting points were used when the problem was wordy and had lengthy descriptions and explanations. Illustrative visualisations enabled learners to articulate in their own way mathematical notations, mathematical equations and expressions that they then used to solve the problem. Organizational pictures provided a useful structural framework for solving the problems.This research suggests that the selected participants indeed used visualisation processes to solve algebraic problems. It is thus important for teachers to harness this aid and make the most use of these visualisation processes when teaching Algebra.
- Full Text:
- Date Issued: 2017
- Authors: Josef, Joseane
- Date: 2017
- Subjects: Visualization , Mathematics -- Study and teaching -- Namibia -- Case studies , Mathematics -- Study and teaching (Secondary) -- Activity programs , Algebra -- Study and teaching (Secondary) -- Namibia -- Case studies
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/7843 , vital:21309
- Description: Visualisation is gaining visibility in mathematics education research. It is a powerful tool for solving different types of problems in many areas of mathematics, including Algebra - the mathematical domain of this study. The aim of this case study was to analyse the visualisation processes that selected senior secondary school learners used to solve a set of ten algebraic problems.The research was conducted at a secondary school in the Erongo region of Namibia. This is the school where the researcher is teaching. The sample consisted of six selected learners, three from Grade 11 and three from Grade 12. The learners were purposefully selected to participate in this study based on their mathematics performance in class and their willingness to participate. The participants were video recorded as they worked through the ten items of an Algebraic Visualisation Tasks (AVT) worksheet. They were also interviewed about the visualisation processes they employed when solving each of the tasks. The AVT, the videos and the interviews were analysed with the aid of an adapted visualisation template. The findings of the study show that learners used visualisations in all their problem-solving processes. These visualisations were used for different purposes such as starting points, for illustrative purposes, as organizational tools and as simplification tools. Visualisations as starting points were used when the problem was wordy and had lengthy descriptions and explanations. Illustrative visualisations enabled learners to articulate in their own way mathematical notations, mathematical equations and expressions that they then used to solve the problem. Organizational pictures provided a useful structural framework for solving the problems.This research suggests that the selected participants indeed used visualisation processes to solve algebraic problems. It is thus important for teachers to harness this aid and make the most use of these visualisation processes when teaching Algebra.
- Full Text:
- Date Issued: 2017
An investigation of how Visual Technology for the Autonomous Learning of Mathematics (VITALmaths) video clips on mobile phones can be used by student teachers as a visualisation tool in the teaching of Number Sense: a case study
- Authors: Kangwa, Lemmy
- Date: 2017
- Subjects: Mathematics -- Study and teaching -- Zambia -- Case studies , Visualization , Number concept , Mobile communication systems in education , Mathematics -- Study and teaching -- Software , Telecommunications in education , Educational innovations , VITALmaths
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/6930 , vital:21201
- Description: Visualisation is increasingly being recognised as having a significant role in the learning of mathematics especially when students are solving mathematical problems (Thornton, 2001). It is argued that visualisation is a powerful tool for learners to construct mental and physical representations that correctly mirror mathematical relationships and concepts. To gain a thorough understanding of the scope of visualisation, three Visual Technology for Autonomous Learning of Mathematics (VITALmaths) (www.vitalmaths.com) video clips were uploaded on mobile phones of each of the eleven participating student teachers who used them in their teaching practice. This is in cognisance of the educational potential offered by mobile phones and their current pervasiveness in the daily lives of both teachers and learners in Zambia (Zambia. Ministry of Education [M.O.E], 2013]). This study sought to investigate how VITALmaths video clips on mobile phones could be used by student teachers as a visualisation tool in the teaching of Number Sense. The videos of the lessons formed the core of my analysis. The study was conducted at four primary schools by eleven student teachers of a public university in Zambia. The study is framed as a case study and is grounded within the interpretive paradigm. The findings revealed that the student teachers’ use of the video clips in the classrooms for teaching was generally approached from two perspectives: at the beginning of the lesson or at the end. The videos were used at the beginning of the lesson as a means to introduce a topic or an idea, and at the end of the lesson to consolidate what was taught. The videos were also used to enhance the conceptual understanding of Number Sense. The findings also revealed that students encountered both enabling and constraining factors in their use of mobile phones to teach number sense. The overall findings revealed that, if well utilised, mobile phones as visualisation tools had the potential to enhance the teaching of Mathematics in general and Number Sense in particular, and therefore teachers should be encouraged to use them in their teaching.
- Full Text:
- Date Issued: 2017
- Authors: Kangwa, Lemmy
- Date: 2017
- Subjects: Mathematics -- Study and teaching -- Zambia -- Case studies , Visualization , Number concept , Mobile communication systems in education , Mathematics -- Study and teaching -- Software , Telecommunications in education , Educational innovations , VITALmaths
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/6930 , vital:21201
- Description: Visualisation is increasingly being recognised as having a significant role in the learning of mathematics especially when students are solving mathematical problems (Thornton, 2001). It is argued that visualisation is a powerful tool for learners to construct mental and physical representations that correctly mirror mathematical relationships and concepts. To gain a thorough understanding of the scope of visualisation, three Visual Technology for Autonomous Learning of Mathematics (VITALmaths) (www.vitalmaths.com) video clips were uploaded on mobile phones of each of the eleven participating student teachers who used them in their teaching practice. This is in cognisance of the educational potential offered by mobile phones and their current pervasiveness in the daily lives of both teachers and learners in Zambia (Zambia. Ministry of Education [M.O.E], 2013]). This study sought to investigate how VITALmaths video clips on mobile phones could be used by student teachers as a visualisation tool in the teaching of Number Sense. The videos of the lessons formed the core of my analysis. The study was conducted at four primary schools by eleven student teachers of a public university in Zambia. The study is framed as a case study and is grounded within the interpretive paradigm. The findings revealed that the student teachers’ use of the video clips in the classrooms for teaching was generally approached from two perspectives: at the beginning of the lesson or at the end. The videos were used at the beginning of the lesson as a means to introduce a topic or an idea, and at the end of the lesson to consolidate what was taught. The videos were also used to enhance the conceptual understanding of Number Sense. The findings also revealed that students encountered both enabling and constraining factors in their use of mobile phones to teach number sense. The overall findings revealed that, if well utilised, mobile phones as visualisation tools had the potential to enhance the teaching of Mathematics in general and Number Sense in particular, and therefore teachers should be encouraged to use them in their teaching.
- Full Text:
- Date Issued: 2017
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