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
- Full Text:
- Date Issued: 2023-10-13
- 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
- Full Text:
- Date Issued: 2023-10-13
The nature of Visual Representations of multiplication and division exercises in nine Grades 1 to 3 South African textbooks
- Authors: Booysen, Tammy Irene
- Date: 2023-03-29
- Subjects: Mathematics Textbooks , Mathematics Study and teaching (Elementary) South Africa , Visual learning , Multiplication Study and teaching (Elementary) South Africa , Division Study and teaching (Elementary) South Africa , Constructivism (Education)
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/422579 , vital:71959
- Description: Mathematics is a language that is rich in visual representations (Mudaly & Rampersad, 2010). Visual Representations assist us in developing our reasoning skills when solving a problem and our understanding of the relationships between concepts (Ozkan et al., 2018). This thesis focuses on the different visual representations (VR) in South African Foundation Phase mathematics textbooks and workbooks. Textbooks and workbooks play an important role in developing an understanding of mathematical concepts for both teachers and learners (Harries & Spooner, 2000). While teachers generally rely heavily on textbooks, they were a key resource while schools were closed due to COVID-19 lockdown regulations. The theory of Constructivism forms part of the theoretical framework for this study. Constructivism advocates that learners actively construct knowledge through experiences rather than passively receiving knowledge from the outside (Von Glaserfeld, 2001). Vygotsky believed that social interactions create experiences that facilitate the learning and meaning-making process (Vygotsky, 1978). This case study is underpinned by an interpretivist paradigm as it sought to examine the nature of VRs in three Grades 1 - 3 textbooks/workbooks. My research approach is primarily qualitative with descriptive statistics to assist in developing a more comprehensive understanding of the research questions. The study was guided by the analytic tool designed by Fotakopoulou and Spiliotopoulou (2008) which I adapted for Foundation Phase mathematics use. The framework provides insight on the type of VR, VRs relation to content, VRs relation to reality, the function of the VR and dimensionality of a VR. While the workbooks had many more VRs than textbooks, the dominant type of VR in textbooks and workbooks are images. The VRs mostly have a strong relation to content with a realistic relation to reality as they were predominantly 2D representation of a 3D object that had an exemplifying function (type b). , Thesis (MEd) -- Faculty of Education, Education, 2023
- Full Text:
- Date Issued: 2023-03-29
- Authors: Booysen, Tammy Irene
- Date: 2023-03-29
- Subjects: Mathematics Textbooks , Mathematics Study and teaching (Elementary) South Africa , Visual learning , Multiplication Study and teaching (Elementary) South Africa , Division Study and teaching (Elementary) South Africa , Constructivism (Education)
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/422579 , vital:71959
- Description: Mathematics is a language that is rich in visual representations (Mudaly & Rampersad, 2010). Visual Representations assist us in developing our reasoning skills when solving a problem and our understanding of the relationships between concepts (Ozkan et al., 2018). This thesis focuses on the different visual representations (VR) in South African Foundation Phase mathematics textbooks and workbooks. Textbooks and workbooks play an important role in developing an understanding of mathematical concepts for both teachers and learners (Harries & Spooner, 2000). While teachers generally rely heavily on textbooks, they were a key resource while schools were closed due to COVID-19 lockdown regulations. The theory of Constructivism forms part of the theoretical framework for this study. Constructivism advocates that learners actively construct knowledge through experiences rather than passively receiving knowledge from the outside (Von Glaserfeld, 2001). Vygotsky believed that social interactions create experiences that facilitate the learning and meaning-making process (Vygotsky, 1978). This case study is underpinned by an interpretivist paradigm as it sought to examine the nature of VRs in three Grades 1 - 3 textbooks/workbooks. My research approach is primarily qualitative with descriptive statistics to assist in developing a more comprehensive understanding of the research questions. The study was guided by the analytic tool designed by Fotakopoulou and Spiliotopoulou (2008) which I adapted for Foundation Phase mathematics use. The framework provides insight on the type of VR, VRs relation to content, VRs relation to reality, the function of the VR and dimensionality of a VR. While the workbooks had many more VRs than textbooks, the dominant type of VR in textbooks and workbooks are images. The VRs mostly have a strong relation to content with a realistic relation to reality as they were predominantly 2D representation of a 3D object that had an exemplifying function (type b). , Thesis (MEd) -- Faculty of Education, Education, 2023
- Full Text:
- Date Issued: 2023-03-29
Exploring how Grade 10 Biology teachers implement practical activities on food tests in Otjozondjupa Region, Namibia
- Authors: Shoopala, Julia Nelago
- Date: 2022-10-14
- Subjects: Pedagogical content knowledge , Biology Study and teaching (Secondary) Namibia , Biology Study and teaching Activity programs Namibia , Food Testing , Visual learning , Biology teachers In-service training Namibia
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/405401 , vital:70168
- Description: The Namibian examiners’ reports have repeatedly reported that Biology is generally poorly performed in most schools and the topic on food tests has been identified as one of the problematic areas. My assumption is that this could be due in part to the lack of inadequate hands-on practical activities that are conducted in most schools in Namibia. In my view, for students to understand food tests and its associated concepts, they need to do hands-on practical activities. It is against this background that in this study I explored enablers and/or constraints when Grade 10 Biology teachers mediate learning of food tests using hands-on practical activities. This study is underpinned by an interpretive paradigm, within which a qualitative case study approach was employed. For my baseline data, I used a questionnaire and I also interviewed two Grade 10 Biology teachers using semi-structured interviews. Afterwards, the two Grade 10 Biology teachers were observed while teaching the topic of food tests. The lessons were videotaped and thereafter I conducted a stimulated recall interview watching the videos with each of these teachers. Further, we also discussed and reflected as a group on the mediation of learning on food tests. I used Shulman’s Pedagogical Content Knowledge (PCK) as my theoretical framework. Within PCK, I used the five Topic-Specific Pedagogical Knowledge (TSPCK) components by Mavhunga and Rollnick as my analytical framework. A thematic approach to data analysis was employed to come up with categories, sub-themes and themes. The findings revealed that Biology teachers do not have a dedicated laboratory for conducting Biology practicals and would instead conduct them in a common laboratory that is employed for both Physical Science and Biology or in their classrooms. There are inadequate resources to complement hands-on activities and teachers lack the capacity, skills and necessary knowledge needed to deliver practical work. The two participants did not bother to do enough practical work and instead taught Biology as a very theoretical topic, which disadvantaged learners for Paper 3, which serves as an alternative to course work. The study also revealed that teachers failed to conduct Biology practicals. They claimed to be doing so, but there were few practicals/experiments being carried out in those schools. The participants stated that the issue of time to conduct practical experiments for food tests was not sufficient. They complained that the processes were exhausting and tiresome and left them drained as they ran around trying to implement the practical experiments. Furthermore, a large number of learners results in overcrowded Biology classes and makes it difficult for the teachers to control them, which affected the teachers’ ability to perform practical experiments. The participants said that the high number of learners makes it difficult for them to accommodate all students in the laboratories, and that the laboratories themselves do not accommodate many learners. The study, thus, recommends that there should be continuous professional development programmes in schools to assist Biology teachers who have challenges in doing hands-on practical activities. Additionally, improvisation in terms of field trips that deal with biological aspects such as food and chemical manufacturers and hospitals should be carried out to allow learners to gain appreciation of the practical aspects of Biology in education. , Thesis (MEd) -- Faculty of Education, Education, 2022
- Full Text:
- Date Issued: 2022-10-14
- Authors: Shoopala, Julia Nelago
- Date: 2022-10-14
- Subjects: Pedagogical content knowledge , Biology Study and teaching (Secondary) Namibia , Biology Study and teaching Activity programs Namibia , Food Testing , Visual learning , Biology teachers In-service training Namibia
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/405401 , vital:70168
- Description: The Namibian examiners’ reports have repeatedly reported that Biology is generally poorly performed in most schools and the topic on food tests has been identified as one of the problematic areas. My assumption is that this could be due in part to the lack of inadequate hands-on practical activities that are conducted in most schools in Namibia. In my view, for students to understand food tests and its associated concepts, they need to do hands-on practical activities. It is against this background that in this study I explored enablers and/or constraints when Grade 10 Biology teachers mediate learning of food tests using hands-on practical activities. This study is underpinned by an interpretive paradigm, within which a qualitative case study approach was employed. For my baseline data, I used a questionnaire and I also interviewed two Grade 10 Biology teachers using semi-structured interviews. Afterwards, the two Grade 10 Biology teachers were observed while teaching the topic of food tests. The lessons were videotaped and thereafter I conducted a stimulated recall interview watching the videos with each of these teachers. Further, we also discussed and reflected as a group on the mediation of learning on food tests. I used Shulman’s Pedagogical Content Knowledge (PCK) as my theoretical framework. Within PCK, I used the five Topic-Specific Pedagogical Knowledge (TSPCK) components by Mavhunga and Rollnick as my analytical framework. A thematic approach to data analysis was employed to come up with categories, sub-themes and themes. The findings revealed that Biology teachers do not have a dedicated laboratory for conducting Biology practicals and would instead conduct them in a common laboratory that is employed for both Physical Science and Biology or in their classrooms. There are inadequate resources to complement hands-on activities and teachers lack the capacity, skills and necessary knowledge needed to deliver practical work. The two participants did not bother to do enough practical work and instead taught Biology as a very theoretical topic, which disadvantaged learners for Paper 3, which serves as an alternative to course work. The study also revealed that teachers failed to conduct Biology practicals. They claimed to be doing so, but there were few practicals/experiments being carried out in those schools. The participants stated that the issue of time to conduct practical experiments for food tests was not sufficient. They complained that the processes were exhausting and tiresome and left them drained as they ran around trying to implement the practical experiments. Furthermore, a large number of learners results in overcrowded Biology classes and makes it difficult for the teachers to control them, which affected the teachers’ ability to perform practical experiments. The participants said that the high number of learners makes it difficult for them to accommodate all students in the laboratories, and that the laboratories themselves do not accommodate many learners. The study, thus, recommends that there should be continuous professional development programmes in schools to assist Biology teachers who have challenges in doing hands-on practical activities. Additionally, improvisation in terms of field trips that deal with biological aspects such as food and chemical manufacturers and hospitals should be carried out to allow learners to gain appreciation of the practical aspects of Biology in education. , Thesis (MEd) -- Faculty of Education, Education, 2022
- Full Text:
- Date Issued: 2022-10-14
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
- Full Text:
- Date Issued: 2022-04-08
- 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
- Full Text:
- Date Issued: 2022-04-08
- «
- ‹
- 1
- ›
- »