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:
- Date Issued: 2019
- 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:
- Date Issued: 2019
Investigating the use of models to develop Grade 8 learners’ conceptual understanding of and procedural fluency with fractions
- Authors: Albin, Simon
- Date: 2017
- Subjects: Fractions -- Study and teaching (Secondary) -- Namibia , Mathematics -- Study and teaching (Secondary) -- Namibia , Information visualization , Visual learning -- Case studies
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/36288 , vital:24537
- Description: Both my teaching experience and literature of this research study strongly suggested that fractions are difficult to teach and learn across the globe generally, and Namibia in particular. One of the identified contributing factors was teaching fractions by focusing on procedures and not the conceptual understanding. Therefore, this research project developed and implemented an intervention in order to experiment and suggest an alternative teaching approach of fractions using models. The purpose of this research was to: “Investigate the use of models to develop Grade 8 learners’ conceptual understanding of and procedural fluency with fractions”. This investigation had three areas of focus. Firstly, the study investigated the nature of learners’ conceptual understanding of and procedural fluency with fractions before the teaching intervention, by means of administering a pre-test and pre-interview and analysing learners’ responses. Secondly, the study investigated the changes in learners’ conceptual understanding of and procedural fluency with fractions after the teaching intervention, by means of administering a post-test, post-interviews and recall interviews, and analysing learners’ responses. Thirdly, this study investigated the possible influence of the teaching intervention on the changes in learners’ conceptual understanding of and procedural fluency with fractions by analysing the lesson videos and learners’ worksheets, and describe their critical interaction. This study was conducted at a multicultural urban secondary school located in the Oshikoto Region, Namibia. The sample consisted of 12 Grade 8 mathematics learners whose age ranged from 13-16 years old. A purposive sampling method was employed to select both the research site and participants. This research is framed as a case study, and is grounded within the interpretive paradigm and qualitative research. This research revealed that these learners displayed conceptual and procedural difficulties in their engagement with fraction models and fraction symbols, before the teaching intervention. Conceptually, the study found that these learners read fractions using inappropriate names; and learners did not identify the whole unit in the models and therefore identified fractions represented by the fraction models using different forms of inappropriate fraction symbols. Procedurally, the study found that these learners compared and ordered fractions inappropriately using the sizes of the numerators and denominators separately; and learners used the lowest common denominator method inappropriately for adding fractions with different denominators. The research also suggested conceptual and procedural changes in learners’ conceptual understanding of and procedural fluency with fractions and that the intervention seemed to help learners to engage better with fraction models and fraction symbols. Conceptually, the findings suggested that the intervention using area models and number lines, seemed to help these learners to read fractions using appropriate names; to identify the whole unit in the fraction models and to develop a sense of the size of fractions in relation to one whole unit. Procedurally, the learners compared and ordered fractions appropriately using either equal fraction bars, equal number lines, benchmarking or rules for comparing and ordering fractions with the same numerator or denominator; and learners used equal fraction bars to visually represent the lowest common denominator method and to recognise that only equally sized units can be counted together. This research identified four factors as possible influences of the teaching intervention. These factors are namely: identifying both fraction symbols and appropriate fraction names to see fractions as relational numbers; prompting to partition whole units of the fraction models and graphically illustrating fraction symbols to identify the whole unit in the fraction models and to develop a sense of the size of fractions in relation to one whole unit; graphically illustrating fraction symbols using the models to use equal fraction bars and number lines, benchmarking and rules for comparing; and graphically illustrating fraction denominations using equal fraction bars to recognise that only equally sized units can be counted together. This research strongly suggests that the effective use of models has the potential to develop learners’ conceptual understanding of and procedural fluency with fractions in a number of ways.
- Full Text:
- Date Issued: 2017
- Authors: Albin, Simon
- Date: 2017
- Subjects: Fractions -- Study and teaching (Secondary) -- Namibia , Mathematics -- Study and teaching (Secondary) -- Namibia , Information visualization , Visual learning -- Case studies
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/36288 , vital:24537
- Description: Both my teaching experience and literature of this research study strongly suggested that fractions are difficult to teach and learn across the globe generally, and Namibia in particular. One of the identified contributing factors was teaching fractions by focusing on procedures and not the conceptual understanding. Therefore, this research project developed and implemented an intervention in order to experiment and suggest an alternative teaching approach of fractions using models. The purpose of this research was to: “Investigate the use of models to develop Grade 8 learners’ conceptual understanding of and procedural fluency with fractions”. This investigation had three areas of focus. Firstly, the study investigated the nature of learners’ conceptual understanding of and procedural fluency with fractions before the teaching intervention, by means of administering a pre-test and pre-interview and analysing learners’ responses. Secondly, the study investigated the changes in learners’ conceptual understanding of and procedural fluency with fractions after the teaching intervention, by means of administering a post-test, post-interviews and recall interviews, and analysing learners’ responses. Thirdly, this study investigated the possible influence of the teaching intervention on the changes in learners’ conceptual understanding of and procedural fluency with fractions by analysing the lesson videos and learners’ worksheets, and describe their critical interaction. This study was conducted at a multicultural urban secondary school located in the Oshikoto Region, Namibia. The sample consisted of 12 Grade 8 mathematics learners whose age ranged from 13-16 years old. A purposive sampling method was employed to select both the research site and participants. This research is framed as a case study, and is grounded within the interpretive paradigm and qualitative research. This research revealed that these learners displayed conceptual and procedural difficulties in their engagement with fraction models and fraction symbols, before the teaching intervention. Conceptually, the study found that these learners read fractions using inappropriate names; and learners did not identify the whole unit in the models and therefore identified fractions represented by the fraction models using different forms of inappropriate fraction symbols. Procedurally, the study found that these learners compared and ordered fractions inappropriately using the sizes of the numerators and denominators separately; and learners used the lowest common denominator method inappropriately for adding fractions with different denominators. The research also suggested conceptual and procedural changes in learners’ conceptual understanding of and procedural fluency with fractions and that the intervention seemed to help learners to engage better with fraction models and fraction symbols. Conceptually, the findings suggested that the intervention using area models and number lines, seemed to help these learners to read fractions using appropriate names; to identify the whole unit in the fraction models and to develop a sense of the size of fractions in relation to one whole unit. Procedurally, the learners compared and ordered fractions appropriately using either equal fraction bars, equal number lines, benchmarking or rules for comparing and ordering fractions with the same numerator or denominator; and learners used equal fraction bars to visually represent the lowest common denominator method and to recognise that only equally sized units can be counted together. This research identified four factors as possible influences of the teaching intervention. These factors are namely: identifying both fraction symbols and appropriate fraction names to see fractions as relational numbers; prompting to partition whole units of the fraction models and graphically illustrating fraction symbols to identify the whole unit in the fraction models and to develop a sense of the size of fractions in relation to one whole unit; graphically illustrating fraction symbols using the models to use equal fraction bars and number lines, benchmarking and rules for comparing; and graphically illustrating fraction denominations using equal fraction bars to recognise that only equally sized units can be counted together. This research strongly suggests that the effective use of models has the potential to develop learners’ conceptual understanding of and procedural fluency with fractions in a number of ways.
- Full Text:
- Date Issued: 2017
Exploring teaching proficiency in geometry of selected effective mathematics teachers in Namibia
- Stephanus, Gervasius Hivengwa
- Authors: Stephanus, Gervasius Hivengwa
- Date: 2014
- Subjects: Mathematics -- Study and teaching (Secondary) -- Namibia , Geometry -- Study and teaching (Secondary) -- Namibia , Effective teaching -- Namibia , Mathematics teachers -- Education (Secondary) -- Namibia , Education, Secondary -- Namibia , Mathematics teachers -- Training of -- Namibia
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:1976 , http://hdl.handle.net/10962/d1013012
- Description: Quality mathematics education relies on effective pedagogy which offers students appropriate and rich opportunities to develop their mathematical proficiency (MP) and intellectual autonomy in learning mathematics. This qualitative case study aimed to explore and analyse selected effective mathematics teachers' proficiency in the area of geometry in five secondary schools in five different Namibia educational regions. The sample was purposefully selected and comprised five mathematics teachers, identified locally as being effective practitioners by their peers, Education Ministry officials and the staff of the University of Namibia (UNAM). The schools where the selected teachers taught were all high performing Namibian schools in terms of students' mathematics performance in the annual national examinations. The general picture of students' poor performance in mathematics in Namibia is no different to other sub-Saharan countries and it is the teachers who unfortunately bear the brunt of the criticism. There are, however, beacons of excellence in Namibia and these often go unnoticed and are seldom written about. It is the purpose of this study to focus on these high achievers and analyse the practices of these teachers so that the rest of Namibia can learn from their practices and experience what is possible in the Namibian context. The mathematical content and context focus of this study was geometry. This qualitative study adopted a multiple case study approach and was framed within an interpretive paradigm. The data were collected through individual questionnaires, classroom lesson observations and in-depth open-ended and semi-structured interviews with the participating teachers. These interviews took the form of post lesson reflective and stimulated recall analysis sessions. An adapted framework based on the Kilpatrick, Swafford and Findell's (2001) five strands of teaching for MP was developed as a conceptual and analytical lens to analyse the selected teachers' practice. The developed coding and the descriptive narrative vignettes of their teaching enabled a qualitative analysis of what teachers said contributed to their effectiveness and how they developed MP in students. An enactivist theoretical lens was used to complement the Kilpatrick et al.'s (2001) analytical framework. This enabled a deeper analysis of teacher teaching practice in terms of their embodied mathematical knowledge, actions and interactions with students. procedural fluency (PF) and productive disposition (PD), were addressed regularly by all five participating teachers. Evidence of addressing either the development of students' strategic competence (SC) or adaptive reasoning (AR) appeared rarely. Of particular interest in this study was that the strand of PD was the glue that held the other four strands of MP together. PD was manifested in many different ways in varying degrees. PD was characterised by a high level of content knowledge, rich personal experience, sustained commitment, effective and careful preparation for lessons, high expectations of themselves and learners, collegiality, passion for mathematics and an excellent work ethic. In addition, the teachers' geometry teaching practices were characterised by making use of real-world connections, manipulatives and representations, encouraging a collaborative approach and working together to show that geometry constituted a bridge between the concrete and abstract. The findings of the study have led me, the author, to suggest a ten (10) principles framework and seven (7) key interrelated factors for effective teaching, as a practical guide for teachers. This study argues that the instructional practices enacted by the participating teachers, who were perceived to be effective, aligned well with practices informed by the five strands of the Kilpatrick et al.'s (2001) model and the four concepts of autopoesis, co-emergence, structural determinism and embodiment of the enactivist approach. The study concludes with recommendations for effective pedagogical practices in the teaching of geometry, and opportunities for further research.
- Full Text:
- Date Issued: 2014
- Authors: Stephanus, Gervasius Hivengwa
- Date: 2014
- Subjects: Mathematics -- Study and teaching (Secondary) -- Namibia , Geometry -- Study and teaching (Secondary) -- Namibia , Effective teaching -- Namibia , Mathematics teachers -- Education (Secondary) -- Namibia , Education, Secondary -- Namibia , Mathematics teachers -- Training of -- Namibia
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:1976 , http://hdl.handle.net/10962/d1013012
- Description: Quality mathematics education relies on effective pedagogy which offers students appropriate and rich opportunities to develop their mathematical proficiency (MP) and intellectual autonomy in learning mathematics. This qualitative case study aimed to explore and analyse selected effective mathematics teachers' proficiency in the area of geometry in five secondary schools in five different Namibia educational regions. The sample was purposefully selected and comprised five mathematics teachers, identified locally as being effective practitioners by their peers, Education Ministry officials and the staff of the University of Namibia (UNAM). The schools where the selected teachers taught were all high performing Namibian schools in terms of students' mathematics performance in the annual national examinations. The general picture of students' poor performance in mathematics in Namibia is no different to other sub-Saharan countries and it is the teachers who unfortunately bear the brunt of the criticism. There are, however, beacons of excellence in Namibia and these often go unnoticed and are seldom written about. It is the purpose of this study to focus on these high achievers and analyse the practices of these teachers so that the rest of Namibia can learn from their practices and experience what is possible in the Namibian context. The mathematical content and context focus of this study was geometry. This qualitative study adopted a multiple case study approach and was framed within an interpretive paradigm. The data were collected through individual questionnaires, classroom lesson observations and in-depth open-ended and semi-structured interviews with the participating teachers. These interviews took the form of post lesson reflective and stimulated recall analysis sessions. An adapted framework based on the Kilpatrick, Swafford and Findell's (2001) five strands of teaching for MP was developed as a conceptual and analytical lens to analyse the selected teachers' practice. The developed coding and the descriptive narrative vignettes of their teaching enabled a qualitative analysis of what teachers said contributed to their effectiveness and how they developed MP in students. An enactivist theoretical lens was used to complement the Kilpatrick et al.'s (2001) analytical framework. This enabled a deeper analysis of teacher teaching practice in terms of their embodied mathematical knowledge, actions and interactions with students. procedural fluency (PF) and productive disposition (PD), were addressed regularly by all five participating teachers. Evidence of addressing either the development of students' strategic competence (SC) or adaptive reasoning (AR) appeared rarely. Of particular interest in this study was that the strand of PD was the glue that held the other four strands of MP together. PD was manifested in many different ways in varying degrees. PD was characterised by a high level of content knowledge, rich personal experience, sustained commitment, effective and careful preparation for lessons, high expectations of themselves and learners, collegiality, passion for mathematics and an excellent work ethic. In addition, the teachers' geometry teaching practices were characterised by making use of real-world connections, manipulatives and representations, encouraging a collaborative approach and working together to show that geometry constituted a bridge between the concrete and abstract. The findings of the study have led me, the author, to suggest a ten (10) principles framework and seven (7) key interrelated factors for effective teaching, as a practical guide for teachers. This study argues that the instructional practices enacted by the participating teachers, who were perceived to be effective, aligned well with practices informed by the five strands of the Kilpatrick et al.'s (2001) model and the four concepts of autopoesis, co-emergence, structural determinism and embodiment of the enactivist approach. The study concludes with recommendations for effective pedagogical practices in the teaching of geometry, and opportunities for further research.
- Full Text:
- Date Issued: 2014
An enquiry into the formative and summative assessment procedures, and perceptions thereof, of grade 10 mathematics teachers : a Namibian case study
- Authors: Marongwe, Anesu Desmond
- Date: 2013
- Subjects: Mathematics -- Study and teaching (Secondary) -- Namibia
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1385 , http://hdl.handle.net/10962/d1001513
- Description: The purpose of this study was to gain insight into observed discrepancies between continuous assessment and final examination average marks in Grade 10 Mathematics in the Oshikoto region of Namibia. The study is framed as a case study and is grounded within the interpretive paradigm. A mixed methods approach was applied, eliciting both quantitative as well as qualitative data. The study took place in two phases. In Phase 1, continuous assessment and Grade 10 final examination average marks for 62 Junior Secondary Schools for the period 2008-2010 were gathered and analyzed. Schools were characterized in terms of the relationship between their continuous assessment and final examination average marks for each of the three years. Phase 2, which was informed by Phase 1, took the form of structured interviews with a sample of three Mathematics teachers and three principals along with a focus-group interview of twelve teachers in order to investigate more deeply the perceptions of teachers and principals toward assessment policy and practice. The study shows that Grade 10 assessment practice in Namibian schools is far from ideal. Many teachers are not fully conversant with the various continuous assessment components as outlined by policy, and teachers are not confident about setting appropriate continuous assessment tasks. There is a strong perception that continuous assessment marks can easily be inflated and those teachers who gave high continuous assessment marks to their learners were generally perceived as being either incompetent or dishonest. While continuous assessment was seen as an important component of teaching and learning, it is evident that teachers and principals would welcome greater clarity, along with standardization and moderation, with respect to continuous assessment practice.
- Full Text:
- Date Issued: 2013
- Authors: Marongwe, Anesu Desmond
- Date: 2013
- Subjects: Mathematics -- Study and teaching (Secondary) -- Namibia
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1385 , http://hdl.handle.net/10962/d1001513
- Description: The purpose of this study was to gain insight into observed discrepancies between continuous assessment and final examination average marks in Grade 10 Mathematics in the Oshikoto region of Namibia. The study is framed as a case study and is grounded within the interpretive paradigm. A mixed methods approach was applied, eliciting both quantitative as well as qualitative data. The study took place in two phases. In Phase 1, continuous assessment and Grade 10 final examination average marks for 62 Junior Secondary Schools for the period 2008-2010 were gathered and analyzed. Schools were characterized in terms of the relationship between their continuous assessment and final examination average marks for each of the three years. Phase 2, which was informed by Phase 1, took the form of structured interviews with a sample of three Mathematics teachers and three principals along with a focus-group interview of twelve teachers in order to investigate more deeply the perceptions of teachers and principals toward assessment policy and practice. The study shows that Grade 10 assessment practice in Namibian schools is far from ideal. Many teachers are not fully conversant with the various continuous assessment components as outlined by policy, and teachers are not confident about setting appropriate continuous assessment tasks. There is a strong perception that continuous assessment marks can easily be inflated and those teachers who gave high continuous assessment marks to their learners were generally perceived as being either incompetent or dishonest. While continuous assessment was seen as an important component of teaching and learning, it is evident that teachers and principals would welcome greater clarity, along with standardization and moderation, with respect to continuous assessment practice.
- Full Text:
- Date Issued: 2013
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