A critical analysis of how the potential of Dynamic Geometry Software as a visualisation tool may enhance the teaching of Mathematics
- Authors: Mavani, Beena Deepak
- Date: 2021-04
- Subjects: Mathematics -- Computer-assisted instruction , Geometry -- Study and teaching (Secondary) -- South Africa -- Mthatha , Manipulatives (Education) -- South Africa -- Mthatha , Information visualization , Mathematics teachers -- Training of -- South Africa -- Mthatha , GeoGebra Literacy Initiative Project (GLIP) , Dynamic Geometry Software (DGS)
- Language: English
- Type: thesis , text , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/177192 , vital:42798 , 10.21504/10962/177192
- Description: Visualisation in the mathematics classroom has its own pedagogical value and plays a significant role in developing mathematical intuition, thought and ideas. Dynamic visualisation possibilities of current digital technologies afford new ways of teaching and learning mathematics. The freely available GeoGebra software package is highly interactive and makes use of powerful features to create objects that are dynamic, and which can be moved around on the computer screen for mathematical exploration. This research study was conceptualised within the GeoGebra Literacy Initiative Project (GLIP) – an ICT teacher development project in Mthatha in the Eastern Cape, South Africa. The focus of this study was on how GeoGebra could be used as a teaching tool by harnessing its powerful visualisation capacity. In the study, selected GLIP teachers collaboratively developed GeoGebra applets, then implemented and evaluated them. The research methodology took the form of action research cycles in which the design, implementation and evaluation of successive applets determined the data gathering and analysis process. My data consisted mainly of recorded observations and reflective interviews. The underlying theoretical foundation of this study lies in constructivism, which aligned well with the conceptual and analytical framework of Kilpatrick et al.’s (2001) description of teaching proficiency. An in-depth analysis of my classroom observations resulted in multiple narratives that illuminated how teachers harnessed the visualisation capabilities inherent in the software. My findings showed that dynamic visualisation and interactivity afforded by the use of technology are key enabling factors for teachers to enhance the visualisation of mathematical concepts. My analysis across participants also showed that technical difficulties often compromised the use of technology in the teaching of mathematics. The significance of this research is its contribution to the ongoing deliberations of visualisation and utilisation of technological resources, particularly through the empowerment of a community of teachers. The findings recognised that the integration of technology required appropriate training, proper planning and continuous support and resources for the teaching of mathematics. This action research provided insightful information on integrating Dynamic Geometry Software (DGS) tools in mathematics classrooms that could be useful to teachers and curriculum planners. , Thesis (PhD) -- Education, Education, 2021
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- Authors: Mavani, Beena Deepak
- Date: 2021-04
- Subjects: Mathematics -- Computer-assisted instruction , Geometry -- Study and teaching (Secondary) -- South Africa -- Mthatha , Manipulatives (Education) -- South Africa -- Mthatha , Information visualization , Mathematics teachers -- Training of -- South Africa -- Mthatha , GeoGebra Literacy Initiative Project (GLIP) , Dynamic Geometry Software (DGS)
- Language: English
- Type: thesis , text , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/177192 , vital:42798 , 10.21504/10962/177192
- Description: Visualisation in the mathematics classroom has its own pedagogical value and plays a significant role in developing mathematical intuition, thought and ideas. Dynamic visualisation possibilities of current digital technologies afford new ways of teaching and learning mathematics. The freely available GeoGebra software package is highly interactive and makes use of powerful features to create objects that are dynamic, and which can be moved around on the computer screen for mathematical exploration. This research study was conceptualised within the GeoGebra Literacy Initiative Project (GLIP) – an ICT teacher development project in Mthatha in the Eastern Cape, South Africa. The focus of this study was on how GeoGebra could be used as a teaching tool by harnessing its powerful visualisation capacity. In the study, selected GLIP teachers collaboratively developed GeoGebra applets, then implemented and evaluated them. The research methodology took the form of action research cycles in which the design, implementation and evaluation of successive applets determined the data gathering and analysis process. My data consisted mainly of recorded observations and reflective interviews. The underlying theoretical foundation of this study lies in constructivism, which aligned well with the conceptual and analytical framework of Kilpatrick et al.’s (2001) description of teaching proficiency. An in-depth analysis of my classroom observations resulted in multiple narratives that illuminated how teachers harnessed the visualisation capabilities inherent in the software. My findings showed that dynamic visualisation and interactivity afforded by the use of technology are key enabling factors for teachers to enhance the visualisation of mathematical concepts. My analysis across participants also showed that technical difficulties often compromised the use of technology in the teaching of mathematics. The significance of this research is its contribution to the ongoing deliberations of visualisation and utilisation of technological resources, particularly through the empowerment of a community of teachers. The findings recognised that the integration of technology required appropriate training, proper planning and continuous support and resources for the teaching of mathematics. This action research provided insightful information on integrating Dynamic Geometry Software (DGS) tools in mathematics classrooms that could be useful to teachers and curriculum planners. , Thesis (PhD) -- Education, Education, 2021
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A design experiment exploring the influence of visual and kinesthetic tools in learning Grade 8 linear algebra in a Namibian secondary school
- Authors: Kalua, Enos
- Date: 2019
- Subjects: Algebra -- 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/10962/92683 , vital:30724
- Description: Based on a broad literature review, understanding algebra is a challenge among learners in middle schools around the world. Early researches also indicated that algebra is often taught through inherent symbols and procedures. This does not exclude Namibian learners in secondary schools whom I have worked with for over 10 years. Examination reports (2014, 2016 and 2017) serve as evidence that learners performed poorly (below 45%) in the area of algebra, with these reports indicating that teachers need to strongly emphasise the issue of solving linear equations. Therefore, this study presents a proposed design research in an attempt to help learners develop meaningful understanding of linear algebra at Grade 8 level. Eight learners whose ages ranged from 13 - 14 years from one Namibian secondary school in Oshikoto region, in northern Namibia, were the participants in this study. The learners represented different groups of learning abilities, ranging from low learning abilities to high learning abilities. The designed programme for this intervention consisting of eight lessons was planned for three weeks and the lessons were conducted in the afternoon to avoid any interruption with normal learning hours. The study used four tools for data collection, namely, benchmark tests (pre-test and post-test), observation, focus groups and unstructured interviews. The data collected for this study was inductively analysed. The purpose of this study was to determine whether and how the specific visual and kinaesthetic teaching tools (diagrams, expansion box and balance method) used may have contributed to learners’ understanding of algebraic concepts and techniques (variables, expressions and equations). The study used diagrams (geometrical plane shapes) for separating terms, an expansion box for expanding brackets and the balance method for solving linear equations. The study revealed the use of diagrams helped the learners in understanding the separation of variable and constant terms when simplifying expressions through addition and/or subtraction. Moreover, the study also revealed that the use of an expansion box was useful for the learners in understanding expansion of brackets in expressions with more than one term. Regarding the use of the balance method, the study showed that learners were already able to solve linear equations by the transfer method, hence, the balance method was not necessary.
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- Authors: Kalua, Enos
- Date: 2019
- Subjects: Algebra -- 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/10962/92683 , vital:30724
- Description: Based on a broad literature review, understanding algebra is a challenge among learners in middle schools around the world. Early researches also indicated that algebra is often taught through inherent symbols and procedures. This does not exclude Namibian learners in secondary schools whom I have worked with for over 10 years. Examination reports (2014, 2016 and 2017) serve as evidence that learners performed poorly (below 45%) in the area of algebra, with these reports indicating that teachers need to strongly emphasise the issue of solving linear equations. Therefore, this study presents a proposed design research in an attempt to help learners develop meaningful understanding of linear algebra at Grade 8 level. Eight learners whose ages ranged from 13 - 14 years from one Namibian secondary school in Oshikoto region, in northern Namibia, were the participants in this study. The learners represented different groups of learning abilities, ranging from low learning abilities to high learning abilities. The designed programme for this intervention consisting of eight lessons was planned for three weeks and the lessons were conducted in the afternoon to avoid any interruption with normal learning hours. The study used four tools for data collection, namely, benchmark tests (pre-test and post-test), observation, focus groups and unstructured interviews. The data collected for this study was inductively analysed. The purpose of this study was to determine whether and how the specific visual and kinaesthetic teaching tools (diagrams, expansion box and balance method) used may have contributed to learners’ understanding of algebraic concepts and techniques (variables, expressions and equations). The study used diagrams (geometrical plane shapes) for separating terms, an expansion box for expanding brackets and the balance method for solving linear equations. The study revealed the use of diagrams helped the learners in understanding the separation of variable and constant terms when simplifying expressions through addition and/or subtraction. Moreover, the study also revealed that the use of an expansion box was useful for the learners in understanding expansion of brackets in expressions with more than one term. Regarding the use of the balance method, the study showed that learners were already able to solve linear equations by the transfer method, hence, the balance method was not necessary.
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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.
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- Authors: Matengu, Given Kahale
- Date: 2019
- Subjects: Manipulatives (Education) , Information visualization , Visualization , Mathematics -- Study and teaching , Mathematics -- Study and teaching -- Activity programs , Geometry -- Study and teaching
- Language: English
- Type: text , Thesis , Masters , MEd
- Identifier: http://hdl.handle.net/10962/96724 , vital:31312
- Description: The relationship between visualisation processes and using manipulatives in the teaching and learning of mathematics is apparent and yet not so vocal in the literature. This could be because of the common mistaken understanding that because manipulatives are visual in nature, then visualisation processes should be obvious. Literature warns that just because something is visual therefore it is transparent, is incorrect. This study argues that the effective use of manipulatives in the teaching of mathematics helps learners to effectively understand mathematical concepts. Research on the teaching and learning of mathematics suggests that physical manipulation experiences, especially of concrete materials concerning shapes, is an important process in learning at all ages. One such teaching tool, the Geoboard, a physical manipulative that employs visualisation processes when correctly used, is explored in this study. The aim of this interpretive case study was to investigate and analyse the use of Geoboards as a visualisation tool in the teaching of the properties of quadrilaterals. The study focused on visualisation processes and the use of Geoboards through a teaching framework that was informed by the Van Hiele phases of teaching geometry. The study was conducted in the Opuwo circuit of the Kunene region, Namibia, and it involved three selected Grade 7 mathematics teachers, each from a different primary school. It was underpinned by a constructivist theory using the Van Hiele phases of teaching geometry and framed within visualisation processes. The study employed the use of qualitative data collection techniques such as observations and interviews. The analysis of the findings of this study revealed that Geoboards were very useful in demonstrating the visual representations of the properties of quadrilaterals in a cheap and yet novel way in the selected teachers’ classes. Moreover, the use of Geoboards by the selected teachers effectively fostered visualisation processes such as concrete pictorial imagery, dynamic imagery, perceptual apprehension, sequential apprehension, discursive apprehension and operative apprehension. It was also revealed that Geoboards enabled the selected teachers to structure and teach their lessons in a well-planned manner according to the Van Hiele phases, although it was difficult for them to adhere strictly to the hierarchy of the phases.
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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.
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- 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.
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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.
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- 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.
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