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: Uncatalogued
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/424019 , vital:72116
- Description: Visualisation is commonly used as a tool in introducing algebra through visual or kinaesthetic sequences designed to prompt learners' development of a general rule for moving from a term's position to its output value. Fluency in both the concepts and the conventions of elementary algebra are essential to learners, as algebra forms the language in which many advanced mathematical ideas are encoded. Moreover, algebraic fluency is often associated with an ability to think abstractly about arithmetic processes. In many classrooms, however, research has shown that learners often focus on fluency in algebraic conventions rather than concepts, learning how to manipulate expressions without understanding the algorithms they are taught to follow. This trend can be linked to several causes, including teacher-centred mathematics classrooms in which learners are – whether implicitly or explicitly – encouraged to copy formulae and methods in order to ‘get it right in a test’ without necessarily grasping the underlying logical relationships. This case study, therefore, aimed to determine whether there was value in using visual, kinaesthetic models to broaden and deepen learners' use of algebra. To that end, in the context of an extra-curricular mathematics club that aimed to decentre the teachers and demand innovative ideas of the participants, six pairs of Grade 9 learners were tasked with creating visual representations of a linear algebraic expression using coloured building cubes. The responses to this task over the course of five assignments were many and varied and almost universally displayed a sustained internal logic that the learners were able to explain and develop. Most pairs began with a visual list of terms arranged in sets of towers, pyramids or, in one case, a spiral. At the end of the study, all but one of the pairs had settled on a Visual Expression, in which various colours were used to represent elements of the algebra such as the values of the coefficient, the variable and the constant term. The participants' representations grew in complexity over the course of the study and the conformity of the final responses showed that the club was a collaborative space in which learners shared ideas. However, the structure of the Visual Expressions and their own confessions of nerves about ‘getting it wrong’ in the interviews suggest that the participants were stuck in a mindset that led them to seek out and idealise the representation closest to the original algebra, even though that representation revealed little about the structural relationship underlying the expression. , Thesis (MEd) -- Faculty of Education, Secondary and Post-School Education, 2023
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Indigenous technology and culture in the technology curriculum : starting the conversation : a case study
- Authors: Vandeleur, Sonja
- Date: 2010
- Subjects: Curriculum evaluation -- South Africa Curriculum planning -- South Africa Education -- Curricula -- South Africa Technology -- Study and teaching -- South Africa Indigenous peoples -- Education -- South Africa
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:1573 , http://hdl.handle.net/10962/d1003455
- Description: Since the collapse of apartheid and the first democratic elections of 1994, education in South Africa has undergone fundamental transformation and part of this transformation was the reconstruction of the school curriculum. The new curriculum, known as Curriculum 2005 and developed in 1997, introduced Technology as a new learning area. This study is based on the inclusion of ‘indigenous technology and culture’, a new aspect introduced in a revision of Curriculum 2005. The broad goal of the study was to examine and explore pedagogic practice in relation to the inclusion of ‘indigenous technology and culture’ in the revised National Curriculum Statement for Technology. The study was informed by an examination of literature pertaining to philosophy of technology, indigenous knowledge systems and technology education. The review of the literature highlighted the contested nature of ‘indigenous knowledge systems’. Philosophies on the nature of technological knowledge were reviewed in order to explore the meaning of ‘technology’, and a comparative review of curriculum reform in regard to technology education in various parts of the world was conducted. This study presented an attempt to determine the rationale for the inclusion of ‘indigenous technology and culture’ in the revised National Curriculum Statement for Technology in South Africa and to explore and examine what teachers’ existing practices were in this regard. It also examined a process of participatory co-engagement with a focus group of teachers. This process was an attempt to implement ‘indigenous technology and culture’ of the curriculum in a more meaningful way. A case study approach using an in-depth, interpretive design was used. A questionnaire, document analysis, interviews and focus group discussions were used to conduct the investigation. What emerged from the data analysis was that there was unanimous support for the inclusion of ‘indigenous technology and culture’ in the technology curriculum, but implementation had been problematic. This was partly due to difficulties with the interpretation of this aspect in the curriculum as well as a lack of meaningful teaching and learning for various reasons. The study revealed that teachers face multiple dilemmas in implementing ‘indigenous technology and culture’ as an assessment standard. These dilemmas are pedagogical, political, conceptual, professional and cultural in nature. The intentions of the study were to build a comprehensive understanding of ‘indigenous technology and culture’ and to determine how a focus group of teachers were dealing with this new inclusion. The interpretive study concluded with implications and recommendations for further studies.
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A needs analysis for the implementation of a complementary course in mathematics education for teachers of mathematics in Namibia: a case study
- Authors: Ilukena, Alex Mbonabi
- Date: 2009
- Subjects: Education -- Namibia -- Case studies Mathematics -- Study and teaching -- Namibia -- Case studies Mathematics teachers -- Training of -- Namibia -- Case studies Teacher educators -- Training of -- Case studies Teachers -- Training of -- Namibia -- Case studies
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1677 , http://hdl.handle.net/10962/d1003560
- Description: After the introduction of the Basic Education Teacher Diploma (BETD) in Namibia, a number of studies were conducted on how teachers, lecturers, stakeholders and the Namibian public perceived the BETD program and its implementation. However, very few studies focussed specifically on mathematics subject content knowledge and pedagogical content knowledge in the BETD. The purpose of this study was to investigate the need for a complementary course in mathematics education to address the lack of mathematical content and pedagogical knowledge in the Namibian BETD. The study involved five mathematics school teachers, two mathematics college lecturers in the Kavango educational region and a professor of mathematics education at the University of Namibia. These participants were purposefully selected because of their knowledge and experiences with various aspects of the BETD program. The motivation for conducting this study was to gain a better understanding of some of the issues that have been raised about the BETD program, particularly the perceived inadequacy of mathematical subject content and methodology since the inception of the program. The study adopts a qualitative approach in reporting participants’ reflections. The views of the focal educators and documents such as syllabi and course outlines were the main source of data. The findings indicate that despite the training that the three BETD mathematics graduates in this study received, the level of mathematics taught in a complementary course, such as an ACE, would clearly better equip mathematics teachers to teach proficiently and facilitate access to institutions of higher learning such as universities. The results of the study revealed that there was a need for the implementation of a complementary course to the BETD in mathematics education for teachers of mathematics in Namibia. This study also provided valuable insights into what such a course could look like.
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Critical reflective teaching practice in three mathematics teachers
- Authors: Luwango, Luiya
- Date: 2009
- Subjects: Education (Secondary) -- Namibia Mathematics teachers -- Training of - -Namibia Mathematics -- Study and teaching (Secondary) -- Namibia
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1485 , http://hdl.handle.net/10962/d1003366
- Description: This qualitative study reports on critical reflective teaching by three mathematics teachers and how it shapes their classroom practice. The study was carried out in three secondary schools in Rundu in northern Namibia. The study employed a case study method. The selection of teachers was based on their rich practical professional knowledge and exemplary teaching practices. Data collection and analysis was done through an interpretive approach. Interviews and document analyses were the two research tools used, not only for the collection of data but for triangulation also. Interpretations of the findings were validated through member checking. Critical reflective teaching involves thought and action, and it raises teachers’ consciousness of what they do. Through critical reflective practice, teachers scrutinize their beliefs and knowledge of the subject and their practice. Furthermore critical reflective practice may get teachers into a disposition to find alternatives to improve their teaching. In this study, the findings are that participants reflect extensively on their classroom practice. The teachers pointed out that reflection on practice enables them to analyse and evaluate their teaching in line with effective mathematics teaching. They emphasised that critical reflection leads to the identification of weaknesses in teachers’ classroom practice. This culminates in better planning whereby alternative approaches to teaching are exercised. Because of its potential to improve teaching and enhance professional development it is therefore recommended that mathematics teachers be exposed to skills that enhance critical reflective teaching practice. Teachers need to familiarise themselves with the concept of critical reflective teaching in mathematics to meet the demands of superior quality teaching.
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Using the van Hiele theory to analyse geometrical conceptualisation in grade 12 students: a Namibian perspective
- Authors: Mateya, Muhongo
- Date: 2009
- Subjects: Hiele, Pierre M. van Hiele-Geldof, Dina van Geometry -- Study and teaching (Secondary) -- Namibia Mathematics -- Study and teaching (Secondary) -- Namibia
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1820 , http://hdl.handle.net/10962/d1003706
- Description: The study reported here utilised a theory of levels of geometric thinking. This theory was proposed and developed by two Dutch mathematics educators, Pierre van Hiele and his wife, Dina van Hiele-Geldof. The van Hiele theory enables investigations into why many students experience difficulties in learning geometry. In many nations, such as the UK, the USA, Netherlands, the USSR and to a certain extent, Nigeria and South Africa, research evidence has indicated that the overall students’ mathematical competencies are linked to their geometric thinking levels. This study is the first of its kind to apply the van Hiele theory of geometric thinking in the Namibian context to analyse geometrical conceptualisation in Grade 12 mathematics students. In all, 50 Grade 12 students (20 from School A and 30 from School B) were involved in this study. These students wrote a van Hiele Geometry Test adapted from the Cognitive Development and Achievement in Secondary School Geometry test items. Thereafter, a clinical interview with the aid of manipulatives was conducted. The results from this study indicated that many of the School A and School B students who participated in the research have a weak conceptual understanding of geometric concepts: 35% of the School A and 40% of the School B subsamples were at the prerecognition level. 25% and 30% of the School A, and 20% and 23.3% of the School B students were at van Hiele levels 1 and 2 respectively. An equal number of students but different in percentages, 2 (10%) in School A and 2 (6.7%) in School B, were at van Hiele level 3. Only one student from School B attained van Hiele level 4. These results were found to be consistent with those of previous similar studies in UK, USA, Nigeria and South Africa. The findings of this study also highlight issues of how the Namibian Grade 12 geometry syllabus should be aligned with the van Hiele levels of geometric thinking as well as the use of appropriate and correct language in geometrical thinking and problem solving.
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An analysis of the influence of question design on pupils' approaches to number pattern generalisation tasks
- Authors: Samson, Duncan Alistair
- Date: 2008
- Subjects: Mathematics -- Study and teaching Number theory -- Problems, exercises, etc Algebra -- Study and teaching Arithmetic -- Foundations Pattern perception
- Language: English
- Type: Thesis , Masters , MEd
- Identifier: vital:1421 , http://hdl.handle.net/10962/d1003302
- Description: This study is based on a qualitative investigation framed within an interpretive paradigm, and aims to investigate the extent to which question design affects the solution strategies adopted by children when solving linear number pattern generalisation tasks presented in pictorial and numeric contexts. The research tool comprised a series of 22 pencil and paper exercises based on linear generalisation tasks set in both numeric and 2-dimensional pictorial contexts. The responses to these linear generalisation questions were classified by means of stage descriptors as well as stage modifiers. The method or strategy adopted was carefully analysed and classified into one of seven categories. A meta-analysis focused on the formula derived for the nth term in conjunction with its justification. The process of justification proved to be a critical factor in being able to accurately interpret the origin of the sub-structure evident in many of these responses. From a theoretical perspective, the central role of justification/proof within the context of this study is seen as communication of mathematical understanding, and the process of justification/proof proved to be highly successful in providing a window of understanding into each pupil’s cognitive reasoning. The results of this study strongly support the notion that question design can play a critical role in influencing pupils’ choice of strategy and level of attainment when solving pattern generalisation tasks. Furthermore, this study identified a diverse range of visually motivated strategies and mechanisms of visualisation. An awareness and appreciation for such a diversity of visualisation strategies, as well as an understanding of the importance of appropriate question design, has direct pedagogical application within the context of the mathematics classroom.
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