Thesis (MEd)--Cape Peninsula University of Technology, 2019. / The mathematics results in South Africa are alarmingly low, with a number of high school learners unable to compute basic operations. International test results show South Africa consistently ranks low in comparison to other countries whilst Singapore continues to perform well. Some schools in South Africa have decided to adopt the Singaporean method of teaching mathematics, known as Singapore Maths, in the hope of improving learner results. This study seeks to understand how two different curricula, South African and Singapore, provide opportunity for the development of algebraic thinking in the Foundation Phase. There is ongoing research which suggests a link between algebraic thinking (Early Algebra) and a deeper conceptual understanding of mathematics (Blanton & Kaput, 2003). This study comprises a qualitative case study of two schools using different curricula and textbooks to teach algebraic thinking with a special focus on patterns and functional thinking. Data were gathered using document analysis of curriculum and textbooks; learner tests; semi structured interviews with class teachers and focus group interviews with Grade 3 learners from each curriculum group. The analysis process involved pattern matching and building explanations related to each data collection instrument using Blanton, Brizuela, Gardiner, Sawrey and Newman-Owen’s (2015) levels of sophistication in learner’s thinking about functional relationships. The results of the study suggest that although South African learners have the potential to think algebraically, they are not, however, always offered the opportunities to do so. The importance of suitable mathematical activities and scaffolding is highlighted and the critical need for professional development for teachers in which the importance of Early Algebra is defined and explained. It is imperative that the curriculum and textbooks activities are relooked at to address the development of algebraic thinking in the early grades and shift the focus from an emphasis on arithmetic relationships to thinking in generalised ways about functional relationships.

published_or_final_version / Education / Master / Master of Education

This was a qualitative study carried out with one grade 8 multicultural, multiethnic, mathematics class. This research study began with the idea of finding out whether the learners home language (especially Zulu Xhosa) could be linked to algebraic misconceptions. The 40 learners (participants) in my study had just been introduced to algebra. I chose the school and participants through “convenience sampling”. This made sense since I am an educator at this particular school. I had explained the meaning of the word "variable" in depth. The concepts "like terms" and "unlike terms" had been explained. The index laws for multiplication and division of the same bases had been discussed. It was within this context that the algebra worksheet was given to the learners, in the first cycle. I examined the algebra errors made by the grade 8 learners after marking the worksheets. I linked the errors to past literature on algebraic misconceptions as well as to Bernard's (2002b) error classification list. The conclusion was that the learners were making common errors which were not affected by their home language. I spent time on reflection since the outcome was not exactly what I had anticipated (that is, I had harboured strong suspicions that English second language learners would commit more algebraic errors than the English home language learners). I then considered a possible link between culture and algebraic misconceptions. Videotaped lessons were used for this purpose. However, observations of these videotaped lessons did not produce much data. I honestly could not reach a conclusion. This formed the second cycle of my action research. Prompted by the obvious lack of interaction in the video recordings from my teaching, I changed my focus to what I, the teacher, did during the lessons, and how these actions may or may not have supported some of the algebraic misconceptions. I reflected on my teaching method and recognized the need to change to a more interactive teaching style. I needed to give the learners the space to think for themselves. I would merely facilitate where necessary. In the third cycle, I drew up a set of problems which matched the new teaching style (interactive teaching).The lessons during which the new set of problems were discussed and solved, were videotaped. These videotaped lessons were analyzed and a completely different picture emerged. The learners were absolutely responsive and showed a side of them that I had not seen before! This study came to be an action research study because I went through three cycles of reflecting, planning, acting and observing and then reflecting, re-planning, further implementation, observing and acting etc. / Thesis (M.Ed.)-University of KwaZulu-Natal, Edgewood, 2010.

Mathematical Literacy (ML) has only recently been introduced to learners, and research in South Africa concerning learners’ conceptual understanding in ML is not widely available. However an important predictor of learners’ success or difficulties in concepts is the success or difficulties that in-service teachers experience themselves. It is therefore important for us as mathematics educators to identify areas in Mathematical Literacy that teachers are struggling to learn and apply. With this in mind, the study sets to explore teachers’ perceptions about, and performance in Mathematical Literacy tasks based on algebraic concepts. This study is located within the principles of the qualitative research case study approach. The combination of data collection techniques has allowed me to identify broad trends across the group as a whole as well as differences within the participants of the group itself. The participants of the study were a class of 17 students who were completing the ACEML programme at UKZN. Four sources of data were used. Firstly, data was generated from teachers’ reflections about certain tasks, the solution of which required the use of algebra. A second data collection instrument was an open-form questionnaire and the third instrument was two unstructured interviews with two teachers. The final instrument was the analysis of the teachers’ examination scripts. For this study, teachers from this group were classified along the lines of whether they were qualified to teach mathematics or not. The theoretical framework for the study was derived from the OECD/PISA (2003) cycle of mathematisation which specifies 5 aspects of mathematisation, together with the theory of reification. For the purpose of this research, a participant was considered as a “mathematics specialist” if s/he studied mathematics up to tertiary level, while a participant was considered as “non-mathematics teacher” if s/he studied mathematics only up to Grade 12 level. The findings reveal that although the teachers conveyed varying understandings of the ML curriculum, they believed that knowledge of basic algebra was necessary and adequate for them to deal with ML problems. Furthermore the teachers believed mathematical teaching experience contributes to improved problem solving in ML and that ‘practice and familiarity’ helped teachers improve their problem solving skills in ML. They also voiced a concern that the pace of the programme constituted a barrier to their success. Within the group, it was found that Mathematics specialist teachers performed better than the non-Mathematics teachers. All teachers found the mathematisation aspects of solving the mathematical problem and of reinterpreting the mathematical solution to make sense of the real-life problems, challenging, while the non-Mathematics teachers experienced problems with all five aspects of mathematisation. The findings of the study suggest that teachers need help in moving from lower levels to higher levels of mathematisation. Opportunities for mathematical modeling experiences need to be incorporated in the part-time in-service contact courses like ACEML. Further research is needed to inform education authorities about whether the use of teachers with only grade 12 mathematical knowledge to teach ML is advisable. / Thesis (M.Ed.)-University of KwaZulu-Natal, Edgewood, 2010.

The purpose of this exploratory study was to look at the importance of Indiana's Algebra I academic standards in relationship with essential, entry-level work skills as perceived by employers and guidance counselors in northern and northeastern Indiana. Counselors and employers were sent electronic (e-mail and web-based) or hard (paper) copies of a survey instrument. Section II of the survey asked for demographic data, (age, race, gender) as well as information related to the size of the business or school, type of business, and years in current position. Respondents were also asked to rate their knowledge of the Indiana academic standards, the Americans with disabilities Act and Section 504 of the Vocational Rehabilitation Act. In Section III, respondents were presented with the nine Indiana Algebra I standards, along with examples to illustrate and further explain each standard. Employers rated each standard, and the standards combined, on its importance to skills required from people seeking jobs immediately after leaving high school using a Likert scale. Guidance counselors rated the importance of the standards based on what they knew of the labor market within school district and surrounding areas using the same Likert scale. Comments were solicited at the end of the survey instruments. Employers disagreed with the importance of any of the Algebra I standards, or the standards as whole, as entry-level work skills. Their highest mean rating was 2.4. Counselors also disagreed, although on two standards their mean rating was above the Neutral rating. Sample size was deemed too low for generalization, but study does supply a framework for future research. / Department of Special Education

<p>This thesis investigates grade seven learners perceived experiences in learning algebra.Things that learners do and say during algebra lessons and about algebra were investigated. The study was done at one of the previously disadvantaged schools in Cape Town, South Africa.The data were collected through observations, a questionnaire and interviews. Observations were made from the day the topic was started in two grade seven classes. Two different teachers taught the two classes. Focus group interviews were conducted, two group of learners, ten learners from each of the two classes were interviewed. Learners devised a number of strategies for solving problems related to sums and differences. The principal learning difficulties experienced by learners in algebra related to the transition from arithmetic conventions to those of algebra, the meaning of literal symbols and the recoginition of structures. It became obvious then that developing algebraic thinking is not necessarily dependent upon algebraic notation and that the presence of algebraic notation says little about the level of problem solving.</p>

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.

Orientador: Dario Fiorentini / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Educação / Made available in DSpace on 2018-08-18T11:52:55Z (GMT). No. of bitstreams: 1 Fernandes_FernandoLuisPereira_M.pdf: 1983189 bytes, checksum: de4eccc8b6aef9d80acef45634dab44f (MD5) Previous issue date: 2011 / Resumo: O objetivo principal desta pesquisa foi compreender como se dá o desenvolvimento do letramento algébrico de alunos do sétimo ano do Ensino Fundamental de uma escola pública no interior do estado de São Paulo. Foram escolhidas duas classes, nas quais eu era o professor responsável. Trata-se de uma pesquisa sobre a própria prática desenvolvida em um ambiente exploratório-investigativo, de natureza qualitativa, sobretudo interpretativa. A coleta de dados contou com a participação de uma auxiliar de pesquisa. O material de campo foi constituído pelo diário de campo do pesquisador e narrativas produzidas pela auxiliar de pesquisa, relatórios e cartazes produzidos pelos alunos e gravações em áudio e vídeo, sendo algumas delas transcritas. Como categorias de análise, temos (i) a mediação realizada pelo professor em diferentes momentos das aulas, dando atenção especial ao momento de socialização/sistematização das apresentações no processo de letramento algébrico; (ii) a constituição de um letramento algébrico, tendo como foco de análise e interpretação as crenças, valores e saberes mobilizados pelos alunos e a produção de sentidos e a negociação de significados desenvolvidas por eles durante a atividade investigativa e a socialização/sistematização de suas produções. Podemos afirmar a aprendizagem não se restringiu às tarefas contempladas. Além de os alunos se apropriarem de um modo de produzir sentido e estabelecer relação com a atividade algébrica, concebida como prática social de leitura e escrita de textos que mobilizam signos próprios da álgebra, também se apropriaram do modo de relacionar-se com o conhecimento matemático, mediante a uma dinâmica exploratório-investigativa. / Abstract: The main objective of this research was to understand how the literacy development of students in algebraic seventh year of elementary school of the public school in the state of Sao Paulo. Two classes were chosen, in which I was the teacher. This is a survey about their own practice developed in an exploratory environment, investigative, qualitative, interpretive above. Data collection included the participation of a research assistant. The field material was created by the diary of the narratives produced by the researcher and research assistant, reports and posters produced by students and in audio and video recordings and some of them transcribed. As categories of analysis, we have (i) mediation carried out by teacher at different times of the classes, giving special attention to the moment of socialization / systematization of the presentations in the literacy process algebraic, (ii) to establish an algebraic literacy, focusing on analysis and interpretation of the beliefs, values and knowledge mobilized by the students and the production of meaning and negotiation of meanings carried by them during the research activity and socialization / systematization of their productions. We affirm the learning was not restricted to tasks covered. Besides the students take ownership of a way to produce meaning and establish relationship with the algebraic activity, conceived as a social practice of reading and writing texts that mobilize themselves signs of algebra, also seized on the way to relate with mathematical knowledge, through a exploratory-investigatory dynamic. / Mestrado / Ensino e Práticas Culturais / Mestre em Educação

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This thesis presents findings made during an examination and comparison of the individualized unit and traditional group methods used to teach ninth grade algebra in Hillsboro, Texas.