This doctoral research was concerned with the role of language and its implications for the learning, teaching and assessment of mathematics for children aged 4-9 years. Earlier research by the author had established language and assessment as bridges enabling learning although they had the potential to increase the divide between teacher and learner. Reflection raised the question on how children achieved in mathematics despite potential difficulties with language and assessment. Review of the literature concluded that resources and sociocultural norms were also bridges between learner and teacher. A model was established of the relationships and processes between all perceived variables that provided an external, theoretical structure to be evaluated against structuralist, pragmatic and integrational linguistic approaches and empirical outcomes. The overarching approaches adopted were institutional ethnography and case study. An appropriate methodology was devised whereby sophisticated ICT equipment captured all visual and speech events during classroom interactions. Frequency analysis at word level, content analysis at utterance level and discourse analysis at total speech level triangulated with content analysis of interviews and evaluation of documentation completed the empirical research. Data analysis revealed five registers of children’s talk. Evidence suggested that the peer-peer ‘conditioned talk’ used in focused group work was the most effective for learning as it enabled them to discern the small steps in the inferential leaps in discourse made by their teachers, work out problems together, inform their peers, share findings and reinforce each others’ learning. Learners’ language showed aspects of structural, pragmatic and integrational linguistics, confirming a conclusion of the literature review that the various linguistic approaches discussed should be used to support and not exclude each other. The contribution made to knowledge is the ethnomethodology provided by the model, ICT resource and the five registers of talk revealed by the linguistic approach to discourse analysis. Teachers would be able to understand nuances of language used by their pupils and acquire essential skills and tools to put into effect the personalised learning agenda. Peer-peer observation of teachers would be an appropriate platform for the observation of the different registers used by learners, the resources that generate those registers, and their most effective use to close the gap between natural language and the subject specific language of mathematics.

Early in 2009 it became apparent from articles in the newspapers that first year mathematics students were not performing as well as the students of previous years. There was great concern regarding the insufficient transition from secondary to tertiary mathematics, as well as the preparedness of first year students for university studies. This research focuses on the different factors that are potential causes of the underachievement of first year mathematics students. Students‟ and lecturers‟ beliefs are shaped by their experiences, the impact of continuous perceptions from the world around them, the present dominant paradigm, as well as the beliefs of their teachers. The different views of the nature of school mathematics show how a worldview has an effect on these views and the implications of this on the teaching of mathematics in secondary, as well as tertiary institutions. The paradigm shift from the modern era to the post-modern era caused an awareness of and interest in the construction of meaningful mathematical understanding. The gap between first year students‟ and lecturers‟ beliefs regarding the nature of mathematics and how mathematics is learned became apparent. The changes in the thoughts about the structure of mathematics were investigated and a better understanding of the processes through which mathematical understanding develops emerged. This brought insight into the gap between the reasoning abilities of incoming students from secondary schools and the reasoning needed to succeed in university mathematics. The theoretical study of the global theories of Piaget and Van Hiele gave insight into conceptual development through different stages and that a person should be on an appropriate conceptual level to make sense of what they learn. If not, then rote learning is likely to occur. The local theory of Tall implies that to facilitate understanding of a concept in mathematics, one should go through three worlds of mathematics: the embodied world, symbolic world and the formal world. The embodied view helps someone to give deep meaning to a concept, otherwise one can be trapped in the symbolic world and not be able to move on to the formal world of mathematical thinking. The theoretical investigations led to an empirical study in three phases. Phase 1 was an investigation into the views of mathematics held by the students and the lecturers. In phase 2 an investigation was done to establish the students‟ preferences on how they learn mathematics and how mathematics should be taught, using the Index of Learning Styles (ILS) questionnaire of Felder and Silverman. The results were compared with the way lecturers want their students to learn and how they themselves prefer to teach. Phase 3 included a classification of the questions in the first mathematics test written at tertiary level and subsequent analysis of the answers of students to obtain information on the type of reasoning required from students at tertiary level, as well as the reasoning abilities of the students. The empirical study assisted in understanding the problematic transition from secondary to tertiary mathematics with regard to the nature of mathematics, the beliefs on teaching and learning of mathematics, as well as the reasoning skills that the students possess when entering university. / Thesis (Ph.D. (Natural Sciences Education))--North-West University, Potchefstroom Campus, 2013

Early in 2009 it became apparent from articles in the newspapers that first year mathematics students were not performing as well as the students of previous years. There was great concern regarding the insufficient transition from secondary to tertiary mathematics, as well as the preparedness of first year students for university studies. This research focuses on the different factors that are potential causes of the underachievement of first year mathematics students. Students‟ and lecturers‟ beliefs are shaped by their experiences, the impact of continuous perceptions from the world around them, the present dominant paradigm, as well as the beliefs of their teachers. The different views of the nature of school mathematics show how a worldview has an effect on these views and the implications of this on the teaching of mathematics in secondary, as well as tertiary institutions. The paradigm shift from the modern era to the post-modern era caused an awareness of and interest in the construction of meaningful mathematical understanding. The gap between first year students‟ and lecturers‟ beliefs regarding the nature of mathematics and how mathematics is learned became apparent. The changes in the thoughts about the structure of mathematics were investigated and a better understanding of the processes through which mathematical understanding develops emerged. This brought insight into the gap between the reasoning abilities of incoming students from secondary schools and the reasoning needed to succeed in university mathematics. The theoretical study of the global theories of Piaget and Van Hiele gave insight into conceptual development through different stages and that a person should be on an appropriate conceptual level to make sense of what they learn. If not, then rote learning is likely to occur. The local theory of Tall implies that to facilitate understanding of a concept in mathematics, one should go through three worlds of mathematics: the embodied world, symbolic world and the formal world. The embodied view helps someone to give deep meaning to a concept, otherwise one can be trapped in the symbolic world and not be able to move on to the formal world of mathematical thinking. The theoretical investigations led to an empirical study in three phases. Phase 1 was an investigation into the views of mathematics held by the students and the lecturers. In phase 2 an investigation was done to establish the students‟ preferences on how they learn mathematics and how mathematics should be taught, using the Index of Learning Styles (ILS) questionnaire of Felder and Silverman. The results were compared with the way lecturers want their students to learn and how they themselves prefer to teach. Phase 3 included a classification of the questions in the first mathematics test written at tertiary level and subsequent analysis of the answers of students to obtain information on the type of reasoning required from students at tertiary level, as well as the reasoning abilities of the students. The empirical study assisted in understanding the problematic transition from secondary to tertiary mathematics with regard to the nature of mathematics, the beliefs on teaching and learning of mathematics, as well as the reasoning skills that the students possess when entering university. / Thesis (Ph.D. (Natural Sciences Education))--North-West University, Potchefstroom Campus, 2013

The purpose of the study was to investigate second language learner‟s proficiency in mathematical language. The study was conducted using the Grade 12 learners in a senior secondary school in the district of Mthatha, Eastern Cape. The researcher used a case study approach and a questionnaire was used to investigate whether or not learners understood the language rather than to assess their mathematical ability. English was the language in question and used as the medium of instruction (MOI). In this study, aspects of the language used for mathematics are highlighted, the importance of mathematical vocabulary is discussed and reference is made to some recent research concerned with the understanding of mathematical language. The importance of language as a factor in the learning of mathematics is established through a consideration of bilingualism and the writer‟s own research into the understanding of mathematical language. Perhaps the most obvious educational difficulty which students encounter is that of language. Teachers and curriculum planners need to assess whether this is desirable, and to attempt to find ways in which learners in bilingual education programmes can learn mathematics in a meaningful way. Theories on the interaction between language and mathematical thinking, as well as the studies conducted with bilingual students can inform one‟s thinking about this issue. These ideas have important implications for teaching, texts and curriculum development. The study examined some of the above aspects in the context of South African Education.

Darbe nagrinėjamas informacinių technologijų fenomenas dabarties žinių visuomenėje, interneto technologijas siejant su mokymo(si) procesu. Pateikiama mokymosi proceso samprata technologinio kitimo amžiuje. Aptariamas konstruktyvizmu grindžiamo mokymosi aiškinimas (konstrukcionizmo pedagoginė teorija), atskleidžiami esminiai konektyvizmo principai ir užduotys, su kuriomis turi susidurti besimokantysis skaitmeninio amžiaus eroje. Pristatomi matematikos mokymo filosofijos (matematikos filosofijos) keliami klausimai. Galiausiai ieškoma atsakymų į klausimus: kokį vaidmenį kompiuterinės technologijos, pastarojo meto interneto tecnologijos ir paslaugos atlieka matematikos mokymo bei mokymosi proceso metu, kaip jį paliečia, kokias galimybes atveria, kaip keičia šį procesą. / This theme paper analyses the phenomenon of information technologies in relation with the present knowledge-based society and the Web technologies in the process of education. The concept of learning process is discussed in the age of technological change. The constructivism based learning (constructionism learning theory) is presented in this paper too. Besides this, the most essential principles and tasks of connectivism which a learner is supposed to deal with in the digital surroundings are being analysed as well. Moreover, the issues raised by the phylosophy of mathematics education (and the phylosophy of mathematics) are also introduced in this paper. Finally, the reply search to the following issues – what role do computer-based technologies, modern internet technologies and services play in the process of teaching and learning mathematics; how do the above mentioned factors influence a process of education; what are the perspectives of the mentioned processes; how do the discussed issues change the process of teaching and learning mathematics – is analysed in the paper.

The theoretical part deals with approaches to teaching, especially constructivistic, instructivistic and transmissive approach. The goal of this part is to compare the approaches and evaluate them considering the permanency of knowledge and active engagement of students. Simultaneously it aims at evaluating the approaches from the point of their of the mechanical knowledge. The charter dealing with inquiry based mathematics education is focused on teaching related to constructivistic approach. The next chapter describes tool called concept cartoons, which can be used as a mean to research. The theoretical part is focused on the topic of fraction, especially fraction interpretation and models and fraction representation. The second part of the thesis describes a survey realized at basic school. I narrowed the problem to fraction interpretation and representation according to survey of student's work. The next part of research is based on this analysis. Trying to accomplish the aims and find the answers to questions of the survey I chose strategy inspired by action research, because the main goal of the research was to improve and enhance my teaching knowledge and skills. The survey was processed qualitatively. The making of the thesis helped me make it clear, why the topic of fractions is so...

Lärare i Sverige använder i stor utsträckning matematikböcker i matematikundervisningen. Hur lärares uppfattningar påverkar deras undervisning är dock debatterat. Denna studie undersöker därför sambandet mellan lärares användning av matematikböcker och deras uppfattningar om matematik, matematikundervisning och lärande i matematik. Först fick åtta lärare i årskurs 1-3 besvarade en enkät om uppfattningar, och om användning av matematikböcker och sedan intervjuades tre av dem. Resultatet visar att lärarna i olika grad kan ha både en dynamisk och statisk syn på matematik som ämne, och både en konstruktivistisk och traditionell syn på undervisning och lärande i matematik. Resultatet visar även att den lärare som i minst utsträckning tar uppgifter från matematikboken är den lärare som i störst utsträckning visar på en konstruktivistisk syn, medan den lärare som i störst utsträckning tar uppgifter från matematikboken är den lärare som i störst utsträckning visar på en dynamisk och problemlösande syn. Den lärare som befinner sig i mitten vad gäller vart uppgifterna tas ifrån är den lärare som i störst utsträckning håller med om en statisk och platonistisk samt instrumentalistisk syn. Ses uppfattningar om undervisning och lärande i matematik och uppfattningar om matematik som ämne däremot som en enhet går det inte att påvisa ett samband till användningen av matematikböcker. / Swedish teachers tend to use mathematics textbooks in their teaching to a large extent. How teachers’ beliefs impact their teaching is however, debated. Therefore, this study investigated the relationship between teachers’ use of mathematics textbooks and their beliefs about mathematics, mathematics teaching and learning mathematics. Eight 1-3 grade teachers completed a questionnaire measuring their use of mathematics textbooks and their beliefs, three of these were interviewed. The results indicate that the teachers can agree with both a dynamic and a static view of the nature of mathematics, and both a constructivist and traditional view of mathematics learning and teaching. The results also show that the teacher that uses the least amount of tasks from mathematics textbooks is the teacher that to the largest extent holds a constructivist view, while the teacher that uses the most amount of tasks from mathematics textbooks is the teacher that to the largest extent holds a dynamic and problem solving view. The teacher that’s in between the other teachers when it comes to where the tasks are taken from is the teacher that to the largest extent holds a static, platonist and instrumentalist view. However, if beliefs about mathematics, mathematics teaching and learning mathematics are seen as a unit there is no correlation between the use of mathematics textbooks and beliefs.

Made available in DSpace on 2016-04-27T16:58:07Z (GMT). No. of bitstreams: 1 dissertacao_benedito_junho.pdf: 448337 bytes, checksum: 6cad7afc1b66e6f62e4db6ba05f6f36d (MD5) Previous issue date: 2003-11-13 / The aim of this work was to construct a map of the dissertations related to Higher Education produced in the Program of Post-Graduate Studies in Mathematics Education of the Pontifícia Catholic University of São Paulo throughout the 90s decade. Following the analysis of each of the ten dissertations studied, it was possible to categorize them according to the themes investigated and the methodologies used. The results obtained indicated that the majority of the studies concerned the teaching and learning of disciplines associated with pure mathematics and employed as a research strategy the elaboration and application of a didactic sequence based on the methodology of Didactic Engineering / Este trabalho teve como objetivo, fazer um mapeamento das dissertações produzidas no Programa de Estudos Pós-graduados em Educação Matemática da Pontifícia Universidade Católica de São Paulo, na década de noventa, que versavam sobre o Ensino Superior. Após análise de cada uma das dez obras, foi possível categorizá-las principalmente quanto aos temas abordados e metodologias utilizadas. Os resultados obtidos permitiram concluir que a maioria das pesquisas abordou o ensino e aprendizagem de disciplinas de matemática pura , e elegeu como estratégia de pesquisa, a elaboração e aplicação de uma seqüência didática, baseada na metodologia da Engenharia Didática

This study investigated teachers' and learners' perceptions of the causes of poor performance in Mathematics among Grade 9 learners in Durban's Ethusini Circuit of KwaZulu-Natal Province. By using a phenomenographic research design, a sample of 15 Grade 9 learners who take Mathematics, one Grade 9 Mathematics teacher and one Mathematics head of department was selected purposively from three high schools in the Ethusini Circuit. The three schools are one former model C school, one ordinary government school and one private school. Focus group and semi-structured interviews were used to collect data from the participants. The data were analysed by transcribing the focus group and semi-structured interviews; thereafter, the transcriptions were coded into themes and categories. The results of this study indicate that teachers and learners perceive that poor performance in Mathematics in Grade 9 is due to CAPS curriculum challenges, such as overloaded and advanced content, and insufficient time to complete the curriculum. Parental support, English (first additional language) as a language of learning and teaching, methods of teaching (such as the talk and chalk method), and learners' negative attitude towards Mathematics were also cited. From these findings, recommendations for revisiting the CAPS curriculum and other areas of need, as well as topics for future research were suggested. / Dinyakišišo tše di nyakišišitše ka ga maikutlo a barutiši le a baithuti ka ga dilo tšeo di bakago go se šome gabotse ka thutong ya Dipalo gareng ga baithuti ba Kreiti ya 9 ka Sedikothutong sa Ethusini ka Durban ka Phrobentsheng ya KwaZulu-Natal. Ka go šomiša mokgwa wa dinyakišišo wa fenomenokrafiki, sampole ya baithuti ba 15 ba Kreiti ya 9 bao ba ithutelago Dipalo, morutiši o tee wa Dipalo ka go Kreiti ya 9 le morutiši o tee yo e lego hlogo ya lefapha la Dipalo o kgethilwe ka maikemišetšo go tšwa dikolong tše di phagamego tše tharo ka Sedikothutong sa Ethusini. Dikolo tše tharo tše ke sekolo se tee seo se bego se le sa motlolo wa C, se tee sa tlwaelo sa mmušo le se tee ke sekolo sa phraebete. Sehlopha seo se nepišitšwego le dipoledišano tša dipotšišo tšeo di beakantšwego peleng di šomišitšwe go kgoboketša tshedimošo go tšwa go bakgathatema. Tshedimošo e sekasekilwe ka go ngwalolla dipoledišano tša dipotšišo tša sehlopha seo se nepišitšwego le tšeo di beakantšwego peleng; ka morago ga fao, dingwalollo tše di filwe dikhoutu go ya ka merero le magoro a tšona. Dipoelo tša dinyakišišo tše di laetša barutiši le baithuti ba bona gore go se šome gabotse ga baithuti ka thutong ya Dipalo ka go Kreiti ya 9 ke ka lebaka la ditlhohlo tša lenaneothuto la Setatamente sa Melawana ya Tekolo ya Lenaneothuto (CAPS), go swana le diteng tša thuto tšeo di tletšego kudu le tšeo di gatetšego pele kudu, le nako ye e sego ya lekana ya go fetša lenaneothuto. Thekgo ya batswadi, Seisemane (polelo ya gae ya tlaleletšo) bjalo ka polelo ya go ithuta le ya go ruta, mekgwa ya go ruta (ya go swana le mokgwa wa go bolela le go ngwala ka tšhooko letlapeng), le maikutlo ao a sego a loka a baithuti go Dipalo le tšona di filwe bjalo ka dilo tšeo di bakago se. Go dikutollo tše, ditšhišinyo tša go etela leswa lenaneothuto la CAPS le makala a mangwe ao o hlokago šedi, gammogo le dihlogotaba tša dinyakišišo tša ka moso le tšona di ile tša šišinywa. / Lolu cwaningo luphenye imibono yothisha kanye neyabafundi yezimbangela zokungasebenzi kahle esifundweni seziBalo kubafundi beBanga le-9 kuSekhethi yase-Ethusini eThekweni esifundazweni saKwaZulu-Natali. Ngokusebenzisa isakhiwo sokucwaninga esisezingeni eliphezulu, isampula yabafundi abayi -15 beBanga le-9 abathatha iziBalo, uthisha oyedwa weBanga le-9 kanye nomphathi oyodwa womnyango weziBalo bakhethwa ngenhloso ezikoleni ezintathu eziphezulu kuSekhethi yase-Ethusini. Lezi zikole ezintathu esisodwa semodeli C, esisodwa sikahulumeni ojwayelekile kanye nesisodwa isikole esizimele. Iqembu lokugxila kanye nezingxoxo ezihlelwe kahle zisetshenzisiwe ukuqoqa imininingwane evela kubahlanganyeli. Imininingwane yahlaziywa ngokubhalwa kweqembu okugxilwe kulo kanye nezingxoxo ezihlelwe kahle; emva kwalokho, imibhalo ivezwe ngendlela engaqondile ezindikimbeni nasezigabeni. Imiphumela yalolu cwaningo ikhombisa ukuthi othisha kanye nabafundi babona ukuthi ukungasebenzi kahle esifundweni seziBalo kuBanga lesi-9 kungenxa yezinselela yezifundo ezibandakanya isifundo esifundwayo seCAPS, njengokuqukethwe okugcwele kakhulu nokuthuthukile, kanye nesikhathi esinganele sokuphothula izifundo ezibandakanya isifundo esifundwayo. Ukusekelwa kwabazali, isiNgisi (ulimi lokuqala olwengeziwe) njengolimi lokufunda nokufundisa, izindlela zokufundisa (njengeyokufundisa lapho uthisha ebhala ebhodini), kanye nesimo sengqondo esingesihle sabafundi maqondana neziBalo nakho kuveziwe. Ukusuka kulokhu okutholakele, kuphakanyiswe ukuphinda kuphinde kufundwe izifundo ezibandakanya isifundo esifundwayo seCAPS kanye nezinye izindawo ezidingeka, kanye nezihloko zocwaningo lwakusasa ziphakanyisiwe. / Curriculum and Instructional Studies / M. Ed. (Curriculum and Instructional Studies)