Dissertations / Theses on the topic 'Mathematics – Stady and teaching'

Mathematics 436 is the advanced mathematics course offered to students in secondary IV in the province of Quebec. Although the course is designed to challenge students in the advanced stream, it has produced a high number of failures. This study examines the relationship between mathematics anxiety and achievement in Mathematics 436. Fifty-six students from an English high school on the island of Montreal took part in the study. The Mathematics Anxiety Rating Scale for Adolescents was used to measure the level of mathematics anxiety experienced by the students. In addition, grades from the previous year in mathematics were obtained, as well as grades from the present year, and the final examination. The results of the study suggest that students enrolled in Mathematics 436 experience a high level of mathematics anxiety. As well, higher levels of mathematics anxiety experienced by the students are associated with poor performance in mathematics.

This study explores the relationship between university mathematics teachers' beliefs about the nature of reading mathematics and their practices regarding reading mathematics. It is a response to the calls for reform in mathematics education, particularly to the assertion made by the National Council of Teachers of Mathematics in 1989 that not all students can read mathematical exposition effectively and that all students need instruction in how to read mathematics textbooks. It presupposes a collaboration between reading and mathematics teachers to help students learn to read mathematics. The objectives were (1) to examine mathematics teachers' beliefs and practices regarding reading, mathematics, and thereby, reading mathematics; (2) to determine whether the theoretical perspectives implicit in those beliefs and practices could be characterized vis-a-vis the theoretical orientations that inform Siegel, Borasi, and Smith's (1989) synthesis of mathematics and reading; and (3) to determine the relationship, if any, that exists between mathematics teachers' beliefs about reading mathematics and their practices regarding reading mathematics. The synthesis presents dichotomous views of both mathematics and reading: Mathematics is characterized as either a body of facts and techniques or a way of knowing; reading, as either a set of skills for extracting information from text, or a mode of learning. The latter view, in each case, can be characterized as constructivist. The researcher was a participant observer in a university sumner program. The primary participants were fourteen mathematics instructors. Interviews were conducted using a heuristic elicitation technique (Black & Metzger, 1969). Field notes were taken during observations of classroom activities and other non-academic summer program activities. The data were coded using a constant comparative method (Glaser & Strauss, 1967) comparative method. Twelve instructors held conceptions of reading that were consistent with their conceptions of mathematics. Of those twelve, two held conceptions that could be characterized as constructivist; ten held conceptions that were not constructivist. Two instructors held conceptions of reading that were not consistent with their conceptions of mathematics. Of those two, one held a constructivist conception of reading but not of mathematics; one held a constructivist conception of mathematics but not of reading. Teachers' practices reflected their theoretical orientations. The study has implications for teacher education: If teachers' beliefs are related to their practices, then teacher education programs should (1) acknowledge the teachers' existing beliefs and (2) address the theoretical orientations implicit in various aspects of pedagogy.

In this thesis, I describe how interactive journal writing was used to improve the understanding of mathematics, and to foster communication with two groups of remedial grade ten students. Mathematics is a gatekeeper course in high school, and students who are not successful with this subject are at a distinct disadvantage, both in terms of their education and in their future careers. A persistent source of difficulty for these students is related to language; students often struggle both to understand what is being taught, and how to explain concepts or problem solutions in their own words. Interactive journal writing was initiated as a means of addressing this situation, and of meeting the objectives proposed by the Quebec Education Plan, which specifies three closely related competencies: (1) solve situational problems; (2) use mathematical reasoning; (3) and communicate by using mathematical language. There is ample proof in the research literature that communication plays an important role in supporting learners by helping them clarify, refine and consolidate their thinking.
This study demonstrates the importance of allowing and encouraging students to use writing as part of their learning processes. By writing about what they are being taught, students are forced to slow down, examine and reflect on the steps they use to solve problems. Sharing what they write promotes meaningful dialogue and personal engagement, essential ingredients of successful learning.

Over the last few years it became clear that the students struggle with the basic concepts of logarithms and inequalities, let alone logarithmic inequalities due to the lack of exposure of these concepts at high school. In order to fully comprehend logarithmic inequalities, a good understanding of the logarithmic graph is important. Thus, the opportunity was seen to change the method of instruction by introducing the graphical method to solve logarithmic inequalities. It was decided to use an mathematical software program, Omnigraph, in this research.

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.

Thesis (MEd)--Stellenbosch University, 2002.
ENGLISH ABSTRACT: In many countries, due to a growing criticism of the inadequacy of mathematics curricula, reforms have been undertaken across the world for meeting new social and technological needs and many researchers have begun to pay attention to the way mathematics is learned and taught. In the same vein, this study aims to investigate innovative and appropriate teaching strategies to introduce in the Rwandan educational system in order to foster students' mathematical thinking and problem solving skills. For this, a classroom-based research experiment was undertaken, focusing on meticulous observation, description and critical analysis of mathematics teaching and learning situations. In the preparation of the research experiment, three mathematics teachers were helped to acquire proficiency in doing mathematics and to refine their teaching strategies, as well as to enable them to create a mathematics classroom culture that fosters students' understanding of mathematics through the problem solving process. Three classes of 121 students of the second year, their ages ranging from 14 years to 16 years, chosen from three different secondary schools in Rwanda, participated in this research experiment. Students were taught an experimental programme based on solving contextualised algebra problems in line with the constructivist approach towards mathematics teaching and learning. Twenty-four mathematics lessons were observed in the three classes and students' learning activities were systematically recorded, focusing on teacher-students and student-student interaction. The participating teachers experienced many difficulties in implementing new teaching strategies based on a problem solving approach but were impressed and encouraged by their students' abilities to generate different and unexpected ways of solving problem situations. However, the construction of mathematical models of non-routine problems constituted the most difficult task for many students because it required a high level of abstraction, characterising algebraic reasoning. Despite evident cognitive obstacles, a substantial improvement in students' systematic reasoning with respect to the different steps in the problem solving process, namely formulating a mathematical model, solving a model, verifying the solution and interpreting the answer, was progressively observed during the experiment. Many students had to overcome a language problem, which inhibited their understanding and interpretation of mathematical problem situations and deeply affected their active participation in classroom discussions. In this study, small group work and group discussions gave rise to excellent and successful teaching and learning situations which were appreciated and continuously improved up by the teachers. They provided students with opportunities for learning to argue about their mathematical thinking and to communicate mathematically. This kind of classroom organisation created an ideal learning environment for students but an uncomfortable teaching situation for teachers. It required much effort from the teachers to transform the mathematics classroom into a forum of discussion in setting up stimulating and challenging tasks for students, in working efficiently with different groups and in moderating the whole class discussion. It was unrealistic to expect spectacular changes in teaching practices established over years to take place during a period of a month. This type of change requires sufficient time and support. However, teachers did develop a new and practical vision of mathematics teaching strategies focusing on students' full engagement in exploring and grappling with problematic situations in order to solve problems. Teachers made remarkable efforts in internalising and adopting their new role of mediators of students' mathematics learning and in being more flexible in their teaching styles. They learned to communicate with their students, to accept students' explanations and suggestions, to encourage their logical disagreement and to consider their errors and misconceptions constructively. Students' results in the pre-test and the post-test showed their low performance in building mathematical models especially when they had to use symbols but revealed a significant progress in the students' ways of thinking which was observed through the variety and originality of their strategies, their systematic work and their perseverance in solving algebra problems. Students also developed positive attitudes to do mathematics; this was exhibited by their pride and satisfaction to accomplish nonroutine tasks by themselves. Teachers' comments indicated that they work under pressure to cover an overloaded mathematics curriculum and have poor support from educational authorities. For them, mathematics IS socially considered as a difficult subject. For many students, mathematics IS a gatekeeper to access higher levels of education; to fail in mathematics unfortunately implies to fail at school and in life. Students' negative attitudes towards mathematics were mainly due to their repeated failures in mathematics, but also to some mathematics teachers who intimidate and discourage their students. Both educational authorities and teachers should make efforts to rethink an appropriate mathematics curriculum and alternative teaching strategies in order to efficiently prepare students to meet new societal and technological requirements.
AFRIKAANSE OPSOMMING: As gevolg van toenemende kritiek oor die kwaliteit van wiskundekurrikula, is bewegings vir hervorming wêreldwyd geïnisieer om nuwe sosiale en tegnologiese behoeftes aan te spreek en baie navorsing is gedoen oor die wyse waarop wiskunde geleer en onderrig word. In lyn hiermee, is die doel van hierdie studie om innoverende en geskikte onderrigstrategieë te ondersoek om in die Rwandese onderwysstelsel in te voer om leerders se wiskundige denke en probleemoplossingsvaardighede te ontwikkel. Om dit te bereik, is 'n klaskamergebaseerde navorsingseksperiment uitgevoer, met die klem op fyn waarneming, beskrywing en kritiese ontleding van wiskunde leer- en onderrigsituasies. As voorbereiding tot die navorsingseksperiment is drie wiskunde-onderwysers gehelp om vaardighede te verwerf in die doen van wiskunde en om hulonderrigstrategieë te verfyn, asook om hulle in staat te stelom 'n wiskunde-klaskamerkultuur te vestig wat leerders se begryping van wiskunde deur die probleemoplossingsproses ontwikkel. Drie klasse van 121 leerders in die tweede jaar, tussen 14 en 16 jaar oud, is uit drie verskillende hoërskole in Rwanda gekies om aan die navorsing deel te neem. Die leerders is deur middel van 'n eksperimentele program onderrig wat gebaseer is op die oplossing van gekontekstualiseerde algebraprobleme in ooreenstemming met 'n konstruktivistiese benadering tot wiskunde-leer en -onderrig. Vier-en-twintig wiskundelesse is in die drie klaskamers waargeneem en leerders se leeraktiwiteite is stelselmatig opgeskryf, met die klem op onderwyser-leerder en leerder-leerder interaksie. Die betrokke onderwysers het baie probleme ondervind om nuwe onderrigstrategieë gebaseer op 'n probleemoplossingsbenadering te implementeer, maar was baie beïndruk en begeesterd deur hulleerders se vermoë om verskillende en onverwagte planne te beraam om probleme op te los. Die opstelling van wiskundige modelle vir nie-roetine probleme was vir baie leerders die moeilikste taak omdat dit 'n hoë vlak van abstraksie wat kenmerkend is van algebraïese denke verteenwoordig. Ten spyte van kognitiewe struikelblokke was daar nogtans 'n merkbare verbetering in leerders se logiese redeneringsprosesse soos geopenbaar in die toepassing van die verskillende stappe van die probleemoplossingsproses, naamlik die formulering van 'n wiskundige model, die oplossing van die model, verifiëring van die oplossing en interpretasie van die antwoord. Baie studente is gekniehalter deur 'n taalprobleem wat hul begrip en interpretasie van wiskundige probleemsituasies en hul vrymoedigheid om aan klaskamergesprekke deel te neem, aan bande gelê het. Inhierdie studie het kleingroepwerk en groepbesprekings suksesvolle onderrig- en leersituasies geskep wat deur die onderwysers raakgesien en verder uitgebou is. Dit het geleenthede geskep vir die leerders om oor hul wiskundige denke te argumenteer en om wiskundig te kommunikeer. Hierdie soort klaskamerorganisasie het 'n ideale leeromgewing vir leerders geskep maar 'n ongemaklike onderrigomgewing vir onderwysers. Dit het baie van onderwysers geverg om die wiskundeklaskamer in 'n gespreksforum te omskep deur stimulerende en uitdagende probleme aan leerders te stel, deur met verskillende groepe te werk en deur die algemene klaskamerbesprekings te fasiliteer. Dit was onrealisties om binne die bestek van 'n maand grootskaalse veranderinge in onderwyspraktyke wat oor 'n tydperk vanjare posgevat het, te verwag. Hierdie soort verandering benodig genoeg tyd en ondersteuning. Onderwysers het nogtans 'n nuwe en praktiese visie ontwikkel van wiskunde-onderrigstrategieë wat fokus op leerders se betrokkenheid by die ondersoek en oplossing van probleme wat vir hulle uitdagend en nie-roetine was. Onderwysers het daadwerklike pogings aangewend om hul nuwe rolle as mediators te internaliseer en te aanvaar, en om meer soepel onderrigstyle te ontwikkel. Hulle het geleer om met hulleerders te kommunikeer, om leerders se verduidelikings en voorstelle te aanvaar, om logiese argumentering aan te moedig en om foute en wankonsepte konstruktief te benader. Leerders se resultate in die voor- en na-toetse dui op swak vermoë om wiskundige modelle te bou veral wanneer hulle simbole moes gebruik, maar wys beduidende vordering in leerders se denke, wat gemanifesteer het in die verskeidenheid en oorspronklikheid van hul strategieë, hul sistematiese werk en hul voortgesette pogings om algebraprobleme op te los. Leerders het ook positiewe instellings teenoor die doen van wiskunde ontwikkel; dit is getoon deur hul trots en tevredenheid wanneer hulle self nie-roetine take opgelos het. Onderwysers se kommentaar openbaar dat hulle onder druk werk om 'n oorlaaide wiskundekurrikulum af te handel en dat hulle min ondersteuning van onderwyshoofde kry. Hulle sê ook dat wiskunde deur die breë gemeenskap as 'n moeilike vak beskou word. Vir baie leerders is wiskunde 'n hekwagter wat toegang tot verdere onderwys en opleiding beheer; om in wiskunde te faal beteken om op skool te faal en om in die lewe te faal. Leerders se negatiewe instellings teenoor wiskunde was hoofsaaklik as gevolg van hul herhaalde mislukkings in skoolwiskunde maar ook as gevolg van sommige wiskunde-onderwysers wat hulleerders intimideer en ontmoedig. Beide onderwyshoofde en onderwysers behoort pogings aan te wend om te besin oor 'n geskikte wiskundekurrikulum en alternatiewe onderrigstrategieë om leerders meer doeltreffend voor te berei om aan nuwe sosiale en tegnologiese eise te voldoen.

This study investigated the teaching experiences of six elementary preservice teachers (EPTs), three with high mathematics anxiety and three with low mathematics anxiety, during their student teaching semester. The EPTs were selected from an initial pool of 121 EPTs who took the Abbreviated Mathematics Anxiety Scale. The cases were compared in a cross case analysis to highlight mathematics teaching experiences among EPTs. Data sources included EPT and researcher journal entries, interview transcripts, pre-lesson surveys, field notes, lesson plans, and artifacts of observed lessons. Data were coded using Shulman’s content knowledge, Graeber’s mathematics pedagogical content knowledge, and mathematics anxiety characteristics. Findings revealed both similarities and differences across EPTs as related to four major categories: (a) planning and resources used, (b) role of the cooperating teacher, (c) content knowledge, and (d) pedagogical content knowledge. All EPTs used mostly direct instruction and relied on the course textbook and their respective cooperating teacher as their primary resources for planning. Additionally, across participants, the cooperating teacher influenced EPTs’ perceptions of students and teaching. Also, EPTs with high mathematics anxiety were weaker with respect to content knowledge and pedagogical content knowledge. Findings suggest a need to re-design methods courses to address improving the pedagogical content knowledge of EPTs with mathematics anxiety. Findings also suggest a need to develop content specific mathematics courses for EPTs to improve their content knowledge. Future studies could include a longitudinal study to follow highly anxious EPTs who take content specific elementary mathematics courses to observe their content knowledge and mathematics anxiety.

The purpose of this study is to illustrate how teachers, in a private school in Thailand, organize and plan their mathematic teaching, what material they use and how the teachers challenge the students. In the background, we report briefly about Thailand and the school's history. Then a presentation follows of previous research that has been made within our chosen subject. The theories are linked to our research questions, such as mathematic teaching, mathematic material and gifted students, according to Western literature. You can also find a short summary of the curriculum for mathematics, from the government in Thailand. Trough a qualitative method five interviews with mathematic teachers and observations in their classes have been conducted and this result have been analyzed and discussed. Interview and observation guides were used with the purpose to give the respondents room to develop their reasoning with the help of our follow- up questions. Trough our interviews and observations we have come to a result that shows that the teachers organized their teaching according to a Brain based learning pattern and after the governments curriculum. The mathematic material consisted of mathematic books, laminated sticks, bars, number cards and also of computer programs. The mathematic books were not individualized but played a significant part of the teaching. The plastic material was used to concretize the mathematics. The school had the idea that separating the gifted students from the others in the group was positive for their development. The gifted students were challenged by more difficult mathematics meant for older students and had to work at a higher pace than their peers. The results can not be generalized for all schools in Thailand. When the study is relatively small, it only shows how a part of the mathematics teaching is implemented at the school where the study was conducted.

This research provided insights into the mathematical achievement of a cohort of tertiary mathematics students. The context for the study was an entry level mathematics course, set in an engineering programme at a tertiary institution, the Cape Peninsula University of Technology (CPUT). This study investigated the possibilities of providing a bridge between the assessment of students by means of tests scores and a taxonomy of mathematical objectives, on the one hand, and the critical analysis of student produced texts, on the other hand. This research revealed that even in cases of wrong solutions, participant members' responses were reasonable, meaningful, clear and logical.

Thesis (MTech (Mechanical Engineering))--Cape Peninsula University of Technology, 2007
Twelve years into a democracy, South Africa still faces many developmental challenges. Since 2002 Universities of Technology in South Africa have introduced Foundational Programmes/provisions in their Science and Engineering programmes as a key mechanism for increasing throughput and enhancing quality. The Department of Education has been funding these foundational provisions since 2005. This Case Study evaluates an aspect of a Foundational provision in Mechanical Engineering, from the beginning of 2002 to the end of 2005, at a University of Technology, with a view to contributing to its improvemenl The Cape Peninsula University of Technology {CPUn, the locus for this Case Study, is the only one of its kind in a region that serves in excess of 4.5 million people. Further, underpreparedness in Mathematics for tertiary level study is a national and intemational phenomenon. There is thus a social interest in the evaluation of a Mathematics course that is part of a strategy towards addressing the shortage in Engineering graduates. This Evaluation of integration of the Foundation Mathematics course into Foundation Science, within the Department of Mechanical Engineering at CPUT, falls within the ambit of this social need. An integrated approach to cunriculum conception, design and implementation is a widely accepted strategy in South Africa and internationally; this approach formed the basis of the model used for the Foundation programme that formed part of this Evaluation. A review of the literature of the underpinnings of the model provided a theoretical framework for this Evaluation Study. In essence this involved the use of academic literacy theory together with learning approach theory to provide a lens for this Case Study.

This study examined the mathematical thinking of English learners as they were taught mathematics vocabulary through research-based methods. Four English learners served as focus students. After administering a pre-performance assessment, I taught a 10-lesson unit on fractions. I taught mathematics vocabulary through the use of a mathematics word wall, think-pair-shares, graphic organizers, journal entries, and picture dictionaries. The four focus students were audio recorded to capture their spoken discourse. Student work was collected to capture written discourse. Over the course of the unit, the four focus students used the mathematics vocabulary words that were taught explicitly. The focus students gained both procedural and conceptual knowledge of fractions during this unit. Students also expressed elevated confidence in their mathematics abilities.

This study focused on the evaluative listening practices of four teachers who participated in an algebra professional development involving lesson study. This instrumental case study operationalizes the enactment of teacher listening followed by teacher questions and responses to define listening-to-question. Also, questioning-to-listen is operationalized as the enactment of purposefully posing questions to posture oneself to listen to students' mathematical thinking. Because of the tacit aspect of teacher listening and the visibility of teacher questioning, interrelating listening and questioning affords teachers an accessible point of entry into developing listening practices. In response to participants wondering as to when evaluative listening is appropriate in the mathematics classroom, this study discusses six instances of teaching excerpts along a continuum of listening orientations from directive to observational to responsive. The results indicate positive aspects of evaluative listening towards an observational and responsive listening stance. Results of the study also confirm a reliance on low-order gathering information questions as the predominant type of teacher question posed in mathematics teaching. This study reveals the necessity of contextualizing teacher questions to inform appropriate uses of evaluative listening. Future professional development should consider emphasizing positive aspects of evaluative listening in mathematics teaching.

ENGLISH ABSTRACT: In today’s hi-tech global economy the fields of science, technology and engineering are becoming increasingly and undeniably central to economic growth and competitiveness, and will provide many future jobs. Qualifications in Mathematics are crucial gateways to further education and will provide access to the Science, Technology, Engineering and Mathematics (STEM) industries. This study focuses on the optional course in Mathematics, called Advanced Programme Mathematics (APM), which is offered and assessed by the Independent Examination Board in the final three years of high school in South Africa. At present, the South African school system does not adequately prepare students for the transition from school to university Mathematics, and APM has been designed to address this gap. The research question set by this study is: To what extent does the APM course succeed in preparing learners for the rigour of first-year Mathematics in the STEM university programmes? The sample group of 439 students was selected from the 2013 cohort of first-year Mathematics students at Stellenbosch University. First, an analysis of the relevant curricula was undertaken, and then an empirical investigation was done to determine the differences in performance between first and second semester examinations of first-year university Mathematics students who took APM, and those who did not. This was followed by an investigation by means of a questionnaire into the perceptions of students on how effective APM was in easing the transition from school to university Mathematics. The research was designed according to the Framework for an Integrated Methodology (FraIM) of Plowright (2011). From an extensive international literature study, it appears that APM is definitely a predictor of post-secondary success. Since no formal research has been recorded to support this claim, this study aims to provide a sound answer to whether APM is advantageous. The effect size results of this study show that APM marks of students explain 68% of the achievement in first-semester university Mathematics when combined with NSC Mathematics marks in a general regression model. There is a significant difference between the marks of students who took APM and those who did not in first-semester university Mathematics, specifically across the National Senior Certificate (NSC) Mathematics mark categories of 80-100%. APM course-taking leads to confidence in Mathematics, which combined with good domain knowledge of calculus, ease the transition from school to university Mathematics. The study recommends that not only students who intend pursuing a career in the STEM industries should take the APM course, but also those who intend to apply for admission to any other tertiary studies, as the cognitive and other skills provided by APM will give them the required edge to perform well in higher education. Schools are called upon to provide access to APM for mathematically gifted students, and teachers and guidance counsellors should encourage learners to enrol for AMP. This will enable them to share in the manifold academic and personal benefits accruing from the course, and to help alleviate the critical shortage of graduates in careers requiring a strong Mathematics background in South Africa.
AFRIKAANSE OPSOMMING: In die hoë-tegnologie-wêreldekonomie van vandag word die gebiede van wetenskap, tegnologie en ingenieurswese toenemend en onmiskenbaar die kern van ekonomiese groei en mededingendheid wat in die toekoms baie werkgeleenthede sal bied. Kwalifikasies in Wiskunde open beslis baie deure na verdere opleiding en verleen toegang tot die Wetenskap-, Tegnologie- Ingenieurswese- en Wiskunde-industrieë. Hierdie studie fokus op die opsionele kursus in Wiskunde, genaamd Gevorderde Program Wiskunde (GPW), wat deur die Onafhanklike Eksamenraad aangebied en geassesseer word in die laaste drie jaar van hoërskoolonderrig in Suid-Afrika. Tans berei die Suid-Afrikaanse skoolstelsel nie studente genoegsaam voor vir die oorgang van skool- na universiteitswiskunde nie en GPW is ontwerp om hierdie gaping te oorbrug. Die navorsingsvraag wat hierdie studie stel, is: In watter mate slaag die GPW-kursus daarin om leerders voor te berei vir die streng vereistes van eerstejaar-Wiskunde in die Wetenskap-, Tegnologie- Ingenieurswese- en Wiskunde-universiteitsprogramme? Die toetsgroep van 436 studente is gekies uit die 2013-groep eerstejaar-Wiskundestudente aan Stellenbosch Universiteit. Aanvanklik is ᾽n analise van die relevante leerplanne onderneem, waarna ᾽n empiriese ondersoek gedoen is om die verskille in prestasie in die eerste en tweede semester eksamens vas te stel tussen eerstejaar-Wiskundestudente op universiteit wat wel GPW geneem het en diegene wat dit nie geneem het nie. Dit is gevolg deur ᾽n ondersoek deur middel van ᾽n vraelys na die persepsies van studente oor hoe effektief GPW was om die oorgang van skool- na universiteitswiskunde te vergemaklik. Die navorsing is ontwerp op grond van ‘n model vir ‘n geïntegreerde metodologie van Plowright (2011). Dit blyk uit ᾽n uitgebreide studie van internasionale literatuur dat GPW definitief ᾽n voorspeller van post-sekondêre sukses is. Aangesien geen formele navorsing om hierdie aanspraak te ondersteun nog op skrif gestel is nie, poog hierdie studie om ᾽n deurdagte antwoord te verskaf op die vraag of GPW wel tot voordeel van studente is. Die effek grootte resultate van hierdie studie dui aan dat die GPW-punte van studente 68% van prestasie in Wiskunde in die eerste semester op universiteit verduidelik as dit in ᾽n algemene regressiemodel met die Nasionale Senior Sertifikaat (NSS) punte gekombineer word. Daar is ᾽n beduidende verskil tussen die Wiskundepunte van studente wat GPW geneem het en diegene wat dit nie geneem het nie in die eerste semester op universiteit, veral in die NSS-Wiskundepuntekategorieë van 80-100%. Om die GPW-kursus te neem, lei tot selfvertroue in Wiskunde, wat saam met ᾽n goeie kennis van die Differensiaalrekening-domein, die oorgang van Wiskunde vanaf skoolvlak na universiteitsvlak vergemaklik. Op grond van die studie beveel die navorser aan dat nie slegs studente wat ᾽n loopbaan in Wetenskap-, Tegnologie- Ingenieurswese- en Wiskunde-rigtings wil volg, die GPW-kursus behoort te volg nie, maar ook diegene wat vir toelating tot enige ander tersiêre studie wil aansoek doen, aangesien die kognitiewe en ander vaardighede wat GPW ontwikkel, hulle die nodige voorsprong sal bied om goed te vaar in verdere studie. Skole word aangemoedig om toegang tot GPW aan wiskundig begaafde leerlinge te verskaf en onderwysers en loopbaanraadgewers behoort leerlinge aan te moedig om vir GPW in te skryf. Sodoende kan hulle deel in die vele akademiese en persoonlike voordele wat die kursus bied, en help om die kritieke tekort aan gegradueerdes in die studierigtings waar ‘n sterk Wiskunde agtergrond ‘n vereiste is, te help verlig.

Reports and consultative documents published at national level since about 1980 have indicated that British Industry must look to modern technology and also educate and train its workers on a 'broad base', with an 'integrated' approach. Traditionally, and still very much the mode of operation, teaching has been confined within subject boundaries. A research group was established by Professor Bajpai consisting of the author, Mr Rod Bond (Burleigh Community College, Loughborough) and a few others working overseas to investigate a teaching strategy based on an interrelated approach to teaching mathematics. Measurement was chosen as the first topic of investigation using this approach which then formed the basis for further research undertaken by the two research workers of the group whose work is reported in the form of two theses. This thesis aims to show that mathematics is naturally related to science and technology in industrial practice and that when taught in an interrelated way it would be more interesting and have more relevance to real applications in technology-based employment at craft and technician levels. To help establish the case experiments carried out by the author are referred to; these include a few case studies, a questionnaire survey and results analysed from more than five hundred basic mathematics tests. The various kinds of mathematics taught in further education are described and compared with mathematics in a practical context as seen from a case study within an engineering training school. Next a survey of mathematics at work shows that, like the training school, there is a task associated with the mathematics which is also related to science or technology or both. Another case study in the pharmaceutical industry lends further support to the way mathematics is used in industry. Much of the mathematics also seems to be basic and used in association with measurement and a particular task. It was decided by the research group that a tape/slide programme on measurement for students and educators should be developed by the author and tested in different situations. Teaching modules on relevant mathematical topics based on the interrelated approach were constructed for students with strong support from industry in the form of materials and advice. Testing of these modules, in their original and revised forms after feedback, is described. These trials were also carried out in other establishments. Modules based upon the interrelated approach developed by the author formed a basis for promoting the underlying philosophy behind this approach. These were presented to educators in in-service training and staff development programmes in the north western region of the UK with success. Observations and conclusions drawn clearly indicate that this type of method makes mathematics more interesting and relevant for students of different abilities and backgrounds. Finally pointers are given in the thesis as to the wider use and promotion of this approach for teaching mathematics in further education.

According to recent research a focus in teaching mathematics to children is the development of problem solving abilities. Problem solving means the process of applying mathematical knowledge and skills to unfamiliar situations. A case study was done using a problem solving approach to the learning and teaching of mathematics with a sample of teachers registered at the Umlazi College For Further Education. These teachers were familiar with the traditional approach of teaching mathematics through drill and practice methods. The new syllabus that is to be implemented emphasises a problem solving approach to the teaching of mathematics. This study set out to implement a problem solving approach with primary school mathematics teachers so that they would be someway prepared for the innovations of the new syllabus. Workshops were conducted using an action research approach with discourse and practice leading to reconstruction with improvements. Early theorists like Piaget and Bruner offered ways of understanding children's learning, to help the teacher develop his teaching. Dienes introduced an element of play and Dewey spoke of the importance of experience. Dienes and Dewey show the first positive signs of recognising the importance of social interaction in the learning situation. Social interaction lays emphasis on language and discussion in the mathematics classroom. A social constructivist model of teaching and learning was used for the research. This research includes a study of the established ideas on developing a problem solving approach to mathematics teaching. These ideas were incorporated into the workshops that the group of teachers attended. During the workshops teachers were gradually exposed to the essence of problem solving techniques through much group discussion and doing practical exercises, which they could then implement in their classes. The teachers reported back at each subsequent workshop. A non-participant observer evaluated the development at the workshops. The workshops' success was evident from the change in the teachers' attitudes and behaviour as well as their feedback of what transpired in the classroom. They reported on the change in their roles as information suppliers to facilitators where the thinking process was focused on, rather than the importance of a correct answer. In the workshops the teachers themselves moved from passive listeners to active participants. It would appear from this preliminary investigation that through using a problem solving approach in workshops, inservice teachers can benefit constructively from this approach and will attempt to use it in their own teaching.

Bibliography: pages 177-183.
The dissertation is concerned with the production of a systematic analysis of HSRC research reports into mathematics education in South Africa between 1970 and 1980. Drawing on the theoretical language of Dowling (1995), the analysis focuses on the (re)production of voice and message in the reports. This entails an analysis of positioning strategies that il1ark out voices in the texts and distributing strategies that distribute message across voices. Voices include bureaucratic, academic, teacher and learner voices and knowledge and practices that constitute message distributed to voices relate to mathematical knowledge, pedagogic knowledge and curriculum innovation practices. Positioning and distributing textual strategies with respect to learner and teacher sub-voices are related to the (re)production of theories of instruction that constitute models of acquirers, transmitters and pedagogic contexts and define pedagogic competence in particular ways. The (re)production of theories of instruction in turn are related to the reproduction of social relations in the broader society. It is hoped that the analysis illustrates the generality of Dowling's language for analysing texts. The substantive focus of the study is the analysis of the reports and the language developed in the analysis is used to make some suggestive comments about current mathematics curriculum development in South Africa. It is hoped, in particular, that the focus in this study on discourses in mathematics education in South Africa in the 1970s will contribute to the documenting of the history of mathematics curriculum development in South Africa.

This thesis presents the findings of a research project which explored students� learning during an activity-based mathematics programme. The research investigated what students learnt about solving linear equations and examined the role of activities in this learning. The investigation of learning in the classroom was guided by the principles of naturalistic enquiry. A longitudinal study was used to investigate students� learning during a unit of work that that made extensive use of activities and contexts. The longitudinal design of the study allowed the development of algebraic thinking to be investigated. The ideas of both Piaget and Vygotsky suggest that it is necessary to study the process of change in order to understand the thinking of students. A group of four students, two girls and two boys, were studied for twenty-seven lessons with each student interviewed individually within six days of each lesson, using the technique of stimulated recall. All lessons and interviews were recorded for subsequent transcription and analysis. Learning to solve equations formally, using inverse operations, proved to be difficult for all the students. For two of them, their poor understandings of arithmetic structure and inverse operations were impediments that prevented them from doing more than attempt to follow procedures. Two of the students did succeed in using inverse operations to solve equations, but were still reasoning arithmetically. There was little evidence in the data that any of the students got to the point of regarding equations as objects to act on. They consistently focussed on the arithmetic procedures required for inverse operations. Even by the end of the topic the most able student, like the others, was still struggling to write algebraic statements. One of the most striking features of the results was the slow progress of the students. For at least two of the students, lack of prerequisite numeracy skills provided a good explanation of why this was so. However for the other two, poor numeracy did not appear to be a reason. The findings are, however, perhaps not too surprising. For children learning about arithmetic, the change from a process to an object view, from counting strategies to part/whole strategies, seems a particularly difficult transition to make. To move from a process to an object view of equations appears to be a similarly difficult transition. The way in which the students made use of the contexts showed that the activities did not directly facilitate the students to develop an understanding of formal solution processes. The students did not usually make use of the contexts when solving equations, working at the abstract symbolic level instead. Although it was hoped that, by engaging students in meaningful activities, the students would construct understandings of formal solution processes, this did not occur. None of the activities used in the study provided a metaphor for the formal method of solving equations. It is suggested that, for a context to be of great value for teaching a mathematical concept, the physical activity should act as a metaphor for the intended mathematical activity.

Complex systems abound on this planet, in the composition of the human body, in ecosystems, in social interaction, in political decision-making, and more. Analytical methods allowing us to better understand how these systems operate and, consequently, to have a chance to intervene and change the undesirable behavior of some of the more pernicious systems have developed and continue to be enhanced via quickly changing technology. Some of these analytical methods are accessible by pre-college students, but have not been widely used at that level of education. Jay Forrester, the founder of one of the methodologies, System Dynamics (SD), used to study complex system behavior involving feedback, laments the lack of understanding of complex systems evident in short-sited decisions made by legislators -- global climate change and fiscal policies being cases in point. In order to better prepare future decision makers with tools that could allow them to make more informed decisions about issues involving complex systems efforts have been underway to increase pre-college teacher understanding of the SD method. The research described in this dissertation introduces the mathematics education community to the value of System Dynamics modeling in pre-college algebra classes, indicates a path by which a traditional mathematics curriculum could be enhanced to include small SD models as a new representation for elementary functions studied in algebra classes, and provides an empirical study regarding conceptual understanding of functions by students. Chapter 2 indicates the numerous beneficial learning outcomes that empirical studies have shown accompany model-building activities. Chapter 3 indicates the need for students to become familiar with complex systems analysis, how SD modeling (one method of complex systems analysis) aligns with the Common Core State Standards in Mathematics, and the work that has transpired over the past two decades using SD in K-12. Chapter 4 focuses on the importance of the concept of function in high school mathematics, some limitations of exclusive reliance on the closed form equation representation for mathematizing problems and the SD stock/flow representations of some of the elementary functions that are studied in algebra classes. Chapter 5 looks at the issues affecting two traditional teachers and the challenges they faced when trying to reintroduce SD modeling into their algebra classes. Chapter 6 explains the student component of the classroom experiment that was conducted by the teachers who are highlighted in Chapter 5. The analysis of the results of student model-building activities in the two classroom studies that are part of the third paper did not indicate a statistical difference between the two experimental groups and the two control groups. Many environmental and scheduling issues conspired to adversely affect the experiment. However, positive outcomes were evident from the two pairs of students who were videotaped while they built the final multi-function drug model, the final student lesson in the experiment. Research focused on student outcomes is needed to further assess the strengths and weakness of the SD approach for student learning in mathematics.

Children from many culturally diverse backgrounds do not achieve in mathematics at the same rates as their counterparts from the dominant White, European-American culture (Gay, 2010). This so-called achievement gap is an artifact of an educational system that continues to fail to provide equal learning opportunities to culturally diverse children (Ladson-Billings, 2006; Nieto & Bode, 2011). Teachers who employ culturally responsive teaching (Gay, 2010) may help to close this opportunity gap and hence, the achievement gap. This study investigated, "How do elementary teacher candidates perceive teaching mathematics in a multicultural environment"; Using a critical constructivism research paradigm, this qualitative instrumental multiple case study involved a questionnaire, two interviews and a focus group with four elementary teacher candidates enrolled in a one-year teaching licensure program. The study examined elementary teacher candidates' images of mathematics and diverse students and the relationship between those images and their perceptions of teaching mathematics in a multicultural environment. The study concluded that the participants', images of mathematics, learners, and the teaching of mathematics were interrelated. The participants struggled to understand how students' diversity based on group membership (e.g., culture) influences a mathematics classroom and their teaching. However, on the basis of these participants, teacher candidates who hold a conceptual image of mathematics could be more open to adopting culturally responsive teaching than teacher candidates who hold a procedural image of mathematics. The study recommends the integration and modeling of culturally responsive teaching throughout all teacher education coursework.

Literature emphasizes how important it is that procedural and conceptual knowledge of mathematics should be learned in integration. Yet, generally, the learning and teaching in mathematics classrooms relies heavily on isolated procedures. This study aims to improve teaching and learning of partitive and quotitive division, moving away from isolated procedural knowledge to that of procedures with their underlying concepts through the use of manipulatives, visual representation and questioning. Learning and teaching lessons were designed to teach partitive and quotitive division both procedurally and conceptually. The study explored the roles these manipulatives, visual representations and questioning played toward the conceptual learning of partitive and quotitive division. It was found that manipulatives and iconic visualization enhanced learning, and this could be achieved through scaffolding using a questioning approach. It was concluded that manipulatives and iconic visualization need to be properly planned and used, and integrated with questioning to achieve success in the learning of procedural and conceptual knowledge.

This study modeled changes in pre-service teacher efficacy in mathematics and science over the course of the final year of teacher preparation using latent transition analysis (LTA), a longitudinal form of analysis that builds on two modeling traditions (latent class analysis (LCA) and auto-regressive modeling). Data were collected using the STEBI-B, MTEBI-r, and the ABNTMS instruments. The findings suggest that LTA is a viable technique for use in teacher efficacy research. Teacher efficacy is modeled as a construct with two dimensions: personal teaching efficacy (PTE) and outcome expectancy (OE). Findings suggest that the mathematics and science teaching efficacy (PTE) of pre-service teachers is a multi-class phenomena. The analyses revealed a four-class model of PTE at the beginning and end of the final year of teacher training. Results indicate that when pre-service teachers transition between classes, they tend to move from a lower efficacy class into a higher efficacy class. In addition, the findings suggest that time-varying variables (attitudes and beliefs) and time-invariant variables (previous coursework, previous experiences, and teacher perceptions) are statistically significant predictors of efficacy class membership. Further, analyses suggest that the measures used to assess outcome expectancy are not suitable for LCA and LTA procedures.

Thesis (MEd)--Stellenbosch University, 2014.
ENGLISH ABSTRACT: Research has confirmed the educational value of mathematical modelling for learners of all abilities. The development of modelling competencies is essential in the modelling approach. Little research has been done to identify and develop the mathematising modelling competency for specific sections of the mathematics curriculum. The study investigates the development of mathematising competencies during the modelling of number pattern problems. The RME theory has been selected as the theoretical framework for the study because of its focus on mathematisation. Mathematising competencies are identified from current literature and developed into models for horizontal and vertical (complete) mathematisation. The complete mathematising competencies were developed for number patterns and mapped on a continuum. They are internalising, interpreting, structuring, symbolising, adjusting, organising and generalising. The study investigates the formulation of a hypothetical trajectory for algebra and its associated local instruction theory to describe how effectively learning occurs when the mathematising competencies are applied in the learning process. Guided reinvention, didactical phenomenology and emergent modelling are the three RME design heuristics to form an instructional theory and were integrated throughout the study to comply with the design-based research’s outcome: to develop a learning trajectory and the means to support the learning thereof. The results support research findings, that modelling competencies develop when learners partake in mathematical modelling and that a heterogeneous group of learners develop complete mathematising competencies through the learning of the modelling process. Recommendations for additional studies include investigations to measure the influence of mathematical modelling on individualised learning in secondary school mathematics.
AFRIKAANSE OPSOMMING: Navorsing steun die opvoedkundige waarde van modellering vir leerders met verskillende wiskundige vermoëns. Die ontwikkeling van modelleringsbevoegdhede is noodsaaklik in 'n modelleringsraamwerk. Daar is min navorsing wat die identifikasie en ontwikkeling van die bevoegdhede vir matematisering vir spesifieke afdelings van die wiskundekurrikulum beskryf. Die studie ondersoek die ontwikkeling van matematiseringsbevoegdhede tydens modellering van getalpatrone. Die Realistiese Wiskundeonderwysteorie is gekies as die teoretiese raamwerk vir die studie, omdat hierdie teorie die matematiseringsproses sentraal plaas. Matematiseringsbevoegdhede vanuit die bestaande literatuur is geïdentifiseer en ontwikkel tot modelle wat horisontale en vertikale (volledige) matematisering aandui. Hierdie matematiseringsbevoegdhede is spesifiek vir getalpatrone ontwikkel en op ‘n kontinuum geplaas. Hulle is internalisering, interpretasie, strukturering, simbolisering, aanpassing, organisering en veralgemening. Die studie lewer die formulering van ‘n hipotetiese leertrajek vir algebra, die gepaardgaande lokale onderrigteorie en beskryf hoe effektiewe leer plaasvind wanneer die ontwikkelde matematiseringsbevoegdhede volledig in die leerproses toegepas word. Die RME ontwikkellingsheuristieke, begeleidende herontdekking, didaktiese fenomenologie en ontluikende modellering, is geïntegreer in die studie sodat dit aan die uitkoms van ‘n ontwikkelingsondersoek voldoen. Die uitkoms is ‘n leertrajek en ‘n beskrywing hoe die leerproses ondersteun kan word. Die analise het tot die formulering van ‘n lokale-onderrig-teorie vir getalpatrone gelei. Die resultate van die studie kom ooreen met navorsingsbevindings dat modelleringsbevoegdhede ontwikkel wanneer leerders deelneem aan modelleringsaktiwiteite, en bewys dat ‘n groep leerders met gemengde vermoëns volledige matematiseringsbevoegdhede ontwikkel wanneer hulle deur die modelleringsproses werk. 'n Aanbeveling vir verdere navorsing is om die uitwerking van die modelleringsperspektief op individuele leer in hoërskool klaskamers te ondersoek.

The purpose of this investigation was to examine, through the use of survey data, relationships between inputs of schooling and outcomes, as measured by student achievement in mathematics. The inputs of schooling were comprised of a number of variables grouped under each of the following categories: students' and teachers' backgrounds; students' and teachers' perceptions of mathematics; classroom organization and problem-solving processes. Outcome measures included student achievement on test total, problem solving and applications. A related question involved exploration of the appropriateness of using cross-sectional survey data to make decisions based on the relationships found among the input and output variables. To address this question, results from a subsequent longitudinal study, which utilized the same instruments, were examined first with post-test data and second with the inclusion of pre-test data as covariates. Data collected from teachers and students of Grade 7 in the 1985 British Columbia Assessment of Mathematics were re-analysed in order to link responses to Teacher Questionnaires with the students' results in teachers' respective classrooms. Responses were received from students in 1816 classrooms across the province and from 1073 teachers of Grade 7 mathematics. The data underwent several stages of analysis. Following the numerical coding of variables and the aggregation of student data to class level, Pearson product-moment correlations were calculated between pairs of variables. Factor analysis and multiple regression techniques were utilized at subsequent stages of the analysis. A number of significant relationships were found between teacher and student behaviors, and student achievement. Among the variables found to be most strongly related to achievement were teachers' attitudes toward problem solving, the number and variety of approaches and methods used by teachers, student perceptions of mathematics, and socio-economic status. Results also show that student background, students' and teachers' perceptions of mathematics, classroom organization and problem-solving processes all account for measurable variances in student achievement. The amount of variance accounted for, however, was higher for achievement on application items, measuring lower cognitive levels of behavior, than on problem-solving items which measured cognitive behavior at the critical thinking level. Through examination of the standardized beta weights from the cross-sectional and longitudinal models, it was found that prediction of change in achievement based on corresponding change in classroom process variables was similar for both models. Differences, however, were found for variables in the other categories.
Education, Faculty of
Curriculum and Pedagogy (EDCP), Department of
Graduate

A key issue for the success of students entering a first year mathematics course at tertiary level is whether or not they have an integrated understanding and view of the mathematical concepts acquired at school. Various integrated applications from first year mathematics suggest that a compartmentalised view of mathematics would be detrimental to any student's chances of passing mathematics at this level. This study tried to assess whether learners do have an integrated understanding of mathematics at grade 12 level.

The Teaching Abstract Algebra for Understanding (TAAFU) project was centered on an innovative abstract algebra curriculum and was designed to accomplish three main objectives: to produce a set of multi-media support materials for instructors, to understand the challenges faced by mathematicians as they implemented this curriculum, and to study how this curriculum supports student learning of abstract algebra. Throughout the course of the project I took the lead investigating the teaching and learning in classrooms using the TAAFU curriculum. My dissertation is composed of three components of this research. First, I will report on a study that aimed to describe the experiences of mathematicians implementing the curriculum from their perspective. Second. I will describe a study that explores the mathematical work done by teachers as they respond to the mathematical activity of their students. Finally, I will discuss a theoretical paper in which I synthesize aspects of the instructional theory underlying the TAAFU curriculum in order to develop an analytic framework for analyzing student learning. This dissertation will serve as a foundation for my future research focused on the relationship between teachers' mathematical work and the learning of their students.

The purpose of this research study is to describe and analyze the self-reported experiences of exemplary high school mathematics teachers who underwent personal and professional transformations in order to develop and use a standards-based, constructivist (SBC) teaching paradigm in their classrooms. These teachers were all past recipients of the Presidential Award for Excellence in Mathematics and Science Teaching (PAEMST), an award that required them to demonstrate that their mathematics instruction was rigorous in the manner described by the NCTM standards. The following research questions are addressed: (a) What are the paths SBC secondary mathematics teachers who received the PAEMST pursued to become highly effective?, (b) What obstacles and challenges did they encounter and how were these obstacles overcome?, and (c) What sustained them on their journeys? The research methodology used to be a narrative inquiry. Following a wide survey of PAEMST recipients, five volunteer participants were chosen for the study. Data were collected from each participant using a one-to-one interview and the written section of each participant's PAEMST application. A narrative was written for each participant describing the path they had followed to become a highly effective high school mathematics teacher. The narrative was sent to each participant, and a follow-up interview was conducted via telephone amending the narrative to reflect the participant's additions and deletions. From the five amended narratives, eight themes were identified: (a) influences; (b) education; (c) professional development; (d) NCTM standards; (e) teaching style: beginning, current, or end of a career; (f) obstacles; (g) personality traits and personal beliefs; and (h) student influence. Several of the themes were supported by previous research. However, this research study discovered two new findings. First, the five participants had common characteristics and beliefs: (a) belief in their students, (b) persistence, (c) belief that professional development is vital for teacher growth, and (d) passion about mathematics and about conveying that passion to their students. The second research finding pertained to the influence that their own students had on all of the five participants. All the participants purposely sought out their students' thoughts about the classroom curriculum and about the instruction they received. The teachers considered their students part of the classroom learning community, and they honored and acted on their input. Finally, in addition to describing the trajectory of five PAEMST winning teachers, this study offers recommendations for students studying to become high school mathematics teachers, teacher educators, and educational researchers. For these students, their teaching preparation courses need to be taught adhering to the four principles of learning: activity, reflection, collaboration and community. According to this research, the model of teacher preparation courses that emphasize the teaching of the above four principles using a traditional teacher-directed method does not prepare future mathematics teachers for the use of SBC teaching in their classrooms. Suggestions about further research are addressed.

The general goal of the present study was to assess mathematics achievement at the end of Grade 12 in the Dominican Republic, with particular attention to school and regional differences, as well as gender differences. Also, gains in achievement were examined by comparing the achievement of students in Grade 12 to that of students finishing Grade 11. In addition, the performance of Grade 12 students was compared to that of Grade 8 students as assessed in the Teaching and Learning of Mathematics in the Dominican Republic (TLMDR) study and to that of students from other countries in the Second International Mathematics Study (SIMS). The sample included 1271 students in Grade 12 and 1413 in Grade 11, distributed over 49 schools. Three types of schools were sampled, public schools, and two kinds of private schools. They were urban schools located in the twelve largest cities of the country. These cities were grouped into three regions of similar size. The mathematics test consisted of 70 multiple-choice items distributed over two test forms. Students' scores were analyzed to assess how much mathematics students in Grade 12 know. Grade 11 data were used as a surrogate for pre-test scores to estimate gains in achievement. School means were used in an analysis of variance designed to examine the effect of school type and region on mathematics achievement. Males' and females' scores were used to analyze gender differences in achievement at the item level, and within each of the school types and regions in the sample. Grade 12 students' responses to 14 items were compared to those of Grade 8 students. Finally, Grade 12 students' responses to 10 items were compared to those of students from other countries in SIMS. Among the findings of this study were: 1. Students in Grade 12 scored poorly on the mathematics test. Grade 11 and Grade 12 students obtained similar achievement levels which indicated that the achievement gains between the two grades were very small. 2. School type and region were found to significantly affect mathematics achievement, but no interaction effect was found. 3. The comparison of school type means showed that only one type of private school significantly outperformed public schools. This type of school also outperformed the other type of private school. 4. The comparison of region means did not produce the predicted outcome. The pairwise comparisons showed that none of the regions was significantly different from the other, despite the fact that the region factor was significant. 5. The analysis of gender differences in mathematics achievement showed that males performed significantly better than females. At the item level, males outperformed females on only 19 items. Most of these items dealt with geometry, or were at the application level. 6. Gender differences favoring males were found to be independent of school type and region. 7. Comparison between Dominican Grade 12 and Grade 8 students revealed that mathematics achievement improved between the grades for most items. 8. Dominican performance was very poor on the SIMS items and it was far behind that of other countries.
Education, Faculty of
Graduate

Federal dollars are utilized to develop instructional programs for students not demonstrating mathematical proficiency on state standardized mathematics assessments, but there is a lack of empirical data on the effectiveness of two different approaches that were used in the local context. The purpose of this quantitative, nonexperimental, casual-comparative study was to determine if state achievement test scores of students in fourth grade who received instruction from a Mathematics Specialist (MS) during the 2007-2009 academic years demonstrated a statistically significant difference from the mathematics state achievement test scores of fourth grade students who received instruction from Grades 1-8 credentialed teachers supported by a Math Coach (MC) during the 2012-2014 academic years. The theoretical base includes two components: National Council of Teachers of Mathematics Standards and Federal No Child Left Behind educational policy, which focus on standards-based education, curriculum, assessment, and instruction to meet students' mathematical needs. Data was collected from a census sample of 13,671 students' state scores from school years 2007-2008, 2008-2009 (MS) and 2012-2013, 2013-2014 (MC). The research question was whether there is a difference in MS and MC scores. An independent samples t test was used to compare the means of all the scores. The results show that the MS program produced statistically higher math scores than the MC. This supports the limited literature in favor of MS. Positive social change includes supporting increasing the use of the MS program in the local context to increase mathematics test scores and the potential for redistribution of federal funds to develop MS programs nationwide.

abstract: The research indicated effective mathematics teaching to be more complex than assuming the best predictor of student achievement in mathematics is the mathematical content knowledge of a teacher. This dissertation took a novel approach to addressing the idea of what it means to examine how a teacher's knowledge of mathematics impacts student achievement in elementary schools. Using a multiple case study design, the researcher investigated teacher knowledge as a function of the Mathematics Teaching Cycle (NCTM, 2007). Three cases (of two teachers each) were selected using a compilation of Learning Mathematics for Teaching (LMT) measures (LMT, 2006) and Developing Mathematical Ideas (DMI) measures (Higgins, Bell, Wilson, McCoach, & Oh, 2007; Bell, Wilson, Higgins, & McCoach, 2010) and student scores on the Arizona Assessment Collaborative (AzAC). The cases included teachers with: a) high knowledge & low student achievement v low knowledge & high student achievement, b) high knowledge & average achievement v low knowledge & average achievement, c) average knowledge & high achievement v average knowledge & low achievement, d) two teachers with average achievement & very high student achievement. In the end, my data suggested that MKT was only partially utilized across the contrasting teacher cases during the planning process, the delivery of mathematics instruction, and subsequent reflection. Mathematical Knowledge for Teaching was utilized differently by teachers with high student gains than those with low student gains. Because of this insight, I also found that MKT was not uniformly predictive of student gains across my cases, nor was it predictive of the quality of instruction provided to students in these classrooms.
Dissertation/Thesis
Ph.D. Curriculum and Instruction 2013

M.Ed.
High school mathematics learners often take mathematics education for granted. They study mathematics simply because it is included in the school curriculum, and thus required for them to pass so that they can obtain a school leaving qualification. They never really succeed in seeing and understanding the relevance of mathematics to their present and future lives. As a result, they fail to relate and apply classroom mathematics to the external environment. They fail to make mathematical connections that would enable them to be confident users of mathematics as an effective tool for solving problems, a means of communication and a way of supporting reasoning. This suggests that there may be some serious constraints associated with the teachers' instructional approaches, which hinder the learners' meaningful learning and understanding of the relevance of mathematics. Thus, there arises the need to examine the relationship between the teachers' instructional approaches and the learners' understanding of the relevance of mathematics. Such an examination may help to expose the strengths and limitations of the instructional approaches, so that the necessary adjustments can be made in the teaching practice to improve the learning of mathematics.

The purpose of this study was twofold: (a) to investigate the prevalence of mathematics anxiety among freshman Algebra I students in an urban, Midwestern high school, and (b) to find out if a pre-quiz and quiz intervention could reduce mathematics anxiety in one specific class. The Mathematics Anxiety Rating Scale for Adolescents (MARS-A) was the primary quantitative data collection instrument. Qualitative data were collected using the Mathematicsitude Survey, student reflections, and interviews. Findings from the MARS-A showed that 50% of students experienced a significant amount of mathematics anxiety, particularly associated with testtaking. However, there was a large amount of variation among scores. In the treatment class, a strategy of pre-quiz followed by the same or similar quiz the following day was used to build student confidence and thereby lessen anxiety. The strategy did not meet this objective as many students reported greater anxiety levels after the intervention than before. Qualitative probing did show that in some isolated cases the strategy worked very well.
Thesis (M.Ed.)--Wichita State University, College of Education, Dept. of Curriculum and Instruction

This study examines how mathematical modeling activities within a collaborative group impact students' `value creation' through mathematics. Creating `value' in this study means to apply one's knowledge in a way that benefits the individual and society, and the notion of `value' was adopted from Makiguchi's theory of `value creation' (1930/1989). With a unified framework of Makiguchi's theory of `value', mathematical disposition, and identity, the study identified three aspects of value-beauty, gains, and social good-using observable evidence of mathematical disposition, identity, and sense of community. Sixty students who enrolled in a college algebra course participated in the study. The results showed significant changes in students' mathematics dispositions after engaging in the modeling activities. Analyses of students' written responses and interview data demonstrated that the modeling tasks associated with students' personal data and social interactions within a group contributed to students' developing their identity as doers of mathematics and creating social value. The instructional model aimed to balance the cognitive aspect and the affective skills of learning mathematics in a way that would allow students to connect mathematical concepts to their personal lives and social lives. As a result of the analysis of this study, there emerged a holistic view of the classroom as it reflects the Makiguchi's educational philosophy. Lastly, implications of this study for research and teaching are discussed.

Popular culture has had a great deal of impact on our social, cultural, and political worlds; it is portrayed through different mediums, in different forms, and connects the world to ideas, beliefs, and different perspectives. Though this dissertation is part of a larger body of work that examines the complex relationship between popular culture and mathematical identity, this study takes a different perspective by examining it through the lens of mathematical Internet memes. This study was conducted with 31 secondary school participants and used a two-tiered approach (in-depth focus groups and an individual meme activity) at each of the five school sites visited around New York City. Multiple sources of data were used to reveal that students are receiving messages about mathematics from memes in popular culture. In particular, participants described six core themes from the meme inventory: (1) stereotypical views of mathematics; (2) mathematics is too complicated; (3) no effort should be needed in mathematics; (4) mathematics is useless; (5) mathematics is not fun; and (6) sense of accomplishment from mathematics. Participants were also given free rein to create hypothetical mathematics memes. Findings demonstrate that not only are memes being used to depict mathematical stereotypes, thereby reinforcing negative messages, but also support social media practices (liking, commenting, sharing, and creating) that reify negative messages about mathematics with little to no resistance from opposing perspectives. In general, participants described mathematical memes in a specific manner that demonstrates them having influence over students’ mathematical identity but not entirely on the way one may think. Future research implications include explorations of the “new” online mathematical space students are utilizing; to wit, what makes these specific memes go viral? What are common misconceptions? Are commenters learning from their mistakes and other answer responses? Implications for practice include the creation of formal spaces within classrooms and communities for students to debrief their thoughts and sentiments about mathematics, as well as informal opportunities for educators, students, and community members to engage positively about mathematics: because without these interventions the messages found in memes, whether positive or negative, are potentially legitimized through popular culture’s presentations. Moreover, the results of this study also show that students are unaware of the processes and proficiencies of mathematical learning. More specifically, teachers and others must help students understand knowledge is not transmitted by copying notes or that teaching strategies need to account for students being apprehensive to ask questions in a mathematics classroom. Memes can also be used to explore mathematics content, through error analysis and explanation of concepts.

The aim of the present study was to investigate the effect of metacognition-based teaching and teaching mathematics via problem solving on students' understanding of mathematics, and the ways in which the students' beliefs about themselves as doers and learners of mathematics and about mathematics and mathematical problem solving were influenced by the instruction. The 60 hours of instruction occurred in the context of a day-to-day mathematics course for undergraduate non-science students, and that gave mea chance to teach mathematics via problem solving. Metacognitive strategies that were included in the instruction contributed to the students' mathematical learning in various ways. The instruction used journal writing, small groups, and whole-class discussions as three different but interrelated strategies that focused on metacognition. Data for the study were collected through four different sources, namely quizzes and assignments (including the final exam), interviews, the instructor's and the students' autobiographies and journals, and class observations (field notes, audio and video tapes). Journal writing served as a communication channel between the students: and the-instructor, and as a result facilitated the individualization of instruction. Journal writing provided the opportunity for the students to clarify their thinking and become more reflective. Small groups proved to be an essential component of the instruction. The students learned to assess and monitor their work and to make appropriate decisions by working cooperatively and discussing the problems with each other. Whole-class discussions raised the students' awareness about their strengths and weaknesses. The discussions also helped students to a great extent become better decision makers. Three categories of students labeled traditionalists, incrementalists, and innovators, emerged from the study. Nine students, who rejected the new approach to teaching and learning mathematics were categorized as traditionalists. The traditionalists liked to be told what to do by the teacher. However, they liked working in small groups and using manipulative materials. The twelve incrementalists were characterized as those who propose to have balanced instruction in which journal writing was a worthwhile activity, group work was a requirement, and whole-class discussions were preferred for clarifying concepts and problems more than for generating and developing new ideas. The nineteen other students were categorized as innovators, those who welcomed the new approach and utilized it and preferred it. For them, journal writing played a major role in enhancing and communicating the ideas. Working in small groups seemed inevitable, and whole-class discussions were a necessity to help them with the meaning-making processes. The incrementalists and the innovators gradually changed their beliefs about mathematics from viewing it as objective, boring, lifeless, and unrelated to their real-lives, to seeing it as subjective, fun, meaningful, and connected to their day-to-day living. The findings of the study further indicated that most of the incrementalists and the innovators changed their views about mathematical problem solving from seeing it as the application of certain rules and formulas to viewing it as a meaning-making process of creation and construction of knowledge.

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Humanities, School of Education, 2013
Mathematics-for-teaching (MfT) is complex, multi-faceted and topic-specific. In this study, a Financial Mathematics course for pre-service secondary mathematics teachers provides a revelatory case for investigating MfT. The course was designed and taught by the author to a class of forty-two students at a university in South Africa. Eight students, forming a purposive sample, participated as members of two focus tutorial groups and took part in individual and group interviews. As an instance of insider research, the study makes use of a qualitative methodology that draws on a variety of data sources including lecture sessions and group tutorials, group and individual interviews, students’ journals, a test and a questionnaire. The thesis is structured in two parts. The first part explores revisiting of school mathematics with particular focus on compound interest and the related aspects of percentage change and exponential growth. Four cases are presented, in the form of analytic narrative vignettes which structure the analysis and provide insight into opportunities for learning MfT of compound interest. The evidence shows that opportunities may be provided to learn a range of aspects of MfT through revisiting school mathematics. The second part focuses on obstacles experienced by students in learning annuities, their time-related talk, as well as their use of mathematical resources such as timelines and spreadsheets. A range of obstacles are identified. Evidence shows that students use timelines in a range of non-standard ways but that this does not necessarily determine or reflect their success in solving annuities problems. Students’ use of spreadsheets reveals that spreadsheets are a powerful tool for working with annuities. A key finding with regard to teachers’ mathematical knowledge, and which cuts across both parts of the thesis, is the importance of being able to move between compressed and decompressed forms of mathematics. The study makes three key contributions. Firstly, a framework for MfT is proposed, building on existing frameworks in the literature. This framework is used as a conceptual tool to frame the study, and as an analytic tool to explore opportunities to learn MfT as well as the obstacles experienced by. A second contribution is the theoretical and empirical elaboration of the notion of revisiting. Thirdly, a range of theoretical constructs related to teaching and learning introductory financial mathematics are introduced. These include separate reference landscapes for the concepts of compound interest and annuities

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 2002.
In this thesis I consider how mathematics students in a traditional firstyear Calculus course at a South African university appropriate mathematical objects which are new to them but which are already part of the official mathematics discourse. Although several researchers have explained mathematical object appropriation in process-object terms (for example, Sfard, 1994; Dubinsky, 1991, 1997; Tall, 1991, 1995, 1999), my focus is largely on what happens prior to the object-process stage. In line with Vygotsky (1986), I posit that the appropriation of a new mathematical object by a student takes place in phases and that an examination of these phases gives a language of description for understanding this process. This theory, which I call “appropriation theory”, is an elaboration and application of Vygotsky’s (1986) theory of concept formation to the mathematical domain. I also use Vygotsky’s (1986) notion of the functional use of a word to postulate that the mechanism for moving through these phases, that is, for appropriating the mathematical object, is a functional use of the mathematical sign. Specifically, I argue that the student uses new mathematical signs both as objects with which to communicate (like words are used) and as objects on which to focus and to organise his mathematical ideas (again as words are used) even before he fully comprehends the meaning of these signs. Through this sign usage the mathematical concept evolves for that student so that it eventually has personal meaning (like the meaning of a new word does for a child); furthermore, because the usage is socially regulated, the concept evolves so that its usage is concomitant with its usage in the mathematical community. I further explicate appropriation theory by elaborating a link between the theoretical concept variables and their empirical indicators, illustrating these links with data obtained from seven clinical interviews. In these interviews, seven purposefully chosen students engage in a set of speciallydesigned tasks around the definition of an improper integral. I utilise the empirical indicators to analyse two of these interviews in great detail. These analyses further inform the development of appropriation theory and also demonstrate how the theory illuminates the process of mathematical object appropriation by a particular student.

Too often, discussions about Mathematics express feelings of anguish and despair; and, indeed Mathematics results in general in South Africa can be described as dismal. The Department of Education (DoE) reported that in the 2010 National Senior Certificate examinations, 52.6% of learners obtained less than 30% in Mathematics and 69.1% of learners obtained less than 40% (DoE, 2010). This implies that a very small percentage of grade 12 learners would be eligible to further their studies in the fields of Mathematics and science at tertiary level, resulting in a depletion of science and Mathematics-oriented professionals. This study explored the mathematical attitudes and achievement strategies of successful Mathematics learners to overcome the factors that might impede achievement. This study has the potential to improve practice because the findings of the study and recommendations are made implicit in the discussion. In particular this study sought to investigate the following issues: (a) What are secondary school learners' attitudes towards Mathematics? (b) In what ways are these attitudes linked to factors to which the learners attribute their achievement in Mathematics? (c) What strategies do successful Mathematics learners use to overcome the factors that they identify as impeding their performance in Mathematics? This research involved a case study approach. The study solicited both quantitative and qualitative data from the participants. The participants comprised 95 Grade 10, 11 and 12 Mathematics learners. The Fennema-Sherman Mathematics Attitude Scales (FSMAS) questionnaire was used to collect data from participants. The data was analysed using Attribution Theory and Achievement Theory. Two learners, who obtained more than 60% in the 2011 half-year Mathematics examination, from grades 10, 11 and 12 respectively, constituted the focus group. The focus group interview enhanced the study by clarifying the responses to the questionnaire and providing answers to the second and third research questions. The findings of the research include the following: teachers play an important role in shaping learners’ attitudes toward Mathematics; learners are anxious when asked to solve mathematical problems; parents are very encouraging of their children learning Mathematics; the importance of Mathematics for future careers exerted a significant effect on mathematical achievement; and finally the various strategies that learners employ that positively impact on their achievement in mathematics include mastery experience, motivation, private tuition and peer group teaching-learning. The final section of this dissertation discusses the implications of this study for practising Mathematics teachers and suggestions for further research in the area of affect.
Thesis (M.Ed.)-University of KwaZulu-Natal, Edgewood, 2011.

In this paper, students' bilingualism and multicultural experiences are examined as cognitive resources for mathematization. Capitalizing on the view of language as action, and on students' familiarity with certain experiences through direct participation, the study includes a conceptual framework, never used with bilingual mathematics learners, to investigate how bilingual students organize and coordinate actions to solve mathematical problems about familiar and unfamiliar experiences in English and Spanish. The study used a research methodology to investigate two questions: (a) How do bilingual students' mathematize familiar experience problems and unfamiliar experience problems in Spanish and English? (b) What do differences and similarities in bilingual students' mathematization across problems and languages reveal about experience and bilingualism as cognitive resources? Findings show important differences. In problems about familiar experiences, students generated more productive actions, more reflective actions, and less unproductive actions than in problems about unfamiliar experience. As for the bilingualism, students used Spanish and English differently. When solving problems in Spanish, they framed actions more socially by including partners or sharing the action with partners, whereas in English they framed actions more individually, more depersonalized, excluding partners and instead relying on words in problems to justify their individual actions. This suggests that reinventing mathematics and themselves as mathematical thinkers is part of using their bilingualism and experiences as cognitive tools, and attention to how they use each language for each type of problem can reveal substantial knowledge about how bilinguals learn mathematics.
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M.Ed. (Media Studies)
The subject mathematics is for one reason or another regarded as a necessary prerequisite for various job directions by many countries (Del;we, 1988:1). This subject is also valuable as preparation for the contingency that the child in future may take up an occupation requiring knowledge of the subject in question (Young, 1927:14). What matters most is that·a large number of pupils fail mathematics every year in their examination in primary and secondary schools. This is a clear indication that there are problems in the teaching and learning of mathematics which need special attention. Mathematics is a sequential learning subject (Larcombe, 1985:12). By this is meant that new learning content cannot be grasped unless there is insight into and understanding of the previous learning content. This is true because the growth of mathematical understanding develops step by· step with each forward move depending upon the consolidation of previous experience. In the light of the evidence that primary school children are in the concrete reasoning state, most mathematics educators believe that it is desirable to use large amounts of manipulative materials with young children. The understanding of the nature of mathematics is more important in teaching, as the teaching of the subject is influenced and determined by the teacher's understanding of the nature of the subject. The theory, most prevalent among teachers is that mathematics affords the best training for the reasoning powers, and this is its traditional form (Young, 1927:15).

What is good mathematics teaching? The answer depends on whom you are asking. Teachers, researchers, policymakers, administrators, and parents usually provide their own view on what they consider is good mathematics teaching and what is not. The purpose of this study was to determine how beginning teachers define good mathematics teaching and what they report as being the most important attributes at the secondary level. This research explored whether there was a relationship between the demographics of the participants and the attributes of good teaching. In addition, factors that influence the understanding of good mathematics teaching were explored. A mixed methodology was used to gather information from the research participants regarding their beliefs and classroom practices of good mathematics teaching. The two research instruments used in this study were the survey questionnaire and a semi-structured interview. Thirty-three respondents who had one to two years of classroom experience comprised the study sample. They had graduated from a school of education in an eastern state and had obtained their teacher certification upon completing their studies. The beginning mathematics teachers selected these four definitions of good teaching as their top choices: 1) have High Expectations that all students are capable of learning; 2) have strong content knowledge (Subject Matter Knowledge); 3) create a Learning Environment that fosters the development of mathematical power; and 4) bring Enthusiasm and excitement to classroom. The three most important attributes in good teaching were: Classroom Management, Motivation, and Strong in Content Knowledge. One interesting finding was the discovery of four groups of beginning teachers and how they were associated with specific attributes of good mathematics teaching according to their demographics. Beginning teachers selected Immediate Classroom Situation, Mathematical Beliefs, Pedagogical Content Knowledge, and Colleagues as the top four factors from the survey analysis that influenced their understanding of good mathematics teaching. The study's results have implications for informing the types of mathematical knowledge required for pre-service teachers that can be incorporated into teacher education programs and define important attributes of good mathematics teaching during practicum.