published_or_final_version / Education / Master / Master of Education

published_or_final_version / Education / Master / Master of Education

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.

Thesis (MEd)--University of Stellenbosch, 2002. / ENGLISH ABSTRACT: Algebra deals amongst others with the relationship between variables. It differs from Arithmetic amongst others as there is not always a numerical solution to the problem. An algebraic expression can even be the solution to the problem in Algebra. The variables found in Algebra are often represented by letters such as X, y, etc. Equations are an integral part of Algebra. To solve an equation, the value of an unknown must be determined so that the left hand side of the equation is equal to the right hand side. There are various ways in which the solving of equations can be taught. The purpose of this study is to determine the existence of a cognitive gap as described by Herseovies & Linchevski (1994) in relation to solving linear equations. When solving linear equations, an arithmetical approach is not always effective. A new way of structural thinking is needed when solving linear equations in their different forms. In this study, learners' intuitive, informal ways of solving linear equations were examined prior to any formal instruction and before the introduction of algebraic symbols and notation. This information could help educators to identify the difficulties learners have when moving from solving arithmetical equations to algebraic equations. The learners' errors could help educators plan effective ways of teaching strategies when solving linear equations. The research strategy for this study was both quantitative and qualitative. Forty-two Grade 8 learners were chosen to individually do assignments involving different types of linear equations. Their responses were recorded, coded and summarised. Thereafter the learners' responses were interpreted, evaluated and analysed. Then a representative sample of fourteen learners was chosen randomly from the same class and semi-structured interviews were conducted with them From these interviews the learners' ways of thinking when solving linear equations, were probed. This study concludes that a cognitive gap does exist in the context of the investigation. Moving from arithmetical thinking to algebraic thinking requires a paradigm shift. To make adequate provision for this change in thinking, careful curriculum planning is required. / AFRIKAANSE OPSOMMING: Algebra behels onder andere die verwantskap tussen veranderlikes. Algebra verskil van Rekenkunde onder andere omdat daar in Algebra nie altyd 'n numeriese oplossing vir die probleem is nie. InAlgebra kan 'n algebraïese uitdrukking somtyds die oplossing van 'n probleem wees. Die veranderlikes in Algebra word dikwels deur letters soos x, y, ens. voorgestel. Vergelykings is 'n integrale deel van Algebra. Om vergelykings op te los, moet 'n onbekende se waarde bepaal word, om die linkerkant van die vergelyking gelyk te maak aan die regterkant. Daar is verskillende maniere om die oplossing van algebraïese vergelykings te onderrig. Die doel van hierdie studie is om die bestaan van 'n sogenaamde "kognitiewe gaping" soos beskryf deur Herseovies & Linchevski (1994), met die klem op lineêre vergelykings, te ondersoek. Wanneer die oplossing van 'n linêere vergelyking bepaal word, is 'n rekenkundige benadering nie altyd effektiefnie. 'n Heel nuwe, strukturele manier van denke word benodig wanneer verskillende tipes linêere vergelykings opgelos word. In hierdie studie word leerders se intuitiewe, informele metodes ondersoek wanneer hulle lineêre vergelykings oplos, voordat hulle enige formele metodes onderrig is en voordat hulle kennis gemaak het met algebraïese simbole en notasie. Hierdie inligting kan opvoeders help om leerders se kognitiewe probleme in verband met die verskil tussen rekenkundige en algebraïese metodes te identifiseer.Die foute wat leerders maak, kan opvoeders ook help om effektiewe onderrigmetodes te beplan, wanneer hulle lineêre vergelykings onderrig. As leerders eers die skuif van rekenkundige metodes na algebrarese metodes gemaak het, kan hulle besef dat hul primitiewe metodes nie altyd effektief is nie. Die navorsingstrategie wat in hierdie studie aangewend is, is kwalitatief en kwantitatief Twee-en-veertig Graad 8 leerders is gekies om verskillende tipes lineêre vergelykings individueel op te los. Hul antwoorde is daarna geïnterpreteer, geëvalueer en geanaliseer. Daarna is veertien leerders uit hierdie groep gekies en semigestruktureerde onderhoude is met hulle gevoer. Vanuit die onderhoude kon 'n dieper studie van die leerders se informele metodes van oplossing gemaak word. Die gevolgtrekking wat in hierdie studie gemaak word, is dat daar wel 'n kognitiewe gaping bestaan in die konteks van die studie. Leerders moet 'n paradigmaskuif maak wanneer hulle van rekenkundige metodes na algebraïese metodes beweeg. Hierdie klemverskuiwing vereis deeglike kurrikulumbeplanning.

published_or_final_version / Curriculum Studies / Doctoral / Doctor of Philosophy

This study is an investigation into the performance of learners in algebra using the levels of understanding as measured by the ICCAMS diagnostic instrument. The study was conducted in two phases. The first phase of the study consisted of an analysis of the scripts of a sample of 29 learners in Grade 9 who had written the test administered by the Wits Maths Connect-Secondary unit at Wits University. The scripts of the same 29 learners in Grade 10 were analysed to determine the progression within the levels of these learners from Grades 9 to 10. Eighteen learners progressed from a lower to a higher level. During the analysis of the tests it was found that the conjoining error was the main obstacle to some learners in progressing from moving from level 1 to level 3. During phase 2 of the study, a sample of 6 learners was selected from the original 29 learners. These learners completed a written task to investigate errors made in algebra in Grade 11. Interviews were conducted with these learners based on a written task. The analysis of the interviews and written task illustrated the problems learners experienced with level 2 questions, particularly with respect to the conjoining error.

A thesis submitted to the Wits school of Education, Faculty of Humanities, University of the Witwatersrand in fulfillment of the requirements for the degree of Doctor of Philosophy Johannesburg 2017 / MT2017

The purpose of this study was to determine the extent to which the modeling of deductive reasoning and proof-type thinking occurs in a mathematics course in which students are not explicitly preparing to write formal mathematical proofs. Algebra was chosen because it is the course that typically directly precedes a student's first formal introduction to proof in geometry in the United States. The lens through which this study aimed to examine the intended curriculum was by identifying and reviewing the modeling of proof and deductive reasoning in the most popular and widely circulated algebra textbooks throughout the United States. Textbooks have a major impact on mathematics classrooms, playing a significant role in determining a teacher's classroom practices as well as student activities. A rubric was developed to analyze the presence of reasoning and proof in algebra textbooks, and an analysis of the coverage of various topics was performed. The findings indicate that, roughly speaking, students are only exposed to justification of mathematical claims and proof-type thinking in 38% of all sections analyzed. Furthermore, only 6% of coded sections contained an actual proof or justification that offered the same ideas or reasoning as a proof. It was found that when there was some justification or proof present, the most prevalent means of convincing the reader of the truth of a concept, theorem, or procedure was through the use of specific examples. Textbooks attempting to give a series of examples to justify or convince the reader of the truth of a concept, theorem, or procedure often fell short of offering a mathematical proof because they lacked generality and/or, in some cases, the inductive step. While many textbooks stated a general rule at some point, most only used deductive reasoning within a specific example if at all. Textbooks rarely expose students to the kinds of reasoning required by mathematical proof in that they rarely expose students to reasoning about mathematics with generality. This study found a lack of sufficient evidence of instruction or modeling of proof and reasoning in secondary school algebra textbooks. This could indicate that, overall, algebra textbooks may not fulfill the proof and reasoning guidelines set forth by the NCTM Principles and Standards and the Common Core State Standards. Thus, the enacted curriculum in mathematics classrooms may also fail to address the recommendations of these influential and policy defining organizations.

No description available.

This research investigated the impact of "embedded teaching" and "learner-controlled" instruction on student learning of algebra in a controlled computer-tutoring environment. Three versions of a computer tutor were developed to establish three experimental conditions. Condition 1 corresponds to a conventional "lecture-demonstration-practice" in which conceptual knowledge is presented by the computer tutor as a coherent entity prior to engagement in problem-solving activities (Lecture-Demonstration-Practice). Condition 2 reflects "embedded teaching" in which before students begin practice, the computer tutor uses examples to demonstrate problem-solving processes, introducing concepts and principles, as they become relevant (Embedded-Teaching Condition). Condition 3 is a "learner-controlled" instruction in which students engage directly in problem-solving activities without receiving any prior formal instruction, but in which they are provided with instructional assistance and demonstrations upon request (Learner-Controlled Instruction). / Twenty-seven high-school students participated in the experiment over a 1-month period. Students were divided into three groups based on their pre-test scores, each group was then assigned randomly to one of the three experimental conditions. The computer tutor was used as the sole source of instruction. Pre- and posttests were administered to measure the changes in students' algebraic abilities. A multivariate analysis of the pre- and posttest results indicates that overall student performance in all three conditions improved significantly over time, as measured by the ability to construct algebraic representations and the ability to made estimates using the various representations ( F (2, 23) = 46.6, p < 0.01). In particular, students in Lecture-Demonstration-Practice Condition demonstrated a higher level of accuracy (89.51%) than students in the Embedded-Teaching and Learner-Controlled Instruction did (61.1% and 63.3% respectively). Moreover, all students in Lecture-Demonstration-Practice Condition completed the posttest successfully, whereas only 56% of students in the other two conditions passed the posttest. / This research demonstrates that students learn more effectively from instruction that emphasizes the coherent representations of the symbol system of algebra. It is postulated that such coherent representations enable students to make sense of the subsequent examples to be studied and the problems to be solved thus leading to better problem-solving performance. This research has implications for the development of instructional theories and educational computer applications.