Improving mathematics teaching and learning through generating and solving algebra problems

Mudaheranwa, G. (2002-12)

Thesis (MEd)--Stellenbosch University, 2002.

Thesis

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.

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