Improving mathematics teaching and learning through generating and solving algebra problems

[摘要] ENGLISH ABSTRACT: In many countries, due to a growing criticism of the inadequacy of mathematicscurricula, reforms have been undertaken across the world for meeting new social andtechnological needs and many researchers have begun to pay attention to the waymathematics is learned and taught. In the same vein, this study aims to investigateinnovative and appropriate teaching strategies to introduce in the Rwandaneducational system in order to foster students' mathematical thinking and problemsolving skills. For this, a classroom-based research experiment was undertaken,focusing on meticulous observation, description and critical analysis of mathematicsteaching and learning situations.In the preparation of the research experiment, three mathematics teachers were helpedto acquire proficiency in doing mathematics and to refine their teaching strategies, aswell 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 to16 years, chosen from three different secondary schools in Rwanda, participated inthis research experiment. Students were taught an experimental programme based onsolving contextualised algebra problems in line with the constructivist approachtowards mathematics teaching and learning. Twenty-four mathematics lessons wereobserved in the three classes and students' learning activities were systematicallyrecorded, focusing on teacher-students and student-student interaction.The participating teachers experienced many difficulties in implementing newteaching strategies based on a problem solving approach but were impressed andencouraged by their students' abilities to generate different and unexpected ways ofsolving problem situations. However, the construction of mathematical models ofnon-routine problems constituted the most difficult task for many students because itrequired a high level of abstraction, characterising algebraic reasoning. Despiteevident cognitive obstacles, a substantial improvement in students' systematicreasoning with respect to the different steps in the problem solving process, namelyformulating a mathematical model, solving a model, verifying the solution andinterpreting the answer, was progressively observed during the experiment. Manystudents had to overcome a language problem, which inhibited their understandingand interpretation of mathematical problem situations and deeply affected their activeparticipation in classroom discussions.In this study, small group work and group discussions gave rise to excellent andsuccessful teaching and learning situations which were appreciated and continuouslyimproved up by the teachers. They provided students with opportunities for learningto argue about their mathematical thinking and to communicate mathematically. Thiskind of classroom organisation created an ideal learning environment for students butan uncomfortable teaching situation for teachers. It required much effort from theteachers to transform the mathematics classroom into a forum of discussion in settingup stimulating and challenging tasks for students, in working efficiently with differentgroups and in moderating the whole class discussion.It was unrealistic to expect spectacular changes in teaching practices established overyears to take place during a period of a month. This type of change requires sufficienttime and support. However, teachers did develop a new and practical vision ofmathematics teaching strategies focusing on students' full engagement in exploringand grappling with problematic situations in order to solve problems. Teachers maderemarkable efforts in internalising and adopting their new role of mediators ofstudents' mathematics learning and in being more flexible in their teaching styles.They learned to communicate with their students, to accept students' explanations andsuggestions, to encourage their logical disagreement and to consider their errors andmisconceptions constructively.Students' results in the pre-test and the post-test showed their low performance inbuilding mathematical models especially when they had to use symbols but revealed asignificant progress in the students' ways of thinking which was observed through thevariety and originality of their strategies, their systematic work and their perseverancein solving algebra problems. Students also developed positive attitudes to domathematics; this was exhibited by their pride and satisfaction to accomplish nonroutinetasks by themselves.Teachers' comments indicated that they work under pressure to cover an overloadedmathematics 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 inmathematics unfortunately implies to fail at school and in life. Students' negativeattitudes towards mathematics were mainly due to their repeated failures inmathematics, but also to some mathematics teachers who intimidate and discouragetheir students.Both educational authorities and teachers should make efforts to rethink anappropriate mathematics curriculum and alternative teaching strategies in order toefficiently prepare students to meet new societal and technological requirements.

[发布机构] Stellenbosch University