Teaching systems of linear equations in Sweden and China: What is made possible to learn?

Abstract

A starting point for this study was an aim to understand better the relationbetween teaching and learning of mathematics. This interest was based onthe assumption that what is possible for students to learn about mathematicsmust be related to how they experience the mathematical content, which inturn must be related to how the content is handled during the mathematicslesson. Many factors that may have an impact on the teaching and learning ofmathematics are beyond the influence of teachers. However, the handling ofthe content what examples to use, what aspects of concepts and methods toemphasise, what exercises students should work on etc is something aboutwhich mathematics teachers must always make decisions. An intention withthis study was to produce results that could inform practice on the classroomlevel, as well as teacher education, in this respect. Another intention was tocontribute to the development of methods for analysing teaching that aresensitive to, and focus on, the specific content of instruction.Sixteen lessons from six classes in Sweden and China video recordedwithin the Learners Perspective Study were analysed and compared. Theanalysis was based on Variation theory and had its focus on differences in howthe same mathematical content was taught. The concept object of learning was used to denote what teachers try to teach and what students are supposedto learn. In this study the three objects of learning analysed were related tothe mathematics taught in the six classrooms systems of linear equations intwo unknowns, solutions to systems of linear equations in two unknowns and themethod of substitution. The analysis was done from the perspective of a studentin the classroom and aimed at describing what aspects of the objects of learningwere made possible to learn. An aspect was considered made possible toexperience if the corresponding dimension of variation was opened. The alternativewas that the aspect in question was taken for granted during teachingand kept invariant.The analytical approach employed made it possible to detect even subtle differencesin how the teachers handled the content. The teachers thereby madedifferent things possible to learn for the students. The description of these differencespoints out several aspects that could be so familiar to many teachersthat these aspects face the risk of being taken for granted in teaching. Further,findings regarding the use of systematic and deliberate variation in the Chineseclassrooms are discussed.