An APOS analysis of the understanding of vector space concepts by Zimbabwean in-service Mathematics teachers.
Mutambara, Illias Hamufari Natsai.
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University mathematics students often find the content of linear algebra difficult because of the abstract and highly theoretical nature of the subject as well as the formal logic required to carry out proofs. This study explored some specific difficulties experienced by students when negotiating the various vector space concepts. The participants were 73 Zimbabwean mathematics teachers who were enrolled in an in-service programme and who were studying for a Bachelor of Science Education Honours Degree in Mathematics. The Zimbabwean mathematics in-service teachers who were studying these concepts were also teaching some of the concepts at high school. The study was qualitative in nature and it was strengthened by the interpretivist paradigm. Data were generated from the teachers’ written responses to three tasks based on the various vector space concepts. The items in the activity sheets probed the participants on the concepts on vector space, subspace, linear combination, linear independence, basis and dimension. Follow-up interviews on the written work were conducted to identify the participants’ ways of understanding. Thirteen students volunteered to be interviewed and were probed further about the vector space concepts so as to elicit more information on the way they understood the various vector space concepts, and the connection they seemed to make between these concepts. An APOS (action–process–object–schema) theory was used to unpack the structure of the concepts. The main aim of the study was to identify the mental constructions that the students made when learning the various vector space concepts and the extent to which they concurred with a preliminary genetic decomposition.. The study also employed another theoretical framework, Sfard theory, which was used to describe the in-service teachers cognitive difficulties in the learning of linear algebra which were identified as errors and misconceptions with particular reference to the study of vector space concepts. The errors were categorised in terms of conceptual (deeply seated misunderstandings) procedural (related to using related procedures) and technical (calculation or interpretation) errors. In terms of APOS theory, the responses revealed that most in-service teachers were operating at the action and process levels, with a few students using some aspects of object level reasoning for some of the questions. Findings revealed that the teachers struggled with the vector space and subspace concepts, mainly because of prior non-encapsulation of prerequisite concepts of sets and binary operations, and difficulties with understanding the role of counter-examples in showing that a set is not a vector subspace. Most of the students operated at the action level of understanding. The findings revealed that across the items on the concepts on linear combinations, linear independence, basis and dimension, students were comfortable in answering problems that required the use of algorithms, for example carrying out the Gaussian elimination method. However a major hurdle that hindered them from interiorising the actions into a process for the items on linear combination, linear independence and basis was their failure to interpret the solutions to the systems of equations and providing insufficient argumentation in relation to the posed questions. Fifty students struggled with concepts on linear combination and did not provide any evidence in their written responses of moving past an action conception.The results on understanding linear independence revealed that 17 (23%) students were able to make arguments based on the use of theorems that given vectors are linearly dependent without showing the step by step procedures and giving precise descriptions of the procedures used to determine linear independence. There were 46 students who represented their understanding in a manner described as the action conception as they were engaged in a step by step manner in an attempt to show that given vectors are linearly independent. The major drawback that hindered the students to develop their understanding of the concept of linear independence was a failure to distinguish the two terms linear independence/dependence, application of inappropriate theorems and inappropriate methods when solving the problems. Furthermore, the results on understanding of basis and dimension also revealed that the in-service teachers were able to cope with the procedures of row reduction, but struggled to justify whether given vectors formed a basis or not; they also struggled to find the basis of the solution space. Only 9 (12%) of the students were able to develop their mental construction at the process conception of basis of a vector space as they were able to coordinate the two processes of establishing that a given set span the particular vector space and that the set is linearly independent. On cognitive challenges, the study revealed the distribution pattern of the conceptual errors, technical errors and procedural errors varied across the items. The most errors manifested were the conceptual and technical. It is hoped that the identification of such errors and misconceptions will assist other educators in modifying their planning so that long term learning will take place.