An APOS study on pre-service teachers’ understanding of injections and surjections

摘要

Highlights?This constitute an innovative contribution in mathematics education since it is the first APOS study focusing on understanding of injections and surjections.?The study illustrated the importance of a suitable Schema for set theory, without which the representation of the domain and codomain may be inaccessible.?A prerequisite conception of function as an Object was also shown to be essential to start the construction of injections, surjections and bijections.?It is shown that the construction of a Schema of mathematical logic is necessary to interpret and understand the definition of an injection and surjection.?By offering insights into students’ difficulties and their relation to the lack of construction of necessary mental structures, it contributes to pedagogical content knowledge development in Mathematics teacher education courses.AbstractThis study was undertaken to explore pre-service teachers’ understanding of injections and surjections. There were 54 pre-service teachers specialising in the teaching of Mathematics in Grades 10–12 curriculum who participated in the project. The concepts were covered as part of a real analysis course at a South African university. Questionnaires based on an initial genetic decomposition of the concepts of surjective and injective functions were administered to the 54 participants. Their written responses, which were used to identify the mental constructions of these concepts, were analysed using an APOS (action-process-object-schema) framework and five interviews were carried out. The findings indicated that most participants constructed only Action conceptions of bijection and none demonstrated the construction of an Object conception of this concept. Difficulties in understanding can be related to students’ lack of construction of the concepts of functions and sets that are a prerequisite to working with bijections.]]>