Student understanding of span, linear independence, and basis in an elementary linear algebra class

The goal of this study was to investigate student conceptual understanding of basic concepts introduced in a first linear algebra course. Three students taking an elementary linear algebra class for the first time were selected to participate in the study. They were interviewed four times throughout the semester. The interviews were semistructured; they were transcribed and coded to be analyzed.;Based on data analysis, students did not memorize or use definitions when trying to understand a concept or when they encounter a problem. Students created their concept images based on misconceptions which helped them get through the class and do well. After investigating students' concept image of these three ideas, what follows is a list of possible reasons for the confusion: (a) students do not understand the definitions of these concepts, (b) students do not understand the importance of definitions, (c) students rely on their intuition to cope with abstract objects, (d) students are not aware of contradictions in generalizations, and (e) familiar terms replace formal definitions. Some of the misconceptions created by students when dealing with new or abstract concepts cannot be avoided, but we as teachers have the responsibility to know where the difficulties may arise and prepare tasks to help students develop an effective concept image. The creation of such tasks should keep in mind the need to develop an intuitive base for abstract concepts that agrees with the formal mathematical definition of the concepts. Possible suggestions for improving student concept images will be discussed.