A Study On The Cognitive Level Of Vector Space Concepts Of Teacher-training Students From The Perspective Of APOS Theory

Vector space occupies a central and fundamental position in linear algebra,and its teaching and learning is a research focus.To investigate students’ cognitive performance in learning the concept of vector space,this study analyzes students’ cognitive performance in learning the concept of vector space in the four stages of APOS theory under the guidance of APOS theory,and points out the difficulties and reasons that exist when students construct the concept of vector space.The subjects of this research are first-year students majoring in mathematics and applied mathematics in local normal colleges.On the basis of classroom observation,a questionnaire is designed to collect data,and the collected data is quantitatively analyzed.Qualitative analysis leads to the following conclusions:（1）There are stage differences in students’ perceptions of the concept of vector space.Students’ cognitive level decreases as the learning requirements of the four stages increase,and a few students can reach the object stage and intrinsic level of the schema stage.（2）There are major problems with the transition of the stages.On the one hand,viewing binary operations as two separate processes,a misunderstanding of zero and negative elements prevents activities involving binary operations from being internalized as processes.On the other hand,it is difficult to compress the set of originally defined binary operations into a single object,or to invert the object by changing the addition operation or the number multiplication operation to go through the activities and processes again,coordinating with other processes and recompressing them into a new object.（3）The teacher excellence class had better understanding of the concept of vector space than the regular class,and the two had significant differences in the process stage,object stage,and schema stage,and not in the action stage.（4）When students learn the concept of vector space,they have difficulties in understanding abstract concepts and binary operations.The reason is that students’ cognition of the concept of vector space is susceptible to interference from old cognition,mainly from addition and quantitative multiplication operations at the primary level.