Research On The Teaching Of Plane Vector Concept In High School Based On APOS Theory

Mathematical concepts are the cornerstone of mathematics and students are better placed to grasp the essence of mathematical content based on a thorough understanding of mathematical concepts.The APOS theory is one of the few concept teaching theories established for the characteristics of mathematics subjects,which is highly operable to guide the teaching of mathematical concepts.The APOS theory can make up for the deficiencies in teaching concepts related to plane vectors in the traditional teaching model,to provide some rationalization suggestions for teachers' teaching of plane vectors.The main research questions are:(1)How well do high school students understand and master the concepts of plane vectors based on APOS theory?(2)What are the common errors in high school student's understanding of the concept of plane vectors? What are the reasons?(3)What are the proposed strategies for teaching the concept of plane vectors in the context of APOS theory?To study the above problems,firstly,the literature analysis method was used to organize the literature on "APOS theory" and "teaching of plane vector concepts",to clarify the research ideas,to sort out the key points of the research,and to lay the foundation for this study.In addition,based on the analysis of related literature,we compiled a "Table of the content division of plane vector concepts based on APOS theory".Secondly,we used the test paper method to test some sophomore students in high school X.We used the statistical analysis method to compile the test results and the interview method to interview the first-line high school mathematics teachers to obtain the following status:(1)on the whole,as the difficulty of knowledge increases and the number of contents increases,the student's understanding of the concept of plane vectors shows a decreasing trend;(2)from the APOS four stages,the percentage of students' attainment in the four stages decreases step by step,and there is a correlation between students' understanding levels in adjacent stages;(3)in terms of the three representations of plane vectors,students' understanding of geometric representations is the worst.Through collation and analysis,the common learning problems are:(1)students do not have a thorough understanding of the nature of vectors;(2)vector-related concepts such as zero vectors,co-linear equivalence,equal vectors,and opposite vectors are not identified;(3)the transformation between different representations of plane vectors is not flexible enough;(4)students have some difficulties in applying vectors to solve problems.The causes of the learning problems are:(1)students tend to ignore the background of vectors in the learning process;(2)teachers do not give enough time for students to abstract and form concepts independently in the teaching process;(3)there is a lack of problem investigation and experience in the teaching process;(4)the application of analogical ideas is not sufficient,and there is a lack of sorting and analysis of related concepts;(5)the teaching of transformation(6)the ability to make diagrams independently according to the meaning of the problem is weak;(7)the ability to transfer knowledge is lacking;(8)the knowledge network of plane vectors cannot be constructed effectively.Based on the above research,the four-stage teaching strategy of APOS theory is proposed:(1)Action stage: highlight the physical background,reflecting intuitiveness and acceptability;give students time for an independent investigation and experience the abstraction process of the concept.(2)Process stage: focus on the penetration of analogical ideas;focus on mining the connotation of concepts and algorithms.(3)Object stage: emphasize the transformation between different representations of vectors,and penetrate the idea of transformation;pay attention to geometric graphing,and penetrate the idea of combining numbers and shapes.(4)Scheme stage: guide timely reflection,emphasize the construction of knowledge network;enhance students' awareness of applying vectors.Finally,we use the case study method to integrate these strategies into the lesson of "The Concept of Plane Vector",and present it in the form of a case study to further illustrate the guiding and effectiveness of APOS theory on teaching the concept of a plane vector.