Research On Space Vector Teaching Based On APOS Theory

High school mathematics curriculum focuses on improving the core literacy of mathematics,including mathematical abstraction,logical reasoning,mathematical modeling,intuitive imagination,mathematical operation and data analysis.The teaching of space vector in high school contains the core literacy of intuitive imagination,and attaches importance to cultivating students’ awareness of solving solid geometry by using space vector,training students’ intuitive imagination thinking and feeling the value of mathematics.The purpose of this paper is to analyze the space vector of senior high school mathematics by using the methods of literature research,questionnaire survey and interview under the guidance of APOS theory,and to investigate the current situation of space vector teaching for senior high school mathematics teachers and students,analyze the possible problems in their teaching and put forward countermeasures.The specific research process is as follows:First of all,through the survey results of students and teachers,it is found that the problems existing in space vector teaching are:(1)The concept teaching of space vector lacks the creation of difficult situations;(2)Students don’t fully understand the concept and operation of space vector;(3)Students’ orientation of space vector is vague;(4)Students are not clear about the framework of space vector knowledge.Secondly,according to the problems found in the investigation,the four-stage teaching strategy based on APOS theory is put forward:(1)Operation stage: whether teachers create difficult situations directly affects students’ interest in space vector learning,so teachers should pay attention to selecting problem situations: first,they should be related to life,reflecting the application of space vectors in life;Secondly,the situation created by teachers should focus on the level of students’ cognitive development and should not exceed the outline;Finally,let students feel the space vector intuitively through this activity,and know and experience the help brought by space vector learning.(2)Process stage: At this stage,teachers should help students abstract the essence of space vectors from the life background and physical background.Therefore,teachers should pay attention to: first,guide students to think actively,and derive space vectors from plane vectors,so that students can compare the similarities and differences between plane vectors and space vectors themselves;Secondly,let the students study cooperatively,communicate with each other,and learn space vectors from multiple angles and perspectives;Finally,teachers should link the past with the future in the teaching process,not only to let students know why they learn space vectors,but also to let students know how learning space vectors can help their later study.(3)Object stage: After students acquire the essence of a mathematical object,they often only understand it literally,and they are not clear about what role it implies for space vectors and what problems it can solve.Therefore,teachers should pay attention to: first,teachers should demonstrate specific cases,and then guide students to imitate training to deepen their understanding of space vectors;Secondly,in vector operation,it is necessary to grasp the operation law clearly,choose typical examples to explain,and let students imitate training.After this process,students will have a clear direction for related operations.Finally,variant training is set up to help students fully understand the space vector.Through variant training,students can understand the space vector from different angles.In addition,teachers need to make clear the orientation of space vector for students.It is not only the most important mathematical tool to deal with solid geometry,but also the best method to solve other mathematical problems.For example,it is the easiest to deduce cosine theorem by vector method.(4)Schema stage: At this stage,teachers need to organize students to summarize in class,and help students consolidate their knowledge by listing knowledge frame diagrams.A good schema will affect students’ overall grasp of space vectors,and students will gradually apply space vectors to solve mathematical and practical problems in the learning process.Finally,based on the APOS theory,the teaching design of "concept" and "quantity product operation" in space vectors is carried out.Through teaching practice,we can draw the following reflections:(1)Operation stage: creating difficult situations;(2)Process stage: strengthening student participation;(3)Object stage:deepening conceptual understanding;(4)Schema stage: building a knowledge framework.