Senior Middle School Students' Difficulties With The Space Vectors And The Corresponding Teaching Strtegies

The vector, which is incorporated in the senior middle school mathematics textbooks, provides a powerful tool for using algebraic method to solve geometric problems, the stress on which is the tendency of the reform in solid geometry in senior middle schools. There are two ways of representing a vector, namely that of a vector itself and of the coordinates, respectively corresponding to two ways of studying and solving geometric problems - the vector method and the coordinate one. Although the vector method is convenient and easy to grasp, but it is inevitable for the students to encounter difficulties and make mistakes in practice.This paper is a report of an empirical study on the errors the senior middle school students make and the difficulties they encounter in solving solid geometric problems by means of the space vectors. Through six tests conducted to 217 senior middle school students from four classes and interviews with a part of them, it is found that, due to the poor habits of problem solving, the low spatial ability, the weak calculating ability and the superficial knowledge grasping, the main errors are inappropriate establishing of coordinate systems, the incorrect coordinates of points, the failure to find the normal vectors, the unclear routine, erroneous calculations, etc. Therefore, while solving the problems in solid geometry with the space vectors, the subjects have difficulties in setting up the coordinate system, seeking the coordinates of space points and finding space angles and distances with the space vectors. Moreover, it is also difficult for the students to decide when the vector method should be chosen to solve a problem in solid geometry.Based upon the research findings, some teaching strategies are put forward corresponding to the different types of errors from both methods of solving problems and the students' ability to grasp the solid geometry.