A Study On The Vector Understanding Level Of The Senior High School Sophomores In Shanghai

By the dual features of "graphic" and "numeric", vector has an extensive connection with other knowledge, thus it develops as a mainline throughout many sections of middle school mathematics, and it has a unique status in the middle school mathematics. The main work of this paper is:First of all, through modifying Dubinsky’s "APOS Theory" and Tall’s "Facets and Layers of Function Concept" properly, we construct the model of "Facets and Layers of Vector Understanding"(see Figl), this targeted theory model can be used to study students’ understanding of vector comprehensively and deeply.First of all, through modifying Dubinsky’s "APOS Theory" and Tall’s "Facets and Layers of Function Concept" properly, we construct the model of "Facets and Layers of Vector Understanding"(see Figl), this targeted theory model can be used to study students’ understanding of vector comprehensively and deeply.Then, according to the model, we focus on four progressive levels:action level, process level, object level, schema level, and three different representations:graphic, symbolic, coordinate to establish our test. Based on the overall analysis of the test results, the performance of 162 senior high school sophomores in Shanghai are as follows:1. The results indicate a decreasing trend on the level of all the senior high school sophomores’ vector understanding, this also proves that the model we set up in the paper is hierarchical and rational.We can conclude from the test results, in the first three layers, on algebraic, graphic, coordinate representations, about 80%-90% reached action level, about 60%-70% reached process level, about 30%-50% reached object level; moreover, students’ understanding of the coordinate representation is the best, and the decrease in the understanding level of algebraic is the most obvious. In the last layer of vector understanding--the schema level, only 35% of the students have the consciousness of applying the vector method to solve relevant problems, and half of these students have just established initial links between the test question and vector method, yet they can’t complete the task of integrating all the knowledge and transforming flexibility between different representations in the following steps; furthermore, only a few (12%) of the total appear to master systematic and structured vector knowledge,2. Different types of schools have different performance in this vector test, no matter in the low or the high level questions, the performance of the key school are better than the ordinary one.That is to say, in the first three layers, the understanding of the key school in each representation is better than the ordinary school; this difference is much more apparent in coordinate representation; then in the last layer, either in the comprehensive use of different representations or the construction of vector knowledge system, the key one still shows some superiority over the ordinary one.Finally, combined with the case study and interview, further analysis has been made on the students’ performance in some specific test questions, we summed up some of the students’ typical performance, common difficulties and interpretations, and this can bring several vector teaching enlightenment to us.