Studying On Vector Curriculum And Teaching In Middle School

Vector is the important sections of senior high school mathematical curriculum. Vector has its operational method and system unique to number, and helps to develop students' idea about "number, quantity and operation". Vector geometry provides a new method for studying space and figures, and makes for appreciating mechanized thought of modern mathematics. Vector as one of basic modern mathematical concepts can make teachers and students annotate much elementary mathematical knowledge from a new angle and establish the foundation for learning linear algebra theories.In recent years, many researches in the domain of mathematical education are more relevant to the current worldwide introduction of graphics calculators and computer technology into mathematics curricula and depict how the technology can influence students' conceptual advancement and allow new problem-solving approaches. The concept of vector is a multidimensional one and concerns geometrical graphic and algebraic symbolic representations. So it may cause many problems in learning as well as in its comprehension. Moreover, since vector is introduced the senior high school mathematical curriculum, it has come to teachers and students' attention and interest. But in our country there are few demonstrating researches on many important and basal questions existed in vector teaching and learning. For example, "comprehending the concept of plane vector", "describing the perpendicular or parallel relation between a line and a line, a line and a plane, or a plane and a plane with vector language, proving some theorems on positional relations between a line and a plane by vector method", "feeling the function of vector method which is used to solve geometric problems" are all emphasized in general senior high school mathematical curriculum standard. Therefore, we try to exhibit the actuality and questions in vector teaching and learning through four basic investigative problems.Firstly, we are based on the mathematical cognitive theories- SOLO, APOS and use questionnaire and interview to investigate the understanding level of the concept of vector (graphic and symbolic) by 302 second grade senior high school students in Shi Jia-zhuang. We find that for questions on the conception of vector, about 22% students reach the graphic level 4, about 3% students reach the symbolic level 4, only about 2% students reach level 4 in graphic and symbolic aspects. So, most students don't construct the concept of free vector. Results also show that students more tend to deal questions with graphic representation, and regarding vector as a line or a number (or a letter) is the main error type.Secondly, we want to examine the effect of solid geometry teaching which introduce space vector to deal with some questions about metric and positional relations. We used a test of the ability to solve solid geometry problems concerning metric and positional relations, a questionnaire to elicit students' opinions about the characteristic of synthetic method and vector method. Through an analysis of written solutions to solid geometry exercises by 368 third grade senior high school students in Shi Jia-zhuang, we find that there are about 76%students who use synthetic method and vector method simultaneously, about 14% students who only use synthetic method, about 10%students who only use vector method. But the grades of V students are higher than E students'. We also analyze two error types-general error and vector error. For E and EV solver, the error- Logically invalid inference is most. Technical error, misused data, distorted theorem or definitions are in turn. For vector method of V and EV solver, misinterpreted language is most error type. For errors related to the concept of vector, errors in using coordinates are most. The error of module of vector, errors in vector addition and subtraction, errors in dot product are in turn.For the characteristic of two methods, our study indicate that a relatively high percentage of students support that 'a basic method to deal with solid geometry' is the most virtue of synthetic method. 'Don't find assistant line and the angle or line requested in the question is the most disadvantage. For vector method, the most virtue is succinctness, 'computation is much and easy to make mistake' is the most disadvantage.Thirdly, we investigate teachers' mathematical knowledge of vector based on Piaget's theory of schemes from the aspect of connectivity through an analysis of data collected from 23 high school teachers by a questionnaire and interviews. The results show that most teachers don't consider the concept of vector and its transformation with a systemic angle. For example, they can judge whether the linear equations has solutions or not, but they don't show the structure of solution space. They think complex number and matrix are similar to vector, but they can't see the relations among the three concepts.Fourthly, we investigate 1069 senior high school teachers in shanghai, Shenyang, Shi Jia-zhuang through questionnaire. Our research questions are teachers' attitude about vector status in senior high school mathematical curriculum, developing approaches of teachers' PCK, and so on. We find that reducing solid geometry and diversification of problem-solving method are many teachers' basic view about vector' status in senior high school mathematical curriculum.Lastly, through integrating and reflecting on results from four studies, we conclude that: (1) text and teaching didn't lay a strong emphasis on the study of the concept of free vector; (2) Solid geometry teaching with vector method and synthetic method didn't cause the students' thinking conflict, but solid geometry teaching and learning is still under the influence of Euclidean geometry; (3) most teachers' mathematical knowledge about vector is mainly composed of procedural experiences;(4) it is the way that accumulating and reflecting their teaching experiences, or sharing other colleagues' experiences block the teachers' pedagogical content knowledge advance. Hereby, we suggest that: (1) the concept of free vector should appear in the text. (2) The definition of vector in text should be improved. (3) Teacher should show the abstractive progress of the concept of vector from physical contexts to mathematics. (4) Vector coordinate operation should be given more considerations.(5) Exploring and extending a new model of developing teacher' knowledge are imperative under the situation.