Study On The Derivative Understanding Level And The Development Rule Of Students

Mathematical understanding has become another central topic the mathematical educators concerned about after "problem solving" in the world, and it's also become the concern that international education research and practice is focused on. As a field of study and practice direction, mathematical understanding of learning and teaching has recently attracted public attention in the field of education ,and also has achieved much results. Derivatives has become the courses of high school around the world. Most schools have it as an elective curriculum in high school, and this has become a global direction. However, the rule characteristic of students'understanding , derivative's curriculum localization, the course content selection, the teaching strategies determination and how to train the high level inferior mathematics thoughts, those problems have attracted the domestic and foreign mathematics educator's attention. Comprehensive survey domestic and foreign to derivative understanding correlation research, mostly all is the heavy speculation but the light real diagnosis, the real diagnosis research lag develops to the curriculum relatively. The real diagnosis research on derivative's learning and teaching is also necessary at the right moment..This paper uses the methods of the quantifics research and the qualitics research to study derivative's teaching and learning. The main research work is following: established the measurement tool of the derivative understanding level; using the SOLO taxonomy evalue the derivative understanding, constructed mathematics understanding level model; analyzed the students'derivative understanding level and its development rule by the statistical method, inquired about the students'derivative understanding characteristics; also analyzed the influence of the teacher variable to the students'derivative learning. Through our country's actual situation, this paper promulgates the derivative learning and teaching rule, and tries to raise dual supports to Chinese educational reform and mathematical theory & practice.The main results of this paper are following:â… ,The students'derivative understanding RegularityFor the complex computation of derivative, derivative's formalized concept understanding, derivative's application ,and graph operation, and etc. ,there are remarkable differences between university freshman and high school three grades'students. But university student's understanding level is higher. For the average rate of change, instantaneous rate of change, derivative's simple computation, derivative's physics significance and geometry significance, and also using the derivative to study simple function characteristics, there are no remarkable differences between university freshman and high school three grades'students. The ability that the students use derivative to carry on the question solution strengthen along with the grade increases. There are remarkable differences between university four grades and university freshman only on explaining the physics significance and simple optimization questions. The students'derivative understanding has certain imbalance in the overall. Generally speaking, Students'derivative understanding level doesn't necessarily increase along with age growth. Three grades students'understanding level truly increases progressively on operateing skill and modelling ability.The research discovered that, it's rational that the derivative curriculum should be established in the high school. The derivative concept teaching should be direct-viewing primarily not formalized concept teaching in the high school. The instantaneous velocity is the base that the students understand the concept empirical, and it's the effective " platform" of implementation derivative concept . It's suitable to use the physical background teaching derivative in the high school, but does not suit through the formalized limit and the continual concept carrying on derivative teaching.â…¡,Main characteristics of student's derivative understanding1st , Main characteristics of the high school three grades'students'derivative understanding : Students'derivative understanding level of developing is certainly imbalanced in various dimensions; derivative understanding level also stays on the direct-viewing derivative stage, and also has a certain disparity to the formalized abstract stage; there exists differences between students'understanding and explaining derivative concept ability , there also exists certain difficulty in derivative explanation ; the ability of the students actually utilizing derivative is imbalanced, the ability of students inducing the inference is weak, the students have certain difficulty in transition from concrete to the abstract .2nd,The sex, the question backgrounds, the attribute ways affect the student on derivative understanding; from the angle of mathematics ability , there exists imbalance in the different duty performance when the students use derivative carrying question solution.3rd,Characteristics of student's cognition development: three grades'students'thought transition characteristics is obvious in the high school, their cognition development stages at dual stages,namely from concrete operation to form operation transitional stage and truly grasping the form operation stage. The three grades students are in the developing period of form operation in the high school, the university freshmen are in the mature period of form operation; their dialectical logical thinking is in the transition time, displays "the leap" and "the polarization", they are in the development essential time; and the university freshmen tends to the stereotype, they are in the mature periods. Obviously, the content choice of derivative curriculum doesn't suit to be divided by the breadth of the knowledge in the high school, we suggest to divide the content according to students'thought development characteristic and the cognition rule.â…¢,The students'intelligence construction and the factors affecting the teacher to the derivative understanding1st, formation of derivative concept along two principal routes: one is the concept compounds of the average rate of change, the instantaneous rate of change, increasing compared, limit and so on, and summarizes abstractly the derivative formalized definition; forming "latent concept", abstracts the concept essential of derivative attributing from the physical background. Through the creative activity of concrete goal, carry on the derivative concept's "mathematics" construction by question solving to form the derivative concept through the visual attribute and the movement attribute.2nd, two kinds of teachers do not have visible differences on the knowledge of derivative understanding, But they have certain problems on the essence understanding of tangential line and the operation explanation. Teachers'of understanding to derivative affect the students'understanding level of derivative, but this has not achieved the remarkable level.