High School Level of Understanding of Inverse Function and the Law of Development Research
|School||East China Normal University|
|Keywords||High School Students Inverse Function Understanding|
"Mathematical Understanding" has become the focus of research and practice of today’s international mathematics education. Adopting a combination of both quantitative and qualitative approaches, concentrating on the evaluation of the understanding of the high school students, this article reveals their understanding level on inverse function, the development law of this understanding and its characteristics.(1) high school students’understanding level on inverse functionFrom four perspectives of the understanding of the inverse function ontology, high school students have the highest cognitive level about the relationship between the original function and the inverse function; roughly the same level about the concept of inverse function and the existence of it; significantly lower level about the application of inverse function. To be specific:Students’understanding of the definition of inverse function is at a lower level and there are significant differences between individuals:40% of the students are in the pre-structural level (P).50% of the students are in a single structure level (U) (28.3%) and multi-structure level (M) (22.5%); only 9.2% of the students reach the level of associated structures (R). As regard to the understanding of the existence of the inverse function with particular nature. Students’performances are various. Although periodic function has little effect on the understanding of the existence of the inverse function, the monotony and the parity of the function impose a direct impact on the students’understanding of the existence of the inverse function. Students’understanding of " symmetry image about a straight line" and "range domain swap" between the original function and the inverse function is almost the same, implying a good overall command. The relationship between awareness of the role of knowledge can migration of a certain place, a "consistent monotonicity", "the original function as a function of the inverse function odd function is odd," "corresponds to the reciprocal rule" and other cognitive;the overall understanding level of the relationship between the original function and the inverse function is multi-level structure (M). As regard to the application of the inverse function, the overall understanding is at a low level, with the vast majority of students’ understanding is of a single structure (U). (2)the characteristics of high school student’s understanding of inverse function and its development law.①with the higher grade, students’ understanding level of the inverse function is also deepening. Although second-year students’ performance about the form definition is poor, they surpass their first year peers in the angle and depth of inverse function, especially in the aspect of abstract thinking. They have already transferred from concrete operation to the form operation.②the analysis of gender differences on high school students’ understanding of inverse function shows that, girls surpass boys in the aspects such as the specific operations, the use of visual or graphical symbols to determine the existence of the inverse function and other related imaginal thinking; boys are good at the form operation and reasoning and other logical thinking of the inverse function.