Research On The Strategies Of Improving The Mathematical Cognitive Understanding Of Trigonometric Function In Senior High School

Trigonometric function is in a key position in the senior high school mathematics knowledge system and is an important carrier for cultivating students’ core literacy in mathematics,and the content changes in the new senior high school mathematics textbook in the2019 edition are significant.Therefore,it is crucial for teachers to adjust the teaching of trigonometric function in a timely manner and for students to be able to improve their mathematical cognitive understanding of trigonometric function.There are few empirical studies on the mathematical cognitive understanding of the complete knowledge system of trigonometric function,which becomes the entry point and direction of this thesis.Firstly,this thesis identifies the research questions by analyzing a large amount of literature and opinions of front-line teachers;then,based on the Pirie-Kieren’s mathematical understanding theory,this thesis constructs a hierarchical model of understanding of each dimension of trigonometric function,and investigates the current situation,problems and reasons of senior high school students’ mathematical cognitive understanding of trigonometric function by using the literature analysis method,survey and research method and other research methods;finally,strategies and instructional designs for improving mathematical cognitive understanding of trigonometric function in senior high school are given for each stage of Pirie-Kieren’s mathematical understanding theory,and the instructional designs are used to test the effects in classroom practice.The main findings of this thesis are as follows.First,the overall level of senior high school students’ understanding of trigonometric function is at the level of "Formalizing" of Pirie-Kieren’s mathematical understanding theory,followed by the level of "Property Noticing";The understanding of trigonometric function in the three different academic levels differs significantly,except for the differences in the understanding of "Arbitrary angle" and "Radian system";The types of understanding of trigonometric function in senior high school are divided into four types: preliminary understanding,exact understanding,"constructive" understanding,and "creative" understanding.Using the K-means algorithm to cluster the test results,this thesis finds that 45.52% of the students are only at the type of exact understanding,and their level of understanding is on the low side.Second,the problems of senior high school students’ mathematical cognitive understanding of trigonometric function have been found:(1)students are easily influenced by the concept of angles learned in junior high school,and are not able to judge the signs of arbitrary angle,and have difficulties in understanding the representation of angles with the same terminal side;(2)students often confuse the "Unit circle definition method" and the "Terminal side definition method" and often misjudge the sign of the trigonometric function;(3)students have difficulties in understanding the graphical transformation of the trigonometric function,and cannot solve the comprehensive problems of the trigonometric function skillfully;(4)students have mechanical memory of the induction formula and the trigonometric constant transformation formula,and cannot use them flexibly.Third,strategies to improve the mathematical cognitive understanding of trigonometric function are proposed for each stage of Pirie-Kieren’s mathematical understanding theory:(1)at the stage of "Primitive Knowing",teachers should pose problems that break students’ cognitive balance to create cognitive conflicts and promote cognitive development;(2)at the stage of "Image Making",teachers should try to integrate distinctive and rich realistic situations or mathematical history into classroom teaching to help students accumulate valuable representations in their mind(3)at the stage of "Image Having",teachers should combine teaching with modern educational technology to prompt students to form general image;(4)at the stage of "Property Noticing",teachers should use variant teaching to promote students’ mastery of the essential properties of mathematical objects;(5)at the stage of "Formalizing",teachers should select exercises for students to master the general methods of problem solving;(6)at the stage of "Observing",teachers should guide students to reflect and summarize;(7)at the stage of "Structuring",students should construct mind maps and sort out the knowledge system;(8)at the stage of "Inventing",teachers should encourage students to make bold guesses and cultivate innovative thinking.