Teaching Research On Math Problem In High School Based On Variant Theory

Trigonometric functions is a link between geometry and algebra,a bridge between elementary and advanced mathematics,and plays an important role in high school mathematics.Its formulas are diverse and changeable,and the corresponding types of math problems are endless.To some extent,Students' learning of this part is difficult.In view of students' learning difficulties,this research,based on the variant theory,focuses on “taking trigonometric functions as a carrier,formulating theories based on the variant theory and exercises,and Transformation methods and principles that correspond to various types of trigonometric functions”.To be specific,this research studies the following three issues:(1)analysis of the current teaching situation of trigonometric function variants;(2)selection of good trigonometric function's standards and examples;(3)compilation of trigonometric function variant sets.This study helps to improve students' effectiveness in learning trigonometric functions,and has a guiding and enlightening role for teachers in formulating routine exercises.The research methods used in this thesis are the literature research method,interview method,and case study method.First,by reading literature and interviewing teachers,the process of the exercises preparation and the current situation of teaching is clarified;the student's learning situation is determined;the selective criteria and principles of the classical examples are determined.Then,comprehensively sort and analyze math problems related with trigonometric functions in the 2015-2019 college entrance examination(science),and the examples of trigonometric functions in the mathematics textbooks of the People's Education A and Beijing Normal University editions.Finally,the variant method is used to design variant problem groups which correspond to the basic problem types,to formulate instructional design of exercises,and to further modify it according to practical effects and teachers' suggestions.Finally,the final instructional design will be obtained..The conclusions of this study mainly include the following three parts:(1)Through analyzing the interviewing results of front-line teachers,this thesis finds the importance of clear example selective criteria,systematic variant methods and structures,and the basic strategy of lectures when teachers explain math problems.(2)A good math problem should include three conditions: belong to the basic problem,have variant solutions,can be further developed and generalized.The process of selecting basic math problems should combine three dimensions and two perspectives.Three dimensions include the Curriculum Standards,the characteristics of the college entrance examination,and the textbook exercises.Two perspectives consist of the qualitative and quantitative perspectives.Then,the two basic types can be obtained,which are evaluation problems and image and nature problems.The evaluation problems can be divided into three categories: the application of formulas,the conversion of angles and names,and the conversion of signs for trigonometric functions like ?cossin,cossin????.image and nature problems can be divided into four categories: the examination of properties,the transformation of figures,the problems of extreme value,and the problems of zero point.(3)The formulation of the trivial function in high school is mainly based on two methods:transformation of elements and negative hypothesis.The evaluation problems mainly use element transformation method to clearly define the major and non-major elements that constitute the original problems,determine the transformable dimensions of the main elements,and compile a series of variant problematic groups.The main method to vary image and property problems is negative hypothesis,and the element transformation method is an auxiliary method.To be specific,teachers need to consider the negative starting points in the problematic groups,define the attributes of objects before and after variation,and finally form a series of variant problems by changing attributes.(4)The instructional design of exercises should focus on students.Therefore,it's necessary to select appropriate variant problems according to students' actual situation and make sure the levels and the difficulty degree of problems are available for students.Besides,to design easily comprehensible explanation for students,teachers need to take students' possible problem-solving ideas.Finally,the teaching practice and teaching results should be arranged on this basis.