Research On The Evaluation Of The Functional Monotonicity Understanding

Mathematical concept understanding evaluation is a hot topic of research,and the concept of functional monotonicity also plays an important role in the function system.Therefore,it is of certain value to evaluate students' understanding of the concept of functional monotonicity.This study first subdivides the secondary dimensions of Lesh and Landau's framework,which describes the evaluation of mathematical concept understanding,and then uses this framework to evaluate students' functional monotonicity understandingThis study focuses on 217 first grade high school students from an ordinary high school in city Y.Based on a paper-and-pen test,questionnaire survey and interview survey on these 217 students,this study examines students' perception,representation,connection,and application of the functional monotonicity concept,researches students' obstacles to the understanding of functional monotonicity concept and their causes.Also this study analyzes students' learning situation and the correlation between the students' learning situation and their concept understandingThe following conclusions are reached through testing,survey and interview:First of all,students' understanding of the concept of functional monotonicity is general.In terms of concept's perception,students can identify and explain some basic concepts;cite examples of monotonic function and write its monotonic intervals;express concept according to their own understanding.However,students' understanding of the word "arbitrary" in the concept is problematic.Secondly,in terms of conceptual representation,students perform better in representing monotonic function's analytic form,but some problems appear in students' monotonic function images.Thirdly,in terms of the connection of the concept,students are relatively unfamiliar with the relationship between functional monotonicity concept and other concepts,and there are understanding obstacles in this respect.Fourth,in the application of the concept,students can use the functional monotonicity concept to solve a relatively simple and familiar problem,but in other applications,students'performance is not satisfactory.Fifth,by analyzing the results of students' test papers,the author finds that students have the following four understanding obstacles to the functional monotonicity:(1)The word"arbitrary" in the concept;(2)Use the definition to prove the monotonicity of the function;(3)The connection between functional monotonicity concept and other concepts;(4)The conversion of written language into mathematical language.Through interviews,it is found that the causes of understanding obstacles' appearances are as follows:(1)The meaning of the word "arbitrary" in the concept cannot be accurately understood.(2)Students'perception of using the definition to prove the monotonicity of the function is biased.(3)Students' divergent thinking is insufficient.(4)Students' abstract thinking is insufficientFinally,students' test scores have a moderate correlation with their final exam scores.Also,students' learning methods,practice situations and extracurricular mathematical learning time have a correlation with their concept understanding.