A Study On The Teaching Of Some Difficult Concepts Of Junior Mathematics Based On PCK

Mathematical concept is considered as a signiifcant component of the basicknowledge of mathematics. Teachers and students are plagued by the difficultpoints of mathematical concept. Previous research about the difficult points of themathematics textbooks in the middle school published by Shanghai EducationPress is mainly centered on some certain specific difficult points of themathematics, which lacks a systematic research about the dififcult points ofmathematics covering all the grades in the middle school. Although these studiescan also provide the teachers with specific teaching design for some certaindifficult points of the mathematics, most of them can not be migrated tobreakthrough other difficult concepts. Therefore, teachers are still confused in theface of numerous other difficult concepts. In this paper, with "finding out thedifficult mathematic concepts in the middle school which are accepted by theteachers and students，studying the teaching process of diiffcult concepts, andputting forward a set of methods to solve the difficult concept"s as the framework,a research will be made on two issues: what are the current difficult mathematicalconcepts in the middle school mathematics textbooks published by ShanghaiEducation Press, and what is the general strategy to breakthrough these difficultconcepts.In order to find out the difficult concepts in the middle school mathematicstextbooks, at first, the researcher makes a math concept table for each semesterbased on all the mathematics concepts (298) in the eight mathematics textbookspublished by Shanghai Education Press. Then, the researcher selects60difficultconcepts rfom the298concepts, and makes a difficult concept table for theselected difficult concepts. Then, a questionnaire is made to do investigationamong the teachers and students so as to make clear what are the difficultconcepts considered by the teachers and students and what are the causes whichmake them difficult. The study found that secondary function is accredited by theteachers and students as the difficult concept in the middle school mathematics.The identification level about the difficult concepts in the middle schoolmathematics among different types of teachers is very high, so is among the boy and girl students. The difficult mathematics concepts among the teachers andstudents have certain relevance. However, there is also difference among theteachers and students. The reasons that these concepts become difficult ones lie inthe concepts themselves, teachers themselves, students themselves and so on.Centering on the difficult concepts and their causes obtained rfom the survey,the researcher referred to a large number of literatures, listened to a lot of mathlessons, and interviewed some mathematics teachers so as to ifnd out a set ofmethods to breakthrough the difficult concepts of mathematics. Based on theabove work, the researcher mainly studied the teaching process of difficultmathematics concepts and found a general method to breakthrough difficultconcepts. Firstly，at the stage of concept introduction，it can be done by theprovision of real prototype, actual problems and the method of mathematicshistory. Secondly, it is the understanding stage of concepts, which reveals theconnotation of the concepts from multiple perspectives so as to grasp accuratelythe extension of the concepts. At last，it is the summary stage of the concepts，establishing the vertical integration and horizontal connection between theconcepts and forming a ‘‘tre”e structure and “network” structure of the concepts.Finally，the teaching suggestions are as follows. It is suggested that theteachers should reveal the connotation of the concepts from multiple perspectiveswhen the students learn the concepts the ifrst time. During the application stageof the concepts, it is suggested that the teachers should establish an all-roundrelationship between the difficult concepts and other concepts. During the stageof deepening the understanding of concepts, it is recommended that the focus011the use of perception should be transferred to the use of thinking.