Research On Instruction Of Irrational Number Lesson From The Perspective Of HPM:A Case Study

Irrational number is an important content of expanding rational number field to real number field,which promotes the first mathematical revolution and plays an important role in junior high school mathematics.However,the existing research shows that students will encounter a lot of cognitive obstacles in learning,such as students know the formal definition of irrational number,but do not know the nature of irrational number;can not understand the objective existence of irrational number;do not know the actual value of irrational number.In view of the above considerations,this study aims to further clarify the cognitive barriers of students in the learning of irrational numbers,adopt the perspective of HPM,integrate the history of mathematics into the teaching of irrational numbers,organically combine the content of history of mathematics with the teaching content,and give full play to the educational value of history of mathematics.Let students experience the discovery process of irrational number,experience its objective existence,and experience the hard process of the development of irrational number,in order to try to solve the cognitive obstacles that students are prone to appear,and open a new perspective for teachers' education and teaching.This paper studies the following three problems:1.Is the organic combination of mathematics history and mathematics education helpful for students to understand the teaching content of this lesson and better overcome cognitive obstacles?2.What effect does the integration of mathematical history into the teaching of irrational numbers have on students' emotional attitude towards mathematics learning(how acceptable is the level of students)?3.What historical materials related to irrational numbers are suitable for integrating into mathematics class?Aiming at the research problems,this study develops HPM teaching cases,which have experienced two discussions and reflections,two modifications and designs,and finally carried out three teaching sessions in three parallel classes.In the process of action research,data analysis was conducted according to classroom records,student interviews and questionnaires,and the following research results were finally obtained:1.Integrating the history of mathematics into mathematics teaching can really explore the historical process,reveal the essence of mathematics,promote students' understanding of the essence of mathematics,and help students overcome some cognitive obstacles2.Students have a high degree of recognition for the history of mathematics.The content of the history of mathematics can increase the fun of learning and help students overcome the difficulties in learning3.Because students are willing to accept it,there are various historical materials of mathematics that can be integrated in this class.For example,the story of Hippasos and Pythagoras,the origin of the name of irrational number,the origin of mathematical symbol-root sign,etc.can make the mathematics class full of humanity and sense of history,and help students understand the concept of irrational numberTo sum up,it is of educational value to integrate the history of mathematics into the teaching of irrational numbers,which can promote the long-term development of students and the improvement of teachers' professional level.