Teaching The Volume Of Sphere From The Perspective Of HPM

Sphere volume is the traditional content of high school solid geometry.Mr.Liu Dun once said: “In ancient times,the degree of understanding of the knowledge of the circle can be used as a measure of a nation's level of mathematics.”(Liu Dun,1985)Similarly,the awareness of sphere volume is also a reflection of national mathematics level.The accurate calculation of the ball volume needs to be based on calculus.Therefore,on the one hand,the correct formulation of the volume formula of the ball is a very difficult problem before the calculus is created;on the other hand,the correct drawing of the formula of sphere volume is a precious opportunity for the idea of creating calculus.There is a abundant of mathematical ideas and methods in the process of exploring the volume formula of the sphere: mathematicians such as Archimedes and Liu Hui used Mathematical thinking methods such as inductive reasoning and analogical reasoning in their exploration of the accurate formula of the sphere volume.The exploration of the formula of the sphere volume also contains rich humanistic spirit,and the spirit of scientific truth: the mathematicians of ancient Greece,China,India,Japan and so on have been defeated without exception in the process of calculating the formula of the boll volume.However,it has also been repeatedly defeated.The sphere volume calculation results have been developed under the guidance of different method systems.The sphere volume formula from the perspective of multiculturalism makes the students' thinking shine.In the feedback after class,some students use "mathematical thinking is really wonderful" and "mathematical thinking is really overwhelming" to describe their appreciation for the diversity of the spherical volume method.According to the author's research survey,86.2% of students are eager to understand the origin of the sphere volume formula.However,the Shanghai textbook only uses three lines of text to state the results of the ball volume formula;while other versions of the math textbooks have some supplementary content in the reading materials,in general,their statements about the sphere volume are also relatively simple.The 2017 edition of the Ordinary High School Curriculum Standard states that mathematics plays an irreplaceable role in shaping people's rational thinking.Therefore,the three-line ball volume formula teaching method is obviously different from the cultivation of students' rational spirit.In order to implement the concept of curriculum standards,high school students can understand the calculus ideas contained in it and lay the foundation for further mathematics learning.Therefore,this paper conducts an action research on the sphere volume formula from the perspective of HPM.First of all,we made a deep study of the early western sphere volume related textbooks before twenty-first century to outline the historical context of the calculation method of the sphere volume formula.Secondly,based on the integration of the history of mathematics into the theory of mathematics teaching,the corresponding teaching scheme of the sphere volume method is designed and two rounds of teaching practice are carried out on the teaching plan,and the corresponding teaching adjustment is carried out according to the feedback from the classroom practice.Finally,the teaching suggestions and textbook compilation suggestions for sphere volume correlation are given.This study has developed four lesson examples: the sphere volume formula analogy teaching method from the perspective of HPM,the sphere volume formula of Zu Geng principle teaching method from the HPM perspective,the sphere volume formula slice teaching method from the HPM perspective,the expansion teaching of the sphere volume formula under the HPM perspective.This paper discusses the following three issues:1.What do students think about the teaching of mathematics history into sphere volume formula? Does the sphere volume formula teaching from the perspective of mathematics history have a positive impact on the students' attitudes and values?2.Will the integration of mathematics history into sphere volume teaching help students overcome the original cognitive barriers?3.Which historical materials related to the sphere volume fit into the mathematics classroom?Through the four steps of action research planning,action,feedback,and reflection,the author obtained the following conclusions for the above three research questions:1.The sphere volume action study pretest questionnaire shows that students want to understand the mathematical history of the sphere volume.The action research shows that students are willing to understand the history of the occurrence and development of the sphere volume formula.Therefore,the sphere volume formula teaching under the perspective of mathematics history has a positive influence on the students' emotional attitude.2.In the process of sphere volume teaching from the perspective of HPM,the students explored the calculation process of the spherical volume formula;promoted the knowledge of the geometry of the curved edge;exercised analogical thinking;and fostered the basic idea of calculus.3.Since students have a diversity of thinking,the historical materials of sphere volume that can be integrated into the mathematics classroom is diverse.However,the results of the survey show that the students are concentrated on four ways of explaining the volume of ball volume: Kepler's small cones to find the sphere volume related historical data,and the calculation of the sphere counting method tocalculate the volume of the relevant historical materials,the Zu Geng principle to calculate the sphere volume related historical materials,Archimede's analogy and leverage method to find the sphere volume and other relevant historical data.In addition,students have a considerable degree of emotional inclination toward historical events,so historical materials related to mathematicians can be incorporated as appropriate.In summary,the teaching design of integrating mathematics history into the sphere volume formula is of educational significance.High school mathematics teachers can develop related teaching curriculum that integrate mathematics history into the sphere volume formula.