History And Teaching Of The Ellipse

Conic sections are an important part of high school mathematics curriculum, which is also one of the most popular and difficult parts in College Entrance Examination of China. To learn conic sections, students generally start from learning the ellipse. The basic spirit of analytic geometry is to solve geometric problems via algebraic methods, which can be seen from the introduction of the ellipse definition and derivation of its equation. Today's mathematics textbooks of high school tries to introduce the ellipse in the following way:it shows the method of how to draw the ellipse by two nails, a rope and a pencil to students, which enables students to observe and try to get the definition of the ellipse, and then derive the equation of the ellipse. Although this way seems easy and logically correct, it still has its disadvantages according to genetic teaching:First, it doesn't explain to us why we should learn ellipse. Therefore, students will not have enough motivations to learn it. Second, the way to introduce the definition of the ellipse is rather unnatural because it is not based on the knowledge that students have known before. Thus, students rarely know the process how the knowledge of the ellipse is formed.This thesis studies the history of conic sections. By dating back to its history, the author finds out relationship between the origin of the ellipse and the definition in today's textbooks, and also a simple method to derive the equation of the ellipse. Meanwhile, this thesis makes a deep comparison of ways to introduce the ellipse used in 14 different textbooks in history. Then, we reconstruct the history of the ellipse according to the idea of genetic approach to teaching mathematics.Based on the reconstruction of history, we design 3 different ways to teach the ellipse and compare these ways with the 2 ways based on textbooks.What's more, one of the 3 teaching designs is put into practice and has achieved big success.A questionnaire was designed to know students'intuitive understanding of the ellipse. The purpose of this survey was to know students'understanding of the first definition of the ellipse and the way that textbooks used to introduce the ellipse, to know students'understanding of the proposition that one of the sections between a cylinder (or a cone) and a plane creates an ellipse and to know students'acceptance of the way to introduce the ellipse using Dandelin spheres.This thesis also investigated both teachers'and students'tendency of teaching methods used in the introduction of the ellipse definition and its equation derivation. The questionnaire survey was conducted to 723 high school students and 40 high school mathematics teachers; an interview was also conducted to 7 teachers after the questionnaire survey. By analyzing the survey and interview materials, we got the following conclusions.(1) Historical parallelism exists when it comes to students' intuitive understanding of the ellipse. Students usually understand the ellipse as "a squashed circle". Majority of students accept the first definition of the ellipse, but they are confused about the way to introduce the ellipse in textbooks.(2) Students can correctly understand that one of the sections between a cylinder (or a cone) and a plane creates an ellipse, and the 3-dimensional model using Dandelin spheres. It shows that to understand these two points has little to do with whether they have learned solid geometry. That is to say, students understand the teaching design base on Dandelin sphere without difficulties.So it is possible to use this method in classroom teaching.(3)There is an obvious difference on the choice of how to introduce the definition and derive the standard equation of the ellipse between teachers and students. Students prefer to the teaching design based on history of mathematics, while teachers prefer to the ways based on textbooks.(4) When the historical order and the logical order of a concept contradict with each other, textbooks mainly choose to obey the logicl order, which is consistent with the knowledge system. To teachers, they should obey historical order of the concept, combine it with knowledge system and reconstruct history of the concept. In this way, teachers can easily stimulate students'learning motivation based on the knowledge they have known before and promote a deep understanding of the concept.Based on above conclusions, the thesis gives some teaching suggestions. Also,3 teaching designs of the ellipse based on history of mathematics are given to teachers for reference.