Senior Middle School Students' Understanding Of The Arithmetic-Geometric Mean Inequality

Inequalities is mainly one study of inequality relation of the numbers, which is the basis of further math study and an important tool for mastering modern scientific technology. The inequality of average value is an important part of the inequalities. Many countries in the world have their detailed requirements on the teaching of the inequality of average value and make definite rules in their Curriculum Standard about teaching this content, which plays an important role in the math curriculum in middle schools. Meanwhile, there is relatively little research on this field, neither in China or on abroad.This thesis, by the way of questionnaire and interview, aims at understanding high school students'familiarity about the form, proof method, application awareness and know-how of the inequality of average value, as well as their awareness and capability of using the inequality of average value when dealing with exercises and their application skills of the inequality of average value.Through the statistical analysis of the students'questionnaire and interview combined with the teaching reflection of students in the authors'teaching practice, the main conclusions about the inequality of average value are as follows:1. The students can basically master its skin-deep content and form, but not its nature and the flexibility of its application.2. The students do not understand its proof method well. The students master or understand proof method from the algebraic point of view, such as method of difference comparison and method of quotient comparison, better than the proof method from the geometrical point of view, for example, proving by visual graphs. Even so, there are only few students who could make clear the tenable conditions of the equal mark when proving with algebraic method.3. As to those problems that must be resolved or could be resolved by the inequality of average value, students have certain awareness of using the inequalities when they encounter these problems. Most students make the same mistakes in the application, that is, they do not find out the application conditions of the inequalities clear.4. The mistakes the students made when using the inequality of average value include ignoring the tenable conditions of the equal mark when obtaining the range by use of the inequality of average value, failing to resolve the problem due to ignoring the positive value conditions, ignoring the meaning of "fixed value" when obtaining the range by use of the inequality of average value and ignoring the condition that the equal mark must be tenable at the same time when using≥or≤for many times, etc.5. The inequality of average value provided by the text book is the basic form of the magnitude relationship between the dyadic arithmetic mean and geometric mean. It is true that this form is important, but its extension forms could not be overlooked when resolving similar problems.6. Students have certain application awareness about the inequality of average value. However, they do not stand on the inequality of average value for those problems that could be resolved by many methods. The flexibility of their thinking, an important aspect, is worth encouraging.7. Generally speaking, students of grade three do better than those of grade one and students of grade one do better than those of grade two in understanding the arithmetic-geometric mean inequality.The following teaching enlightenments are obtained through research:1. Memorizing is necessary for mastering the form of the inequality of average value, but learning by rote is not advisable.2. Should pay more attention to the proving method of the inequality of average value for better understanding of it.3. The application of the inequality of average value shall accord with certain conditions. When teaching this point, teachers must make those conditions clear to the students, so that the students could have substantive understanding about the inequality of average value.4. All kinds of extension forms of the inequality of average value could not be overlooked.