The Heuristic Teaching Of Quadratic Reciprocity Law And Its Application

Quadratic reciprocity law plays an important role in number theory module learning, which is one of the center of the history of number theory development. Traditional quadratic reciprocity law teaching let students master the content of theorem and understand attestation process, but it ignores the cultivation of students' interests and thinking in mathematics easily, which makes students regard mathematics as a boring and abstract subject and even lose the enthusiasm of learning mathematics.This paper discusses the development of the quadratic reciprocity law based on the heuristic teaching model which includes surveying data, analyzing data, guessing the rules and proving the results of speculation. Using this model to discuss the properties of the quadratic residue, the multiplication rule of quadratic residue, the introduction of legendre symbol, a solution of x2=-1(mod p) and x2= 2(mod p), the content and relevant proof of the quadratic reciprocity law and generalized quadratic reciprocity law theorem respectively. Based on discovery learning of the quadratic reciprocity law, this paper divides quadratic reciprocity law questions in International Mathematics Contest for nearly 14 years into four categories:the application of solving diophantine equation, the application of divisible and congruence, the application of questions about integral function and the application of the research on prime numbers.Through the study of this paper, the heuristic teaching model on mathematics should be taken enough attention. The process of surveying data and sorting data can cultivate students' interests and enthusiasm of learning mathematics; the process of analyzing data, guessing the rules and proving the results of speculation can develop students' abilities of self-study and mathematical thinking, promote students to master knowledge firmly and improve the ability of knowledge extrapolation. The analyses and methods' summations on quadratic reciprocity law questions in International Mathematics Contest for nearly 14 years can provide the examination preparation guide to competition tutors and participating students.