The History And Teaching Of Complex Numbers

The historical evolution of mathematical concepts is similar to the cognitive development of students. That is so called the theory of historical parallelism. Therefore history can help educators to predict and interprete students' cognitive obstacles and mistakes in mathematics and help teachers to design teaching programs which meet students' cognitive development. Starting with this theory, we choose the part of complex numbers from senior high school curriculum and manage to design teaching programs from the historical perspective. Using these teaching programs, we try to help students to overcome those common cognitive difficulties appeared in their studies.First of all, we study the mankind's process of understanding complex number. According to the mankind's understanding process, the history can be divided into three phases: the introduction of complex numbers, the geometric representation of complex numbers, the application of complex munbers. Learning from the previous experiences of overcoming difficulties, combining with the current teaching and learning situation, three pieces of teaching program respectively about the these phases are put forward. Then two questionnaire surveys are conducted to students and teachers. One aims at examining students' understanding of the related knowledge of complex numbers and the acceptance to the teaching programs. The other aims at examining teachers' teaching of complex number and their opinions and suggestions to the teaching program.The analysis reveals that the confusions of students' understanding of complex numbers can be classified into three aspects: 1) Why imaginary unit i was introduced? 2) Do the imaginary numbers really exist? 3) What's the application of complex numbers? This result is similar with that of mankind in history. As to the teaching program, both students and teachers approve it and accept it gladly. A new program is posed after some modifications according to the suggestions from students and teachers.In the end, relevant teaching inspirations are put forward based upon the conclusion of this research.