Design Research On Teaching Of Mathematical Induction Which Based On HPM

In recent years, with the rapid development of international Study Group on the Relations between History and Pedagogy of Mathematics, more and more researchers on HPM walk out of the ivory tower into the teaching, practice research in the field of teaching become one of the most important research directions on HPM, becoming an important way for researchers to create, test and develop the theory of HPM.Freudenthal H, a famous Dutch mathematics educators, said that there are a lot of serious problems in the teaching of mathematical induction, some even violated the principles of pedagogy, and proposed to teach mathematical induction in the way of basing on its history. What’s more, Harel’s research showed that students’understanding of mathematical induction presents historical parallelism. Learned from previous studies, this study used mathematical induction as a carrier, designed instruction of mathematical induction based on HPM, did practice research in the real teaching situations. Based on the relative theories in the field of mathematics education, researcher build a theoretical framework on combed the area of mathematics education theory, on this basis, build a theoretical framework of instructional design in the field HPM, and combined with the method of design research methods to set up pyramid model in the field teaching practice on HPM. With the guidance of pyramid model, teaching practice mathematical induction based on HPM been carried out. in the process of studying, instructional design was revised and completed, and the established framework and model were tested.Four classes from two high schools were selected to be taught mathematical induction, a class is a control class, the others are experimental classes. With the guidance of pyramid model, the instructional design were carried out three times. The research questions are: Whether historical parallelism on Students’ understanding of mathematical induction exists?What are the impact on the levels of students’ understanding of mathematical induction after the HPM teaching, and emotion, values and attitude? What is the impact on teachers’professional development after the HPM teaching?The study collected related data through interviews, questionnaires testing and teaching memoir, analyzed the data by the way of quantitative and qualitative to solve research questions, and drew conclusions. More specifically, historical parallelism on students’ understanding of mathematics education were analyzed by comparing with four pre-tests to; and varieties of the level of understanding on mathematical induction were analyzed by comparing with four after-tests; the differences between four pre-tests and after-tests were analyzed by one-way ANOVA; varieties in the aspects of emotion, attitude and value were analyzed through interviews; the degrees of integration of HPM were analyzed through teaching memoir; teachers’ professional development were analyzed through teacher interviews; lastly, the conclusions on teaching practice of mathematical induction were made.This study showed that:(1)there are significant historical parallelism on the students’ understanding of mathematical induction, and the level of understanding is the level of inductive reasoning or in connection recursive level;(2)comparative analysis of two ways of teaching, found that the levels of students taught by the way of HPM teaching were significantly higher than students taught by using the normal teaching way, and the HPM teaching way is likable by students, because this HPM teaching way could contribute to learning and deepening their understanding;(3)teachers involved in the process of HPM teaching practice, their education conviction occurred slightly changed, instructional knowledge on HPM is significantly increased, and teaching abilities has been improved. The results of this study includes that pyramid model in the field of HPM teaching practice, case study of historical parallelism on students’ understanding of mathematical induction, case study of HPM instructional design on mathematical induction perspective, three research principles of HPM teaching practice, four levels of understanding of mathematical induction, and five research Steps of HPM teaching practice.