Secondary School Students’Understanding Of Mathematical Induction:Structural Characteristics And The Process Of Proof Construction

Few studies have concentrated on the secondary school students’first experiences with mathematical induction (MI). Moreover, emphasis has not been given on the meaning that the students attribute to the structural characteristics of MI. We argue that it is crucial to investigate the aspects of their understanding at this initial level, both from a research and from a teaching point of view.The understanding of MI emerges through the structural relation of MI to a set of natural numbers and through the way MI works in proving recursion problems. In this study, we investigate the meaning students attribute to the structure of MI and the process of proof construction using mathematical induction in the context of a geometric recursion problem. Two hundred and twenty eight students of grade2in a key high school of Zhejiang Province participated in the study. We analyzed students’responses in six written tasks and the interviews with27of them. While in school, MI is treated operationally in school, the students are only asked to use MI to prove some propositions. But when get inspired and guidance, they started to recognize the structural characteristics of MI, and recognized the relationship between the function of each step of MI and characteristics of the collections. In the case of proof construction, we identified two types of transition from argumentation to proof, interwoven in the structure of the geometrical pattern. In the first type, MI was applied to the algebraic statement that derived from the direct translation of the geometrical situation. In the second type, MI was embedded functionally in the geometrical structure of the pattern. Students determined the critical ideas of the algebraic relations between nth and (n+l)th terms by analyzing geometrical pattern, and thus produce a proof.About the teaching of MI, to promote the development of students’ mathematical meaning of MI, we suggest that teaching MI in a meaningful way, focusing on the structural characteristics of MI and the relationship between the number of sets. Besides, geometrical pattern can be an effective situation for the students.