Senior Middle School Students' Understanding About Actual Mathematical Infinity

Actual infinity is a very important concept not only in mathematics, but in philosophy as well, which makes great contributions to the foundation and many theories of mathematics.This thesis makes a study of senior middle school students' understanding of the concept of mathematical actual infinity. Through a questionnaire test and interviews, the students' strategies of comparing the two infinite sets are found to fall into four classes: (1) the two sets are both infinite, therefore they have the same number of elements; (2) a set which is strictly s included in another set has fewer elements that set; (3) two infinite sets are incomparable; (4) if between the two infinite sets there exists a one-to-one correspondence, then these two sets have the same number of elements.Explanations provided by the subjects reveal the parallelism between their understanding of the concept of the actual infinity and that of mathematicians in the history, justifying the historical-genetic principle.The students' responses to the questionnaire also reveal that students' choice of inclusion relationship or one-to-one correspondence relationship is influenced by the representation of the sets, numerically or geometrically.The interviews reveal that the students cannot solve the contradiction between the two results as a result of different representations of sets.At last, a conclusion is drawn from this study and some teaching implications are given.