Undergraduate And High School Students' Understanding About The Limit Of A Number Sequence: An Empirical Study Of Historical Parallelism

The limit of a number sequence is the first concept involving infinity. Many students have difficulties in understanding about the limit, which teachers don't have an effective method to teach. It has been found that there are four major epistemological obstacles in the history of the concept of limit:(1) Using algebra to solve geometry problems; (2) The notion of the infinitely large and infinitely small; (3) The rigor of the notion of limit; (4) Is the limit attained or not?According to the above four aspects, a questionnaire was designed to investigate undergraduate and high school students'understanding about the limit of a number sequence. The questionnaire survey was conducted to 608 first year undergraduate students and 431 high school students. Through an analysis of the obtained results, the following conclusions are arrived at:(1) Students have few difficulties in using algebra to solve geometry problems, they have already overcome this epistemological obstacle. Historical parallelism doesn't exist with regard to this epistemological obstacle. The school mathematics improves students'ability to use algebra to solve geometry problems.(2) Students have lots of difficulties in the understanding about infinitely small, the rigor of the notion of limit and the attainability of the limit. Students'confusions in these three epistemological obstacles indicate the existence of historical parallelism. They haven't overcome these epistemological obstacles. It takes them longer time to get the real understanding about these issues.(3) The independent-samples T Test between undergraduate and high school students shows that undergraduate students do better than high school students.(4) The independent-samples T Test between male and female students shows that male students do better in the understanding about infinitely small, while they are the same in other aspects.(5) The independent-samples T Test between undergraduate students who study Advanced Mathematics A and Advanced Mathematics B shows that they have no significant difference in these four aspects.(6) The independent-samples T Test between undergraduate students who study higher Mathematics A from different majors shows that they don't have significant difference in these four aspects.Based on the above conclusions, some reasons of students'understanding about the limit of a number sequence are analysed, meanwhile several teaching suggestions are proposed. Teachers should know the history of the limit of a number sequence and the epistemological obstacles, pay more attention to the teaching of the infinitely small and investigate students'understanding about the attainability of the limit of a number sequence in different ways.