First-year Students' Understanding About Limits, Continuity And Derivatives

Concept definitions serve as the foundation of mathematics, but they do not derive directly from the objective reality and have the multiple characteristics of abstractness, generalization and simplicity, thus, it is difficult for students to recite concept definitions without omitting a single word. For a student, to obtain concepts means to form concept images. However, sometimes the students' images of a concept are one-sided, twisted and conflicted. If permitted to learn the related concept first, students can better understand the original one. Therefore, concept definitions are fundamental to establish good concept relations, while good concept relations will conversely promote the understanding of concept definitions, which will help students form correct concept images.Functions are research objects of the calculus and limit is its research tool. The way to define the calculus concept differs distinctly from that of elementary mathematics. Although students can find the limits, derivatives and integrals of functions, they are not able to gain a good understanding of the basic concepts of calculus. Through a test, questionnaire survey and interviews, this paper intends to investigate the images of the freshmen with different specialities about the concepts of limit, continuity andderivative of a function at a point x₀ and the relationship among them. It is found thatthe images held by many students from three different specialities are one-sided, conflicted and even inadequate, and at least 50% of the freshmen see the concepts of the limit, continuity and derivative of a function at a certain point isolated, without taking the relationship among them into consideration.