Design Research On Concept Instruction Of Single Variable Calculus

In the context of mass higher education, the ability of college freshmen are generally in lower level than before, and increasingly uneven. It has become a major problem facing the world of higher education. On the other hand, reform on basic education has made tremendous change in the secondary school curriculum. This change also put forward new requirements for the university setting. Mass education and the high school curriculum cause problems increasingly on Calculus teaching in university. Many college students can do simple works on derivative and calculus, but can not understand the idea behind the concept, and as a result, usually have fuzzy understanding of the relationship between concepts. Therefore, how to find the cognitive difficulties of the students on the concepts of calculus and to make the targeted instructional design is the key to the reform of the teaching on Calculus.This research, selecting cases on limit, derivative, differential, Mean Value Theorem, definite integral, focuses on the two questions:(1) What concept images and misunderstandings do college students have on the basic concepts in calculus?(2) How to design the teaching of calculus concepts can help students understand the concepts, and improve their ability to apply basic concepts?This research, based on the construction of principles on concept instruction in calculus, presents a case study on calculus course in three classes of a college of science and engineering freshmen by carrying out the teaching design and teaching experiment. Research methods such as design research, questionnaires, interview, classroom observation, control of quasi-experimental are adopted. There are3teachers and255students participated in the practice. The study consists of three phases.1) Preparation and design:based on literature review and teaching experience we sum up the common errors and problems encountered by students, and the main points in concept instruction. Then we design test volumes for before/after the teaching, and make arrangements of test time, teaching time.(2) Teaching practice: according to the problems found in the pre-test, we correct the original instructional design (prototype) and implement the concept instruction.(3) Review:teachers write teaching reflection on the concept instruction design principles and make second correction; based on the revised principles, the next round of instructional design is carried on. At the end of the study, we conduct a test of the effect of instructional design, primarily through three paths:firstly, making longitudinal comparison among the testing classes according the pre-test and post-test comparison of the specific case; secondly, using midterm and final exams in school uniform arrangements for horizontal comparison; thirdly, at the end of the semester, making the student survey to know students’recognition of the concept instruction.Based on the findings of this study, the following conclusions can be drawn:Firstly, college students’concept image of the fundamental concepts of calculus is one-sided, and some even wrong.(1) Before learning, students can not define the limit by correct words. Some students even take a static point of view, as limit is "when n tends to infinity (tend), the number of series (function) is equal to a."(2) Most of the students usually think of the function curve slope of the tangent at a point when seeing derivative; students mainly consider the angle of the slope as derivative, rather than from the point of view of the rate of change in understanding the definition.(3) The "operational knowledge" makes deeply impression to students. There is confusion in the profound understanding of the significance of differential geometry and linear approximation.(4) some students know the definite integral is the area, but it is not clear exactly which area of the region; Some know the definite integral concept segmentation and approximation substitute of the process, but do not know which amount is divided; some students simply think dx as part of the integral symbol, while ignoring the practical significance of the "differential".Secondly, we constructed principles on concept instruction in calculus and the corresponding instructional design and teaching experiments. The principles are as follows:(1) through the primitive (history, essentially) the introduction of mathematical concepts, with the historical development of elaborate mathematical concepts;(2) by means of geometric or intuitive life examples to help students understand the concepts;(3) paying attention to the elaboration of the concept of the relations between them. Based on the problems in the pre-test, the design focus of each case are as follows:Limit focuses on intuitive way to help students learn, understand and use formal language; Derivative on clarifying the thought of "the rate of change"; Differential on the links between prominent concept to help students in the minds of a concept map; Mean Value Theorem on the history of the theorem in the form to help students understand the concept of strict process; Given integral as a typical representative of the process concept, the design point is to help students in teaching the concept of definite integral decompression, which will be fixed integral concept to migrate to an unknown situation.The research, first in China, systematically studied students’understanding of the fundamental concepts of calculus, and explores students’concept image through a semester teaching practice. The implications of this study for teaching and learning the concept of calculus as well as on research in mathematics education are also discussed. The practical value is that it can help university teachers to know students’ conceptual understanding of the situation, to provide teachers with specific teaching strategies and design reference, and also as material for a university textbook writing.