微積分了沒？
微積分教材嚴格化的反思以及對高中和大一課程的啟示

單維彰 Wei-Chang Shann

Department of Mathematics

National Central University

E-mail: shann@math.ncu.edu.tw

    Many grade-12 students in the course Elective Math A find Calculus actually easier to handle with than the required materials they had in grades 10 and 11. Many math teachers agree with the students. Calculus is generally considered hard and deep since the later part of the 20th century, therefore it is isolated and reserved to the last part of the school math curriculum. We must agree that there are hard concepts in Calculus. However, those are the rigorous aspects of the subject after Cauchy, Weierstrass, and others. The naive Calculus at its early stage in the 17th century was rather intuitive and it was the source of many intriguing mathematical ideas. It is a matter of simple fact that Newton's Principia was published only 50 years after Descartes' La Géométrie. We all know that La Géométrie spawned Cartesian coordinates and the later fertilized Calculus and mathematical analysis. Closer historical investigations suggested that the idea of Calculus occurred to Newton and Leibniz within a decade since they were exposed to coordinate systems and early analytical geometry. Calculus in that era was much more primitive than some materials in grades 10 and 11: space vectors, matrices and linear transformations, much of the contents in probability and statistics, to name a few.
In speaker's opinion, to cut Calculus from the rest of math curriculum was the most unfortunate strategy made for the math education. Because calculus is the key for the sense-making math curriculum. It plays the central role that motivates and links almost all topics of school math: the very idea of rates and ratios and functions, to begin with, and it provides the reason-to-be for many topics covered in high school: the polynomial inequalities, the radian, the trigonometric and exponential functions.
In this session, the speaker will elaborate on the foregoing remarks and propose a curriculum design that incorporates Calculus with the current materials naturally, organically, and painlessly.

Keyword: math education, school math curriculum, calculus