Re-examination of the Calculus m School Mathematics

In the early part of the 20th century, J. Perry and F. Klein insisted that the elementary principles of the calculus should be an essential element of school mathematics, from the standpoint of functional thinking, comprehending the physical phenomena, and mental training. And, since 1946 we have taught the calculus for all high school students as the capstone experience of school mathematics. But the need for reforming the school calculus has increased by the current trend of mathematics-educational thoughts, stressing the intuitive understanding, problem solving, application, and the importance of discrete mathematics. The present study was undertaken to closely examine the phase of the calculus in school mathematics. The limiting approach to calculus now in use gives rise to a trouble in understanding the basic concepts of calculus using the symbols of the old infinitesimal method, and comes into question because of the alienation from the infinitesimal method in physics and the claim for the applicable mathematics fraught with relations to reality. This study is going to propose that the infinitesimal method is an appropriate approach to school calculus on the ground that it is harmonious with the historical genetic principle, the concrete mathematical approach fraught with physical relations and modern mathematical thinking in terms of the actual infinities. The formalistic deductive approach to mathematics in the college of education has brought forth the attitude of mathematics teachers who prefer rigorous systematic approach to school mathematics turning their backs to the historical origin, the goal of teaching, the intuition and the application. In order to dissolve the serious problem of 'double-forgetting' in mathematics teacher education caused by the formalistic approach to mathematics, this study is going to propose the historical genetic approach to the calculus in the college of education, taking examples by O. Toeplitz and I. Lakatos. The growing importance of discrete mathematics for information processing resulted in the need for changes in the school calculus. But the development of the mathematics curriculum appropriately accentuating the positive aspect in both the calculus and the discrete mathematics is needed, because they are complementary not alterative