Enrichment or Acceleration

THERE has been much difference of opinion among secondary mathematics teachers during the last several decades as to just how the curriculum should be changed. There appears to have been, however, general agreement that changes were necessary. This agreement has given rise to the so-called accelerated programs in which students are enabled to study the usual (but modernized) high school courses at an earlier stage. In many cases this acceleration allows the student to be exposed to a full year of calculus while still in high school. Many of the students who have followed this route are unable to qualify for Advanced Placement credit and are required to repeat the first year of calculus as college freshmen. This indi cates that we have allowed some students to enter an accelerated program who are not really ready. When this happens, have we really done our students any good? Or, have we succeeded only in making the first-year college mathematics course repetitive? We are confronted with the task of ade quately preparing those students who are average or above in ability but not quite ready for the accelerated program. After some years of teaching at both the high school and college level, the author has reached the opinion that we, as teachers, can enrich the secondary program to such an extent that when a student reaches college he is ready for calculus but, except for the truly outstanding student, has not studied it. A competent secondary school math ematics teacher will encounter no diffi culty in enriching a course if he tries to introduce, at a simplified or intuitive level, topics that he knows will be studied at a later time. During the second year of algebra, for example, it is possible to in troduce parabolic interpolation at an extremely low level. This could lead to better understanding of Newton's formula for approximating roots of a polynomial equation, which is customarily introduced about the twelfth year, and eventually to LaGrange's formula, which is usually not encountered until a course in differential equations is taken. Many secondary teachers introduce Newton's method as a mechanical process that involves only the ability to find the first derivative of a polynomial and to compute algebraically. This is done by using the fact that, by starting with a fairly good estimate of a zero r (call it Xi), you may obtain successively closer ap proximations x2, ?3, Xiy . . . by using the recursion formula f(xn) . 100 \ Xn+i = Xn --? (fi = 1, 2, 3, ... ) J \%n)