Written: April 1, 2011
I am very proud to be a member of the Mathematical Association of America (MAA) because of its brave and very wise decision to recommend to the US Department of Education (DOE) to illegalize the teaching of Calculus except to those who really need it. I believe that this decision was inspired, at least in part, by the preachings of its former president, who observed that, paradoxically, the number of students who go into science or engineering declined ever since it became fashionable to take Advanced Placement Calculus, that traumatized millions of bright high school students, and made them hate math and science for the rest of their lives. Another factor was probably the recent wonderful book, Loving and Hating Mathematics, by Reuben Hersh and Vera John-Steiner, that convincingly argued that mathematics, and especially calculus, should not be used as a filter to weed-out people, like doctors and lawyers, who would never have to integrate or differentiate, not even x2, but since our society only needs so many doctors or lawyers (we already have too many, especially of the latter kind!), it is used as an arbitrary and artificial filter. I agree! Let applicants to medical school do hundred push-ups in one minute instead. It is more important that doctors would be physically fit than that they would know how to integrate by parts.
I am very disappointed, though, at the American Mathematical Society (AMS) for aggressively opposing the above liberal and brave recommendations of the MAA, ostensibly claiming that Calculus is a great mental discipline, and while doctors and lawyers may not need calculus per se, it is good for developing their brains. The real reason of course, is that they are worried about the demand for mathematics professors. If Calculus would only be taught to the few students who actually may need it, there would be even fewer academic math jobs in the future, which would mean that there would be fewer mathematics professors, which would mean that there would be fewer members of the AMS, which would mean that their dues-revenues would drastically decrease. As usual, they are only worried about the bottom line!
Let me suggest a compromise, though, that would not reduce the demand for math faculty. Replace calculus classes by classes on how to solve Sudoku puzzles, and how to play video-games that involve mazes and other challenging combinatorial problems. It would be also fun for all those poor burnt-out calculus professors, who would welcome this change in the curriculum.
I beg to differ, though, on one point with the MAA's suggestions. Not even engineering and physics majors (and especially not chemistry and certainly not mathematics majors) need traditional continuous calculus, with its long-winded and tedious definitions and theorems, that are all obsolete in today's digital age. Instead we should teach them discrete calculus. The Fundamental Theorem of Discrete Calculus is much more user-friendly than its continuous namesake, and only takes few lines to prove. Here it is.
FTDC: Let i be a fixed integer, and for n ≥ i, let S(n):=a(i)+a(i+1)+ ... +a(n),
then: S(n)-S(n-1)=a(n) .
Proof: S(n)-S(n-1)=(a(i)+a(i+1)+ ... +a(n))-(a(i)+a(i+1)+ ... +a(n-1)) (by definition of S(n) and analogously S(n-1)
=((a(i)+a(i+1)+ ... +a(n-1))+a(n))-(a(i)+a(i+1)+ ... +a(n-1)) (by associativity)
=((a(i)+a(i+1)+ ... +a(n-1))-(a(i)+a(i+1)+ ... +a(n-1)) +a(n) (by commutativity)
=0+a(n) (by the theorem A-A=0 for every A, applied to A=a(i)+...+a(n-1))
=a(n) (by the theorem 0+A=A for every A, applied to A=a(n)). QED.
Also replace differential equations by difference equations, and just teach them how to use numerical solvers.
I hope that these wise recommendations would be widely adopted as soon as possible, and maybe in thirty years I will not be afraid to tell the passenger sitting next to me in the airplane that I am a mathematics professor and to hear the predictable answer: "I have always hated math, especially calculus".