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 x^{2}, 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".

Opinions of Doron Zeilberger