Written: Jan. 6, 2013

In some sense, this is a continuation of
Opinion 127, but
with more insight gotten from the *trenches*, after teaching
Second Semester Calculus
for scientists and engineers.

Guru Don Knuth (who would celebrate 75-years in 4 days, Happy birthday Don!) famously said, about computer programming (i.e. teaching computers to do stuff) ``Premature optimization is the root of all evil", and as Knuth taught us so well, we have to start with the most naive approach, and then step-by-step make it better.

Analogously with students. There is such a great gap between what we assume that the
students know and can understand, and what they actually know and are able to understand.
When we teach calculus for example, we take it for granted that they know high-school algebra
and precalculus inside out. True, in order to enroll in Calc2, they were *supposed* to
know Calc1, which means that that they were supposed to know precalculus etc.
But they **don't!** Even those that get an A have a very shaky understanding of very
*basic* high-school algebra (that is **much** more important than calculus).
95 percent of even the A students get an A because they do the symbol-manipulations
correctly, and know how to reproduce the solutions in the review problems.
Those who are B and C students, are not as good fakers. This is like the
Turing Test, have a calculus student simulate *understanding*, but
very few actually understand what is going on.

And why should they? Calculus, and even high-school algebra, are **very abstract**.
For the prof., and even TA, it is all ``obvious'', so most of the instructors can't even
understand how it is "not possible to understand". The few professors (like myself) and TAs that
are actually (acutely!) aware of students' lack-of-understanding (and empathize with it)
are still bound by the demanding, largely obsolete, curriculum, teaching students
to do, mostly by rote, things that computer algebra systems can do so much better today.

The epitome of ignoring the abyss between the expected and actual
levels of students' understanding, and the great hypocrisy of our
system, here at Rutgers (and very possibly elsewhere), are extremely
challenging "workshops" that are fun for the TAs, but are way over the
heads of even the strongest students. What happens is that the TA
"guides" them to write a report, and those who can ape the guidelines
as faithfully as possible, get a good grade. Very few students actually
*understand* what is going on.

In fact, quite a few of them believe that 1/(a+b)=1/a+1/b, and log(a+b)=log(a)+log(b), and that int(1/f(x),x)=log |f(x)|+C (for any f(x), not just x), etc. etc. They have no nonsense-correction instincts that every future scientist and engineer should have. And it is not their fault!

It is not even *our* individual fault. It is our *collective* fault.
We should get a *reality check*, and gear our teaching level, and curriculum (making
full use of computers!) to the level that the students can understand and do well in.
Because of the great gap (I would call it ABYSS) between the actual level of the
students, and what we expect from them, often the average of the Final exam is
a failing grade. Since you can't fail everyone, professors start curving, and then
water boils at 0 degrees centigrade.

Curving should be prohibited, but the level of instruction should be geared to the actual level of understanding, and background, of the students. And don't be bashful about reviewing! Adding fractions, and algebraic simplifications, and teaching them that sqrt(a+b) is NOT sqrt(a)+sqrt(b) are much more important than partial fractions and integration by parts. Computers are such wonderful tools, and one should make careful use of them, and use them to illustrate the concepts.

It is amazing that many students, eventually, still come out OK, but this is
*in spite* of our dysfunctional system of instruction, not thanks to it,
and if we
would know to adjust the level of teaching to the *actual* level of
the students' background and abstraction level, and not to what they are "supposed to know and understand", they would be much better off!

Added Sept. 14, 2014: Read the interesting feedback by Brian Leair.

Opinions of Doron Zeilberger