"Calculus is so last century"? Hardly!

Two months ago a WSJ op-ed ('Calculus Is So Last Century') by Tianhul Michael Li caught the eyes of many mathematicians as well as physicists (and I can assure you, many space and solar physicists). In it Li basically argued that calculus is primarily of the last century as it "is the handmaiden of physics - invented by Newton to explain planetary and projectile motion."

Adding:

"While its place at the core of math education may have made sense for Cold War adversaries engaged in a missile and space race, 'Minute Man' and 'Apollo' no longer occupy the same prominent role in national security and prosperity they once did."

Which makes one wonder what alternative universe and planet this guy is living on? Surely not the one that I'm on!

A feature story in Saturday's NY Times, for example, expatiated on the latest developments including "hypersonic targeted warheads" not only under development by the U.S. but China and Russia as well. The U.S. has a non-nuclear version but the take is that the Russians and Chinese are developing nuclear warheads- independent of missiles- that can be hurled into space then come down on an enemy with little or no warning time. This is in contrast to Cold War ICBMs that general gave about 30 minutes advance notice, unless launched by submarines)

Most alarming is the drive by a number of nations, including the U.S., China and Russia, to develop low yield nukes which - as the article put it- could make their use much more likely than the "mutually assured destruction"  (MAD) model of the past allowed. Indeed, the Russian already have a military posture firmly in place that permits the use of tactical nukes (with 10- 25 kt warheads) in the event that NATO forces overwhelm Russian positions in Eastern Europe. (And cruise missiles are also being outfitted with low yield nukes.)

The current U.S. proposal to "modernize" its nuclear warheads is also seen as possibly destabilizing the existing system and rousing the Chinese and Russians to action in their own warhead development.

Far from national security having switched away from H-bomb Cold War worries, we are evidently entering a new era wherein nuclear "swords of Damocles" literally hang over our collective heads, in the form of targeted hypersonic warheads, or even nuclear-armed satellites,

None of these systems will use linear algebra (the math Li mainly advocates teaching) to reach their targets, but plain old brute differential calculus, of a form I described in earlier blog posts, e.g. from May 6, 2013 (the rocket problem).

Surely then, Calculus isn't passé if it is still central to the ultimate form of nullifying any national security or 'prosperity': nuclear war,

Meanwhile, in fields as diverse as astrophysics, astronomy, quantum mechanics and plasma physics - not to mention climate science -  differential calculus is as critical as ever. While it is true algorithms and numerical simulations, models have been developed in all these fields, it is also true that differential calculus underscores that development and continued tweaking of those models depends on the actual math, not just crunching numbers into numerical or statistical data sets.

Li is correct when he asks at the outset 'Can you remember the last time you did calculus?'  But that question could also be asked about linear algebra or finite mathematics, or a course in multivariate statistics.  The point is that any form of advanced learned in high school or college will fall by the wayside if it is not used in some way later on, irrespective of how it was taught.

Li argues that he isn't saying calculus shouldn't be taught, as it is "great mental training" which is most gracious of him. But if nuclear warheads are going to be set off in detached form and land on my head, I'd be curious as to the math underlying the dynamics. I don't want to just read about it in the NY Times.

Where I do agree with him more, is when he criticizes the "single drive toward calculus in high  school and college" which "displaces other topics more important for today's economy and society".  These include "statistics, linear algebra and algorithmic thinking" (such as revealed in finite math.)

I concur because I've always been skeptical of the high school AP Calculus curricula and whether students are really of sufficient mental maturity that the courses serve any useful purpose (apart from the usual academic 'feather' in the cap to expand their choice of college). And I never saw the reason for pre-med students to be taking calculus. In each case then, perhaps some finite math course or statistics would better serve these populations as opposed to differential and integral calculus.

Calculus then, ought to be here to stay, certainly for most college math and science majors and perhaps even some others (e.g. Philosophy) interested in exploring the role of quantum mechanics in modern expositions, say involving quantum nonlocality and entanglement.

Why Is LEARNING College Math So Difficult For So Many Students?

This is the note sounded in a recent article in the Notices of the American Mathematical Society, Vol. 61, No. 6, p.597) by Prof. Igor Rivin of Temple University. While reminding readers of the special burdens of the university professor: including research, administration responsibilities as well as teaching (he never mentions the role of adjuncts - now becoming ever more prevalent) he then takes aim at "service teaching" which he distinguishes from "specialized teaching".  The latter is preferable and more rewarding because it's directed at graduate students and math majors. Hence, one expects a much higher level of abstract thought, computation and ability to show proofs.

Service teaching, on the other hand, is directed at the unwashed hoi polloi, who enter college and need 1-2 years of some form of college math, perhaps including college algebra, calculus and linear algebra. In the words of Prof. Rivin:

"(My) thoughts will center primarily on service teaching which for me, combines some of the most depressing aspects of my job."

But WHY so depressing when in theory the exposure of young, eager, college-age minds to math ought to be supremely rewarding? While my major wasn't in math, I did have nearly enough mathematics background (through advanced calculus and numerical analysis)  to qualify as a math minor and taught math for the equivalent of 7 years, including in Peace Corps as a volunteer, see e.g.

http://brane-space.blogspot.com/2012/11/looking-again-at-groups-sub-groups.html

and at college/university level. I had few problems but that may be because the students were (mostly) in the Caribbean where math teaching standards are much higher.

Anyway, Rivin's answer is as follows:

"Why depressing? Consider the following: the vast majority of service courses are concerned with differential and integral calculus and linear algebra. These are both rather deep subjects, as evidenced by the fact mathematics has been practiced for thousands of years by rather talented people before the basic principles of calculus were laid down in the late 17th century. It took another two hundred years of extensive work to make the foundations of the subject truly solid."


The above latter point is well taken. Indeed, the historical process of serial integration leading to the foundations had first to assemble the separate theorems  in a logical sequence (in order to create the first calculus course.) Think of the fundamental theorem of calculus, the mean value theorem, Rolle's theorem, the theorems of Pappus and others. All had to be subsumed and approached in a sound didactic form for calculus to be imparted to college students. (High school "AP" courses in calculus, by contrast, are mostly focused on the 'mechanics' - i.e. of differentiating, integrating and applications and not so much on the underlying theorems).

Rivin continues:

"Linear algebra, as used today, is an even later bloomer. The current machinery of matrices and linear transformations was not put into a truly modern form until the beginning of the 20th century. We have no choice but to agree that these subjects are quite deep and require some considerable technical skill to use successfully."

So,  okay, let's concede the subjects will be tough going for the average college student, What is the nature and background of these students? Rivin again (ibid.):

"WHO are we teaching them to? In a public university our students are, in the main, somewhat above average products of the U.S. public secondary school system. This means that their technical ability is already quite severely taxed by arithmetic with fractions. Their abstract reasoning skills are essentially non-existent..."


Wow! If this is indeed the case these students would be in over their heads with college algebra and trig, before they hit the first chapter of a calculus or linear algebra text. But in fact he highlights a key complaint of many college math teachers that most 1st and 2nd year students (mainly business majors, as well as med students required to take a year of these subjects) simply find themselves in over their heads.  Prof. Rivin confirms this take with his next remark:

"A consequence of all this is that it is well nigh impossible to teach them what we purport to teach them. Higher mathematics requires a certain level of abstraction, and even if we commit the crime of forgetting that and define calculus as a 'collection of computational techniques without understanding' , the students' technical weakness renders even that aspect essentially worthless.

They cannot compute. The result is that our calculus and linear algebra classes consist of a collection of trivial examples, which the students must memorize by rote. This has the consequence of not teaching the students anything but the fear and hatred of mathematics."


Which is truly sad, because one is then limited as a teacher in doing the interesting stuff and having the students participate actively, say like working out the orbit of a planet or computing the position of Jupiter on March 15, 2050. It's little wonder Rivin is depressed.

That leaves us asking: What is the solution? From Prof. Rivin's take it may well be not to impose calculus or linear algebra on pre-med or business majors at all. As he observes:

"The majority of the students never use calculus in their future lives- small wonder since they don't actually know any. But they never had any intention of using advanced mathematics even before taking the courses. They are required to take the courses because of the (not unreasonable) belief that mathematics should be a part of every college-educated person's intellectual make up.

The result is that the loathing of mathematics is part of the intellectual makeup of a sizeable majority of Americans."

All of which echoes the complaints of many to the 'Common core' Algebra II requirement for high schools. See e.g.

http://brane-space.blogspot.com/2013/08/kids-hate-algebra-ii-why.html

It appears to me that one solution, which may not be palatable to math didacts or education idealists, is that we temper our expectations,  reduce college "service courses"  to the rudiments, and also acknowledge that calculus and linear algebra are simply asking too much of non-math majors.  We then leave the full, 'meaty' calculus and linear algebra courses to the specialists- for whom "trivial examples" and rote memory need not form the course basis.

The only alternative seems to be to retain the watered-down,  difficult courses and thereby create ever more American math haters: an unfortunate lot that will likely not then even pick up a book - even a science fiction novel - if they happen to spot one equation on one page.

See also:

http://brane-space.blogspot.com/2013/01/college-math-prof-and-why-us-education.html