Number Theory is the best example. Since the ancient Greeks first became interested in the odd behavior of Prime Numbers (around 300 BC), the Primes came with an ironclad guarantee that they had no practical value and never would. That guarantee of uselessness lasted for almost 23 centuries.

Yet strangely, in every mathematically rich generation, the greatest mathematicians, like Gauss and Euler, were always lured by Number Theory and Prime Numbers and enriched it with brilliant new discoveries -- all completely impractical.

Then, suddenly, in the 1970s, a new kind of secret code was invented based on pairs of large prime numbers. The world's biggest, fastest supercomputers can't break such codes if they keep trying for decades, perhaps centuries. And yet such "twin-prime" codes are extremely easy to use to send perfectly secure messages between computers that know the prime number keys.

Prime numbers -- and specifically, the ability to determine if a huge integer is Prime or Composite (factoring) -- are now subjects of tremendous practical importance for financial institutions, corporate computer security, and governments.

And this is the way it is throughout mathematics -- subjects that seem to have no practical value for centuries, suddenly prove to have extremely practical applications. There is a fundamental intimacy between mathematics and the Real World which is filled with such surprises.

And this is the mysterious relationship between mathematics and nature which should be communicated to math students. And to science students.

Another example: Pascal's Triangle, a curiosity of Pure Mathematics, which, 300 years later, turned out to be the secret of the Laws of Biological Heredity (Mendel).

There are dozens of other mathematical surprises which turned the theoretical into the extremely useful and practical. And there are scores of practical applications which pushed the creation of new theoretical and pure tools in mathematics, the most ancient and famous being astronomy, which led Newton to develop the calculus. Twentieth-century physics (as has been noted) has proven to be amazingly rich in "practical" applications for previously impractical pure math tools. Relativity relies entirely on the strange 18th and 19th-century curiosity of non-Euclidian geometry, where extended parallel lines meet once or more than once. The square roots of negative numbers -- a detested and rejected impossibility for centuries -- are now the standard tool engineers use to describe alternating electric current.

Elmer Elevator

Just to establish a common ground let me mention that my PhD was in Applied Mathematics, more accurately, it dealt with a numerical solution of the stamp problem. As a programmer, I'd look grotesque fighting for the nebulous purity of mathematics.

The only thing I am against is the claim that, say, the "baseball mathematics" is an universal attraction to students, and that while learning to compute the batting averages students somehow unawares learn mathematics as well.

Perhaps this was just a result of our frustration. We had to take an introductory course in thermodynamics without knowing what the total differential is. Or a course in analytical mechanics without knowing what a variation is (for the least action principle). So we kept asking our math professors, why do we have to study this and why do we have to study that - just like the guy who asked for Uses for factoring trinomials. And the only answer we ever got was that our curriculum was designed by the physics department. This was probably true, but we considered it as an unfair and evasive maneuver.

>The only thing I am against is the claim that, say, the "baseball
>mathematics" is a universal attraction to students, and that while
>learning to compute the batting averages students somehow unawares
>learn mathematics as well.

I could not agree more. Do we really need this marvelous tool to calculate batting averages? Perhaps you noticed that I have been praising MS Excel in some of my replies (it could be some other spreadsheet). Still, if you look at the Excel examples or tutorials, it is all about plotting sales in the 4th quarter, etc. It really turned me away for some time before I realized what a marvelous tool a spreadsheet program is.

I have a feeling that nowadays, if confronted with a similar circumstance, you would behave differently. You would probably just look up the definition of the total diffirential, right?

But regardless, the example just says that the success of student-teacher communication depends partly on the teacher's ability and methodology, partly on the curriculum, and partly on the student's mindset.

>Or a course in
>analytical mechanics without knowing what a variation is
>(for the least action principle). So we kept asking our math
>professors, why do we have to study this and why do we have
>to study that

The question does not sound right to me. Why did not you ask what is a variation? For, assume you new what a variation was, how would this help with the question of this or that being worth studying?

>- just like the guy who asked for >href="https://www.cut-the-knot.org/htdocs/dcforum/DCForumID3/239.shtml";]Uses
>for factoring trinomials.

I believe strongly that such a question has no satisfactory answer because, for one, it demonstrates scholastic immaturity of the questioner. Also, none of the answers could be 100% sincere. Factorization is used here and there - so what? Does not Mathematica or a calculator can do the job? Do not, for every job that might need factorization now and then, there exists a hundred jobs that never need one?

Thirdly, it is simply impossible to give sensible answers to all such questions. Sooner or later the teacher won't have an answer and this one moment will ruin the whole course.

>And the only answer we ever
>got was that our curriculum was designed by the physics
>department. This was probably true, but we considered it as
>an unfair and evasive maneuver.

It was probably correct and sincere, though.

>>The only thing I am against is the claim that, say, the "baseball
>>mathematics" is a universal attraction to students, and that while
>>learning to compute the batting averages students somehow unawares
>>learn mathematics as well.
>
>I could not agree more. Do we really need this marvelous
>tool to calculate batting averages? Perhaps you noticed that
>I have been praising MS Excel in some of my replies (it
>could be some other spreadsheet). Still, if you look at the
>Excel examples or tutorials, it is all about plotting sales
>in the 4th quarter, etc. It really turned me away for some
>time before I realized what a marvelous tool a spreadsheet
>program is.

Anything becomes a marvelous tool once you got curiosity. In the spring I had a chance to watch two young sisters in a science museum. One (10 years old) tried to figure out what each stick or button was for. The other (12) had the only goal of grabbing those sticks first. Once she succeeded, she would step aside and let the younger sister play.

Excel is a marvelous tool for any one who has a goal and curiosity to achieve that goal. For others, to study how to use it is as much annoying as any other subject.

>I have a feeling that nowadays, if confronted with a similar
>circumstance, you would behave differently. You would probably
>just look up the definition of the total diffirential, right?

Yes I would. But at the time we were just in the middle of the elementary calculus, dy/dx was an unbreakable synonym for y'(x), barely tolerated. There was still a long way to get to the concept of the total differential and even if I looked it up, I would not understand.

>The question does not sound right to me. Why did not you ask
>what is a variation? For, assume you new what a variation
>was, how would this help with the question of this or that
>being worth studying?

The analytical mechanics professor gave some explanation, but not rigorous. It is an unusual math concept and before using it, knowing some of its properties is necessary. Incidentally, the concept did not show up in any subsequent math courses. The only book giving a rigorous treatment of variation that I ever found (unfortunately, a year or two after the analytical mechanics course) was the ancient

M. A. Lavrenteev, L. A. Lusternik, Course in Variation Calculus, (Gos. izd. techniko-teoreticeskoi literatury, Moskva 1950).

I still have it. Of course, once I found this one, I stopped searching.

Regards, Vladimir