Gian-Carlo Rota - A few lessons

Came across Gian-Carlo Rota on my browsings through the internet. Somewhere to somewhere. And so glad I stumbled across these. (Three pieces of writings (1), (2), (3) - a few excerpts follow). Although it is grounded in quite high level Math education, still some fundamentals can be carried and applied anywhere.

Following from 1997. Perhaps all institutions where the most gifted congregate to learn have similar notes, still, inspiring to see them collected in his writeup. His focus on very high levels of performance, opting for more demanding work, "knowing-how" subjects and permanent long term value subjects. A few excerpts:

The most brilliant students will invariably work out all the problems and let other students copy, and I pretend to be annoyed when I learn that this has happened. But I know that by making the effort to understand the solution of a truly difficult problem discovered by one of their peers, students learn more than they would by working out some less demanding exercise.

By and large, "knowing how" matters more than "knowing what." Half a century ago, the philosopher Gilbert Ryle discussed the difference between "knowing how" courses are those in mathematics, the exact sciences, engineering, playing a musical instrument, even sports. "Knowing what" courses are those in the social sciences, the creative arts, the humanities, and those aspects of a discipline that are described as having social value... To be sure, the content of "knowing what" courses if often the most memorable.... (But) - Where you can test, you can set a high standard of proficiency on which everyone is agreed; where you cannot test precisely, proficiency becomes something of a judgment call.

You don't have to be a genius to do creative work. It is demoralizing to give a young person role models of Beethoven, Einstein, and Feynman, presented as saintly figures who moved from insight to insight without a misstep. Scientific biographies often fail to give a realistic description of personality, and thereby create a false idea of scientific work....The drive for excellence and achievement that one finds everywhere at MIT has the democratic effect of placing teachers and students on the same level, where competence is appreciated irrespective of its provenance, Students learn that some of the best ideas arise in groups of scientists and engineers working together, and the source of these ideas can seldom be pinned on specific individuals. The MIT model of scientific work is closer to the communion of artists that was found in the large shops of the Renaissance than to the image of the lonely Romantic genius.

You must measure up to a very high level of performance. One learns a lot more when taking calculus from someone who is doing research in mathematical analysis than from someone who has never published a word on the subject...What matters most is the ambiance in which the course is taught; a gifted student will thrive in the company of other gifted students. An MIT undergraduate will be challenged by the level of proficiency that is expected of everyone at MIT, students and faculty. The expectation of high standards is unconsciously absorbed and adopted by the students, and they carry it with them for life.

The world and your career are unpredictable, so you are better off learning subjects of permanent value.

The hidden curriculum: The future belongs to the computer-literate-squared. In their second year, they catch on to the fact that their required courses in computer science do not provide the whole story. Not because of deficiencies in the syllabus; quite the opposite. The undergraduate curriculum in computer science at MIT is probably the most progressive and advanced such curriculum anywhere. Rather, the students learn that side by side with required courses there is another, hidden curriculum consisting of new ideas just coming into use, new techniques and that spread like wildfire, opening up unsuspected applications that will eventually be adopted into the official curriculum.

Mathematics is still the queen of the sciences. 

Following from another article about things he wished he'd learnt.

On lecturing:

Every lecture should state one main point and repeat it over and over, like a theme with variations.

Make sure the blackboard is spotless. By starting with a spotless blackboard you will subtly convey the impression that the lecture they are about to hear is equally spotless.

Generally this can be picked and carried and replicated in anyone's life:

Use the Feynman Method -  Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!”

About introduction to papers (and I find prefaces, introductions, table of contents highly enlightening especially when one seems to wish to read whatever one comes across!!)

Keep lengthy introduction, summarizing the history  of the subject, giving everybody his due, and perhaps enticingly outlining the content of the paper in a discursive manner, will go some of the way towards getting us a couple of readers.

And finally, the third paper on differential equations, which is quite beyond me. But it closes with a beautiful note. I really like his resounding note throughout the three pieces about pitching or aiming at levels slightly higher than one's current reach.

TEACH CONCEPTS, NOT TRICKS

What can we expect students to get out of an elementary course in differential equations? I

reject the “bag of tricks” answer to this question. A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality, and the justifications I have heard of it, citing poor preparation of the students, their unwillingness to learn, and the possibility of assigning clever problem sets, are lazy ways out.

In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives, such as the universal occurrence of the exponential function, stability, the relationship between trajectories and integrals of systems, phase plane analysis, the manipulation of the Laplace transform, perhaps even the fascinating relationship between partial fraction decompositions and convolutions via Laplace transforms. Who cares whether the students become skilled at working out tricky problems? What matters is their getting a feeling for the importance of the subject, their coming out of the course with the conviction of the inevitability of differential equations, and with enhanced faith in the power of mathematics. These objectives are better achieved by stretching the students’ minds to the utmost limits of cultural breadth of which they are capable, and by pitching the material at a level that is just a little higher than they

can reach.

***

I looked him further, and his prose 'Indiscrete Thoughts'. From there, following on beauty in mathematics:

Mathematicians are concerned with the truth. In mathematics, however, there is an ambiguity in the use of the word "'truth.'" This ambiguity can be observed whenever mathematicians claim that beauty is the raison d'etre of mathematics, or that mathematical beauty is that feature that gives mathematics a unique standing among the sciences. These claims are as old as mathematics, and lead us to suspect that mathematical truth and mathematical beauty may be related. Mathematical beauty and mathematical truth share one important property. Neither of them admits degrees. Mathematicians are annoyed by the graded truth which they observe in other sciences.  

What is beautiful, is that which is enlightening: (Ooh!! That can be taken directly and applied to rest of the life and world too.)

The property of being enlightening is objectively atzributed to certain mathematical statements and denied to others Whether a mathematical statement is enlightening or not may be the subject of discussion.... Enlightenment is a quality of mathematical statements that one sometimes gets and sometimes misses, like truth. A mathematical theorem may be enlightening or not, just like it may be true or false.

Enlightenment is what keeps the mathematical enterprise alive and what gives mathematics a high standing among scientific disciplines.  

Mathematicians seldom explicitly acknowledge the phenomenon of enlightenment for at least two reasons. First, unlike truth, enlightenment is not easily formalized. Second, enlightenment admits degrees: some statements are more enlightening than others. Mathematicians dislike concepts admitting degrees, and will go to any length to deny the logical role of any such concept. Mathematical beauty is the expression mathematicians have invented in order to obliquely admit the phenomenon of enlightenment while avoiding acknowledgment of the fuzziness of this phenomenon. 

We acknowledge a theorem's beauty when we see how the theorem "fits" in its place, how it sheds light around itself, like a Lichtung, a clearing in the woods. We say that a proof is beautiful when it gives away the secret of the theorem, when it leads us to perceive the inevitability of the statement being proved.

***

Inspired by Rota: Perhaps, for an ideal education, a range of know-how courses, in varying degrees of depth - and a set of know-beauty or know-excellence courses across a wide range of human endeavors. Perhaps!!

Know-how to include lifeskills to high Maths, to problem solving, coding/algo thinking, to skills around software, to just a very deep engagement with something very difficult but with clean fundamentals (science, maths) which shows or guides learning how to learn. (with lifelong skill update focus)

Know-what to include excellence and beauty from literature, poetry, arts, architecture, lives, people, endeavors, businesses, models and frameworks. (with lifelong exposure to excellence and beauty focus). 

A spiralling outwards path of knowledge and excellence, ever continuing while a person lives.

Perhaps. :)