Archive for the 'Teaching' Category

December 26, 2016

“Una volta archiviati i vari professor Vinson, comincerai ad avvicinarti sempre di piu’ — ammesso che tu lo voglia, e che sappia cercare, e attendere — al genere di conoscenze che finira’ per occupare un posto molto, molto importante nel tuo cuore. Tra le altre cose, scoprirai di non essere stato il primo a sentirsi confuso, e spaventato, e perfino disgustato dai comportamenti umani. Non sei affatto solo, in tutto questo, e scoprirlo sara’ emozionante e stimolante. Tanti, tanti altri uomini hanno provato lo stesso turbamento morale e spirituale che provi tu ora. Fortunatamente, alcuni di loro hanno messo quei turbamenti per iscritto. Tu imparerai da loro… se lo vorrai. Cosi’ come un giorno, se avrai qualcosa da offrire, qualcun altro imparera’ da te. E’ un magnifico accordo reciproco. E non e’ istruzione. E’ storia. E’ poesia.”

“Non sto cercando di dirti che solo gli uomini colti e studiosi possono dare al mondo un contributo prezioso. Non e’ cosi’. Dico pero’ che gli uomini colti e studiosi, sempre che siano brillanti e creativi — cosa che purtroppo accade di rado — tendono a lasciare di se’ una traccia infinitamente piu’ preziosa di coloro che sono semplicemente brillanti e creativi. Tendono a esprimersi con maggior chiarezza, e hanno di solito la passione di seguire i propri pensieri fino in fondo. Inoltre, cosa ancora piu’ importante, nove volte su dieci sono molto piu’ umili dei pensatori non dediti allo studio.”

“C’e’ poi un’altra cosa che lo studio accademico ti regalera’. Se lo porterai avanti per un tempo significativo, comincera’ a darti un’idea delle dimensioni della tua mente. Che cosa sia in grado di contenere, e che cosa magari no. Dopo un po’ ti sarai fatto un’idea dei pensieri che stanno bene addosso a una mente della tua taglia. Questo puo’ innanzitutto farti risparmiare moltissimo tempo che altrimenti perderesti a provare idee che non ti si confanno. Comincerai a conoscere le tue reali misure, e a vestire la tua mente di conseguenza.”
~ “The catcher in the rye”, Salinger.

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The traditional mathematics professor

June 4, 2011

« The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to turn his back on the class. He writes a, he says b, he means c; but it should be d. Some of his sayings are handed down from generation to generation. “In order to solve this differential equation you look at it till a solution occurs to you”. “This principle is so perfectly general that no particular application of it is possible”. “Geometry is the art of correct reasoning on incorrect figures”. “My method to overcome a difficulty is to go round it”. “What is the difference between method and device? A method is a device which you use twice”. »
~ “How to solve it”, G. Polya.

June 4, 2011

« The future mathematician learns, as does everybody else, by imitation and practice. He should look out for the right model to imitate. He should observe a stimulating teacher. He should compete with a capable friend. Then, what may be the most important, he should read not only current textbooks but good authors till he finds one whose ways he is naturally inclined to imitate. He should enjoy and seek what seems to him simple or instructive or beautiful. He should solve problems which are in his line, meditate upon their solution, and invent new problems. By these means, and by all other means, he should endeavor to make his first important discovery: he should discover his likes and his dislikes, his taste, his own line. »
~ “How to solve it”, G. Polya.

June 1, 2011

« The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.
[…]
The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach. »
~ “How to solve it”, G. Polya.

On teaching

April 15, 2011

« Vigorous mental work may be an excercise as desirable as a fast game of tennis. Having tested the pleasure in mathematics [the student] will not forget it easily. »

« The best that the teacher can do for the student is to procure for him, by unobtrusive help, a bright idea. »

« A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution. »

~ “How to solve it”. G. Polya.

January 21, 2011

« “What was that?” enquired Alice. “Reeling and Writhing, of course, to begin with”, the Mock Turtle replied; “and then the different branches of Arithmetic — Ambition, Distraction, Uglification, and Derision”. “I never heard of ‘Uglification'”, Alice ventured to say. “What is it?” The Gryphon lifted up both its paws in surprise. “Never heard of uglifying!” it exclaimed. “You know what to beautify is, I suppose?” “Yes”, said Alice doubtfully: “it means — to — make — anything — prettier”. “Well, then”, the Gryphon went on, “if you don’t know what to uglify is, you are a simpleton”. Alice did not feel encouraged to ask any more questions about it: so she turned to the Mock Turtle, and said “What else had you to learn?” “Well, there was Mystery”, the Mock Turtle replied, counting off the subjects on his flappers — “Mystery, ancient and modern, with Seaography: then Drawling — the Drawling-master was an old conger-eel, that used to come once a week: he taught us Drawling, Stretching, and Fainting in Coils”. »

~ “Alice’s adventures in Wonderland”, L. Carroll.

Student wisdom

December 18, 2010

At the very beginning of his final exam, a student writes: “It was not one of my better days”.
At the end of his final exam, a student writes: “3 minutes left? WTF…”.

Proving NP-Completeness

November 20, 2010

« To prove NP completeness: (1) Ask if the problem can be discretized, e.g. ability to break analog audio into discreet samples. If no, may be NP complete. (2) Is there a finite way to solve this (even if looping/recursion involved)? Can the problem always be solved this way? E.g. using recursive calls to find Fibonacci numbers at nth place. If no, NP complete (though further analysis may make this into sub-function of larger algorithm). (3) Is the solution always the same no matter what tweaks are made? If no, the problem has many routes to solve it. E.g. FFT analysis since FFTs can vary in accuracy of frequency analysis, allowing some speed. » ~ From an undergraduate midterm exam.