8

In 1969 Lang wrote an article for the Columbia Daily Spectator, Don't Blame Us if You Flunk Math (Volume III, Number 4, December 8, 1969). The phrasing of the subline illustrates how much the times have changed:"A fifteen minute quiz raises questions what kind of people should be taught what kinds of math at Columbia". He provides a basic introductory test ...


6

Mathematicians rarely describe the process which led them to their discoveries. One notable exception was Euler. Some books on the subject written by great mathematicians are: J. Hadamard, The Mathematician's Mind. The Psychology of Invention in the Mathematical Field (English translation). G. Polya, a) Mathematics and plausible reasoning. b) Mathematical ...


5

Students learned theorems, propositions and their proofs. A teacher would call a student to the blackboard and ask to reproduce a proof. A shorter test would be just to state the theorem. Many students memorized theorems and proofs by heart. A good (but not popular) teacher would make his own picture on the blackboard, with his own notation, different from ...


5

This question comes up often, but there is no up to date scholarly study of it that I know of. The most comprehensive recent accounts (and they are rather brief) seem to be Jeff Miller's Earliest Uses of Symbols of Operation and Peterson's post on Math Forum, parts of which are aped on various pop-sites, often without attribution. But Peterson admits:"I have ...


4

The first mathematician who did that was Archimedes, in his The Method of Mechanical Theorems (usually known as The Method).


4

Cédric Villani's Living theorem does exactly that. A great read for scientists, even non-mathematicians.


2

but for how long has PEMDAS been widely taught in high school mathematics classes? I assume this will differ for different parts of the world, so please include what countries or regions you can speak for. The first appearance of an explicit PEMDAS rule in Dutch is in an appendix of a textbook on algebra for the military academy (1838), aimed at military ...


2

Klaus Weltner and Martin Ingelman-Sundberg (from the Department of Physics at the University Frankfurt) in an ultimately unpublished paper, “Physics of Flight — reviewed”, submitted to the European Journal of Physics in 2003, argue that the original source for the misconception of “equal transit time” was, somewhat unwittingly, aerodynamics pioneer Ludwig ...


2

This explanation has probably been independently 'rediscovered' many times, and it is unlikely to be able to point to a particular origin. Ackroyd says in 'Babinsky's Demonstration: The Theory of Flight and Its Historical Background' (Journal of Aeronautical History, 2015): Imagine that two adjacent air elements ... A and B ... are about to reach the ...


2

The asymmetric shape (not its explanation with "equal transit time"!) comes from the first mathematical theory that explained the lifting force (Chaplygin's formula, known in the West as "Blasius theorem"). I suppose that the theory of "equal times" was developed in attempts to explain the Chaplygin-Joukowski theory to non-mathematicians:-) Under certain ...


2

The popularity of the equal transit-time fallacy is a bit more complicated than a mistake spreading from a single source. It is simple, intuitively appealing (blowing over an airfoil is often invoked, along with an erroneous picture of flow lines around an asymmetric wing), and gets things done quickly. Just like the "explanation" of seasons by the Earth's ...


1

Alexander Grothendieck, without a doubt one of the most creative mathematicians in the last 100 years, wrote a voluminous manuscript Récoltes et Semailles about (among other things) his approach to mathematics. Since he later requested that it be withdrawn from the public, it is hard (but probably not impossible) to find the full text on the internet. One ...


1

Perrin originally defined the Avogadro number as the number of atoms in one gram of hydrogen (later it was redefined as the number of atoms in 12 grams of carbon-12). As such, it was certainly discoverable from measurements. However, picking grams, or hydrogen was not "natural". Neither was picking circles and diameters in the definition of $\pi$. Once they ...


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