Learning without understanding: this is well illustrated by Feynman (excerpt from his autobiography "Surely You're Joking, Mr. Feynman!"):
"I often liked to play tricks on people when I was at MlT. One time,
in mechanical drawing class, some joker picked up a French curve (a
piece of plastic for drawing smooth curves - a curly, funny-looking
thing) and said, "I wonder if the curves on this thing have some
special formula?"
I thought for a moment and said, "Sure they do. The curves are very
special curves. Lemme show ya," and I picked up my French curve and
began to turn it slowly. "The French curve is made so that at the
lowest point on each curve, no matter how you turn it, the tangent is
horizontal."
All the guys in the class were holding their French curve up at
different angles, holding their pencil up to it at the lowest point
and laying it along, and discovering that, sure enough, the tangent is
horizontal. They were all excited by this "discovery" even though they
had already gone through a certain amount of calculus and had already
"learned" that the derivative (tangent) of the minimum (lowest point)
of any curve is zero (horizontal). They didn't put two and two
together. They didn't even know what they "knew."
I don't know what's the matter with people: they don't learn by
understanding, they learn by some other way-by rote, or something.
Their knowledge is so fragile."
Quote from harrytrader:
Btw I have made a basic faq on normal law notably for those who makes confusion between MATHEMATICALITY and PHYSICALITY: PHYSICALITY (PHYSICS / REALITY) HAS NOT TO CONFORM TO MATHEMATICALITY when the PREMISCES OF MATHEMATICAL THEOREM ARE NOT FULFILLED !!!
http://www.econometric-wave.com/faqs/probability/home.html.html
Is Normal Law always true ?
Normal Law is qualified as "Natural Law" because of the "Central Limit Theorem" which says that the sum of n random variables belonging to any random law as long as it has a mean and a variance, will tend towards the Normal Law as n grows. Physically it is a good approximation when among multiple causes none is preponderant so that the deviation from the main cause will be symetrical. When some causes become preponderant so as to perturbate the usual (normal) behavior of the phenomena, it is not astonishing that normal law can be inadequate. This depends on the degree of approximation needed. When a multiple-causes phenomena exhibits non-normal behavior, this is indicative of abnormal forces: anomalies detection are often the main decision criteria in some applications.