A few of the oftenmissed, simple but subtle, elements of effective use of knowledge. These are things especially well worth honing to make sharp tools of: (Here's a beginning of a set of tools for everyday use to address some "obvious yet unobserved" problems. Let's develop this into something useful.) 
"Buzzsaw
certainty"
Boolean logic Models and metaphors; examples and exemplars Statistical reasoning Conservation laws; symmetries; mutual reciprocity Proportions and ratios; orders of magnitude Extrapolation to unnatainable limit Symbolic representation Suspicion of all comparatives and superlatives Systematic generation of potential alternatives Relevance and irrelevance Operations upon operations; negation of negation


That's the exemplar. An example is the debate in the physics teaching literature over the common definition of energy, "Energy is the capacity for doing work." In a betweensessions conversation at an annual national AAPT meeting this comment was overheard: "I don't see how my fellow physicists could be so stupid!" She was refering to someone who had argued vigorously in favor of the faulty definition. (It has the same logical errors as does "A vegetable is a potato," an inverted implication and the improper substitution of an implication for an equivalence.) The establishment of just
what does constitute "capacity for doing work" is a matter of observation,
of meticulous experiment . . . and a matter of careful definition of terms.
All interwoven with careful adherence to rules of logical consistency.
(Don't invert implications, for example.) That leads to "theory,"
and explains why the word "theory" and "theater" come from the same roots.
A theater is a place where we observe. Theory is not "a stab
in the dark," a thing of minimal certainty. It's a structure of bricks
of observation with mortar of buzzsaw certainties of logic (which
are often at the edges of human comprehension). [TOP
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You should also notice that many people develop intuitive sense of what is correct and what is not...no Venn diagrams or truth tables needed. Wason's card selection puzzle is a good exemplar for the misunderstanding of logical implication. A common misunderstanding of this one is to think "implication" refers to "not explicitly stated; implied." The Boolean relationship is then simply missed. [DISCUSSION] The notion that E = mc^{2}
tells us that mass can be converted to energy confuses Einstein's (Boolean)
equivalence with a (Boolean) mutual exculsion. Einstein says that
mass and energy are different ways of looking at the same "thing."
Like the two sides of a sheet of paper, if one side one becomes smaller,
then the other becomes smaller: some was removed. "Convert mass to
energy" means mass decreases and energy increases: some "thing"
is either mass or it is energy, but not both at the same time. That
is a mutual exclusion. (Very important: energy and
mass
in
E = mc^{2} have their physics meanings, not their colloqual meanings.)
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These people see reality
in the abstract. They recognize when they are looking at abstract
concepts in the real world. They can use those concepts and even
extend them. They can invent them. Today, cognitive psychologists
recognize a strong role for exemplars in our reasoning. [TOP
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The gambler plays the state
lottery apparently unaware of the concept of "expectation value" of an
investment. [GAMBLING]
Apparently unaware of the nature of stochastic processes or the meaning
of complete randomness. Unaware that randomizing processes have a
valuable place in calculating reallife events—"Monte Carlo" techniques.
Unaware that people who understand statistics don't play the lotteries;
they run a lottery or play the stock market. [TOP
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( HERE is a typical question used for testing understanding of ratio and proportions.) A PBS news announcer once
commented that "Anyone who can give some meaning to all those astronomical
national budget numbers that even my grandmother can understand deserves
a Nobel prize." The next day he reported one of those "billions"
numbers and pointed out that someone had calculated the individual's share
as $5. No mention on his part of Nobel prizes. He didn't seem
to recognize that portioning out those big numbers to individual share
gives them very easy to understand meaning. He seemed not to understand
how to do such a calculation. He had encountered an "obvious" application
of ratio and proportion and had missed "seeing" it. [TOP
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Extrapolation to the unnatainable
limit of zero width of the little segments in the integration process (as
we did when calculating cone responseX)
is a key to calculus. It's also a key to understanding acceleration
and the bouncing ball problem[LOOK
AGAIN]: accleration
is the derivative of velocity with respect to time, the other main
concept of calculus. [TOP
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