Hyperlinking thinking
A few of the often-missed, 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.)

"Buzz-saw 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

"Buzz-saw certainty"
This kind of certainty derives from logic, not from observation.  So, it can't establish "truth" in the sense of agreement with observation.  It can establish logical consistency that avoids looking silly from the viewpoint of someone who "sees" the logic.  Martin Gardner's buzz saw puzzle is our exemplar. [THE PUZZLE]  If a person continues to seek a five-cut solution after you explain the answer, you wonder about his understanding.  If he continues to seek a five-cut solution after he explains the answer . . . well, isn't that a bit silly?

Science is an art of not seeming silly...
To someone who sees something we've missed.

Mathematics is the science of seeing...
Those things we never suspected exist.

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 between-sessions 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  buzz-saw certainties of logic (which are often at the edges of human comprehension).    [TOP OF PAGE]

Boolean logic
This is the core of an elementary logic course.  You can learn Venn diagrams.  You can learn rules about "undistributed middle," and "illicit major," and truth tables.

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 = mc2 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 = mc2 have their physics meanings, not their colloqual meanings.) [TOP OF PAGE]

Models and metaphors; examples and exemplars
Many highly productive scientists are known for their intuitive models: Maxwell for his spinning wheels, ball bearings and rods to help  him understand vector derivatives of electromagnetic fields; Feynman for the Feynman diagram which visually depicts two conservation laws; Einstein for his real-world tests of the most abstract of hypotheses.

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 OF PAGE]

Statistical reasoning
The modern radical constructivist is prone to stating, "I don't know anything."  "We can be certain of nothing."  The modern statistician is prone to scratch his head in puzzlement at such statements.  Do they mean nothing has a probability of precisely 1?  Do they mean they can take no action on any knowledge, always rendered immobile by indecision?  Do they mean if one isn't perfectly sure, then one is perfectly unsure: that probabilities all have values of either 0 or 1?  These are "flip-flop" notions; very common and very wrong. [PSEUDO-REASONING]

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 real-life eventsó"Monte Carlo" techniques.  Unaware that people who understand statistics don't play the lotteries; they run a lottery or play the stock market.  [TOP OF PAGE]

Conservation laws; symmetries; mutual reciprocity
Conservation laws and  symmetry principles  include some of the most powerful concepts of modern physics.  Conservation laws come from observations.  Symmetry principles come from logic, including the logic of mutual reciprocity.  (Newtons law of action and reaction, for example.)  [TOP OF PAGE]

Proportions and ratios; orders of magnitude
Much useful physics is recognizing that many things in the world change proportionally with changes of other things, at least to a good approximation.  Current in an electrical conductor is (approx) proportional to voltage across the conductor.  Heat flow is (approx) proportional to temperature difference.  Acceleration of an object is (approx) proportional to the net force on it.  There is, hopefully, a sense of realness of proportion and ratio, a sense that makes "obvious" many implications when one encounters itóbut it is a sense that is widely missed.  Piaget had his "bug on a board" problem which he considered to look at underlying causes of difficulties with ratio and proportion.  My "mislaid camera" exam question is the bug-on-a-board in thin disguise. THE QUESTION

( 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 OF PAGE]

Extrapolation to unnatainable limit
Jean Piaget used ability to "see" Newton's first law of motion (an object retains its velocity unless a force acts to change it) without being "taught" as a indicator that certain genetic potential for reasoning has been actually achieved by children.  He set up an experiment with balls rolling across various surfaces which provide different amounts of frictional drag.  The task was to extrapolate to the unnattainable limit of no friction.  About 5% of the children were successful.  And, in a study by Jack Clement, about 5% of college physics students "saw" the principle by the end of the course: the majority retained the misconception, showed false by Newton, that "motion implies a force."  Piaget and Clement observed the same level of understanding; yet one taught and the other didn't.  (Put another way: teaching was irrelevant.)

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 OF PAGE]

Symbolic representation
Language represents experience.  Algebra uses letters to represent numbers.  Primitive peoples often confuse the representation with what it represents (the "referent").  For example, they may feel that when a disease is given a name, recovery has been substantially advanced.  But these historical "primitives" are from no more than several centuries ago at most, much too recent to suggest evolutionary or genetic advance since then.  Modern peoples often confuse the representation, money, with what it represents, value.  The representational aspect of algebra is missed by many, rendering algebra meaningless and useless.  Mathematical science rests on a foundation of symbolic representation.  [TOP OF PAGE]

Suspicion of all comparatives and superlatives
Comparatives and superlatives are properties of scalar measure, stemming from rank order.  When the measure is multicomponented, any ordering is multidimensional; rank order is not unique.  When this is not seenóand it only rarely is seenósome scalar measure must be chosen from the infinity of possible scalars to give the rank order.  Oversimplification is a near certainty. [REVISIT IT]  "All men are created equal" cannot be a statement about rank orders, or one person being "superior to" or "inferior to" another, or about people being "the same."  Its meaning must lie elsewhere. But many people do often criticize "equality of people" with statements that refer to "being the same," or "identical, like clones." To another person, who is not blind to multi-dimensional ordering, such statements are simply silly.  [TOP OF PAGE]

Systematic generation of all alternatives
When faced with a problem, such as "What factors influence sunburn and how do they work?[LOOK]" we must not rely on what we see at the moment or what we would like to see, or like to believe[LOOK].  We must look at all the possibilities, and that requires we be systematic.  Science has its roots here.  Pseudoscience lacks a good root system in this ground.  [TOP OF PAGE]

Relevance & Irrelevance
Consider the sunburn problem [LOOK], or look about you at day-to-day discussions of "controversial" issues.  Controversy is frequently built on foundations of differences in different peoples' abilities to sort through multiple influences and separate the relevant from the irrelevant.  A little acerbic Boolean critique almost always has the power to crumble some of the foundation of controversy.[TOP OF PAGE]

Negation of negation
"I could care less!" [TOP OF PAGE]


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