A bigpicture look at how to attain mastery in almost anything.
EDITED 8/30/15
INTRODUCTION
I routinely tell my cello students that mastering a particular musical concept or technical passage isn't about whether the item in question is easy or hard, it’s about whether or not it can be made familiar.
I then remind them that cello playing is roughly as complex as simultaneously rubbing your stomach and patting your head, while also jumping up and down, tapdancing, and saying the alphabet backwards!
Nevertheless, the complex act of cello playing is also very much like walking, or driving, or running up a flight of stairs, or any of a thousand other familiar human actions, in that any of them can be familiarized over time, if you so desire.
If a compound skill is too complex to grasp at a glance (if it outstrips your realtime cognitive capacity), then common sense suggests that it should be reduced to a series of smaller, more easily solvable puzzles. Once these smaller steps have been solved, then they can be layered together into more and more complex combinations until mastery is achieved. The catch is that there are four distinct forms of familiarity, and if at any point your developing understanding should become fixed in the wrong form, then your overall performance will suffer.
In this post, I introduce the four forms of familiarity and demonstrate their relationship to one another.
ROAD MAP
Draw yourself a 2x2 table, and label the rows "negative" and "positive" and the columns "static" and "dynamic."
static  dynamic 
negative 


positive 

This gives us the four forms, one per quadrant: negative/static, negative/dynamic, positive/static, and positive/dynamic.
Let's explore each form in turn, using the alphabet game "Can you guess what letter I am thinking of?" as a recurring example.
Technical note: In our example, the domain containing the answer (hint: it's in the alphabet) is known from the start, but you must still start the game without any information from within that domain. We are using a small, bounded, discrete domain for clarity, but the same informational principles also apply across unbounded, unspecified, continuous, or uncountably large domains.
QUADRANT I: NEGATIVE/STATIC
The simplest way to begin collecting information is to try guessing a letter.
"Is it A?" "No."
Congratulations! You have obtained your first piece of negative/static information!
static  dynamic 
negative 


positive 

Information from the negative/static quadrant is cheap, easily collected, offers a lowresolution view of the true solution, and comes in the form "it isn't this" or "don't do that." For some situations ("Don't raise your bow shoulder"; "Thou Shalt Not..."), including those uniquely strange parental instructions given to small children ("Hey! Stop eating the cat!"), this is sufficient.
But negative/static information doesn't really tell us very much about what TO do...
Some additional information can be extracted from the negative/static quadrant by making multiple guesses ("is it B? ...Q? ...X?"), but this strategy has its limits. The cheapness of each separate piece of information comes at a collective attentional cost, and that cost grows exponentially higher as problems grow more complex. Remembering which guesses have been tried becomes harder and harder as the number of uncoordinated guesses mounts. We quickly approach our cognitive limits, winding up right back where we started!
The problem has again become too complex to grasp at a glance.
Fortunately, that complexity is easily reduced by a shift into the second quadrant:
QUADRANT II: NEGATIVE/DYNAMIC
Ordered groups of negative/static information ("Is it A? ...B? ...C? ...D? ...E?" etc.) can be efficiently collapsed into dynamic ranges ("Is it before F?" "After M?"), substantially reducing the cognitive load of the assembled information. Therein lies the value of the second quadrant! Where the negative/static quadrant can take up to 25 pieces of information to guess the letter in question and requires all 25 to specify it, the negative/dynamic quadrant can specify the letter in two and guess it in no more than seven.
static  dynamic 
negative 


positive 

Information found in the negative/dynamic quadrant ("It isn't those," or "Don't bother with that region") is also relatively cheap, provided you trust the information source. Its compact smoothness is useful in realtime situations, where conditions are often continuous rather than discrete. "Don't play that passage in the lower half of the bow" is far easier to remember than "Don't play that passage in this part of the bow, or in this part, or in this part...".
The mental shift from the first quadrant to the second is so easy as to be practically automatic, and in situations where exact answers are unnecessary the results it produces are often “good enough.” Most daily habits live in the second quadrant.
But negative/dynamic information still isn't all that precise, nor is it streamlined enough for every situation. While it is effective in simple oneaxis problems such as our letter puzzle, it is far too unwieldy to handle complex problems with lots of parameters, such as residential construction, professionallevel auto racing, or orchestral string playing.
Sometimes life calls for a more powerful class of solutions.
QUADRANT III: POSITIVE/STATIC
The third quadrant is where the Eureka! moment is found.
When enough negatives have been nailed down, a powerful phase change occurs. Suddenly, the collected negative information crystallizes, collapsing all at once into a single, positively defined solution:
"The letter is W!"
As this phase change occurs, clouds of nolongerneeded negative definitions dissipate like smoke. An essential clarity is revealed, and the mind frees itself from unneeded clutter. The harder the problem—the more parameters that are in play—the more satisfying the mindclearing "Aha!" moment is when it finally arrives.
static  dynamic 
negative 


positive 

Information from the positive/static quadrant is densely packed and can be cognitively expensive to acquire, but it is incredibly easy to use. Its compressed precision ("do it this way") makes it invaluable in realtime situations where decisive competence is needed. (“This goes here, and that goes there.”)
Many people stop once they reach this level of familiarity (“It is the way it is, we’ve figured it out, discussion over”), but this form is not the end of the story either.
Why?
Because despite human hopes to the contrary, the world never stands still under our feet. Change is ubiquitous, even when it is very slow, and the world has a way of shifting out from under our preassembled expectations when we least expect it. Perfect circles conceived in the abstract tend to become fuzzy approximations when drawn in real life. People act very differently in different situational contexts. Language, culture, and political parties evolve over time.
Even at the shorter time scales of daily life, many of our most engaging human activities are dynamic in nature. Conversing with other people requires that we keep up with a flow of shifting topics. Your cello bow may be perfectly placed on the string, and your posture may be completely free of tension, but you are not making any music unless your bow is moving. Driving a winding road through the mountains at high speed requires anticipating just where in your lane you need to be, in combination with just when and how hard to gun the engine into the turns. Static solutions are not always enough.
The fourth quadrant comes into play when aspects of the dynamics themselves turn out to be predictable.
QUADRANT IV: POSITIVE/DYNAMIC
If the third quadrant is the quadrant of competence, the fourth is the quadrant of mastery. The fourth quadrant is entered when a whole series of positive/static data points can be collapsed into a single elegant equation, or heuristic: the simpler, the better. Once again, any unneeded information dissipates like smoke.
If the letter game were a repeated affair (“W...” “H...” “E...” “...”) with the eventual goal of guessing an entire quotation (“W H E N I N T H E C O U R S E O F H U M A N E V E N T S...“) then the equation would call up the entire quotation as a familiar whole, thereby eliminating the need to serially discover its individual letters.
[Note: Fourth quadrant equations are continuous. That is, they apply not just at the sampled points, but at all possible points within the domain they specify, and perhaps beyond. This is hugely important.
For example, while the third quadrant can tell me that my bow is perfectly (but differently) balanced here, and here, and here, and here, the fourth quadrant collects those points into a single unifying curve, comprehensively integrating the changes in bow balance, speed, pressure, positioning, and overall posture over time and space. This newfound freedom from technical micromanagement allows me to direct my limited attention to the higherlevel task of making beautiful music, which is the whole point.]
static  dynamic 
negative 


positive 

Whether the fourth quadrant equation describes a rigorously formal relationship, a nonverbal sense of events, or a purely physical action is irrelevant. Its establishment confers an intuitive fluency in any domain, an easy, flexible mastery of the material that leaves the mind clear and the body relaxed. The positive/dynamic quadrant is what enables great musicians to engage in onthespot virtuosic improvisation, and chess masters to see many moves ahead, and you not to trip over your own feet as you run up a flight of stairs, and politicians to manipulate the emotions of their constituents. Psychologist Mihaly Csikszentmihalyi (link goes to TED talk) refers to being engaged in this performance experience as “flow.”
Making the transition from the third quadrant to the fourth is not so much a matter of raw computing power—you gain back much more than you spend—as it is an act of imagination. You simply have to learn to look past the static logic of a local solution to imagine how it might vary (or not!) across wider domains, and then test for fit.
This is an important reallife skill to cultivate, because all too often things that work well here turn out not to work at all well over there.
Music students routinely rediscover this fact of life every week, when pieces that [they say] they can play perfectly well at home crash and burn in their lessons, or in competitions, or in recitals. Giving a speech to thousands of people feels different from practicing in front of the mirror the night before. Inexperienced travelers quickly learn that different cultures can have wildly varying standards of appropriate behavior. But, once you have latched onto an essential continuity, a stable equation that works across a whole range of possibilities, you can easily adapt to whatever circumstances are on the ground, intuitively flowing around obstacles that would stop a less fluent understanding in its tracks.
Once acquired, mastery can be deepened through layering. Our fourth quadrant equation can borrow from the second quadrant the idea of ranges, so that the curve it describes gains a margin of error. Another fourthquadrant equation can be layered in that draws an increasing fraction of your conscious attention to the situation whenever you approach the edges of your error margin. Further higherlevel dynamic equations can even be written that integrate two or more existing but previously unrelated equations, like, say, “tapdancing” and “saying the alphabet backwards,” or the relative motions of your bow shoulder, first finger, and the small of your back. The possibilities are endless.
Not all such fourth quadrant connections will be useful. Not all of them will be meaningful. Not all of them will be true. And almost none of them will truly apply equally well across all domains in the real world (failing to recognize this sometimes leads to nasty surprises). But any time you find anyone demonstrating intuitive mastery of even a moderately complex activity, the fourth quadrant is in play.
So, go and find it!