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One step just far enough

In my BC Calc class in high school, my teacher had "Be sure brain is in gear before commenting" posted on the wall. (Yes, you certainly could ask stupid questions in her room.) Since class started at 07:35 or so, coffee was about the only thing keeping me from wondering why she doodled those little squiggles each day.

The phrase also makes for a semi-complete analogy (I have not taken it very far, but it does look promising).
Most of the effort to think, to pay attention, is expended on focusing mental power - ensuring that mental energy is not wasted on stuff other than the problem at hand.
    The way a transmission works involves ramming a spinning thing towards another thing to get it spinning. The typical method is to press the two together (even in fluid drum transmissions - there's just a fluid buffering the stresses). The clutch is a mechanism for engaging and disengaging the transmission. For better transmission of energy (why it's called a transmission), better contact is needed, bringing the system closer to continuity - since a continuous rod is typically lossless over distances (assuming rigid stuff comprising the transmission). Better contact is achieved by ramming the two ends harder - less slippage that way.
    Now to take the analogy to a second order of separation. This is nearly directly analogous to taking your two hands and pressing them together hard so they don't rub past each other easily. Holding that position for long periods of time is hard work, and the pressure weakens. Push too hard and your hands will, on occasion, suddenly slip apart (due to tremors in muscles flexing out of sync with each other - why an outstretched hand typically trembles a bit, amplified).
    Now the background is set to complete the analogy - general cases set and second order of separation conditions defined.
      General solution: transmission pressure :: hand pressure over time, with hand pressure = clutch engagement force.
      Boundary conditions defined:
        hands slip at low pressure
        hands immobile at moderate pressures - this is the function for normal operation
        hands slip suddenly (sudden large displacement) at high pressures

So when we work to maintain a certain focus, we are straining to focus as much energy into the task at hand.
    Too little effort, and our mind slips a bit, our focus drifts. This is typical of how we think when distracted.
    When we're on task, our attention is focused such that we get our work done with little wasted energy.
    Strain too hard, and the excess energy is wasted. If there is no outlet for it, our mind slips, disengaging and causing a net loss in focus - little is accomplished.

When do we have this excess energy?
Too much caffeine makes the mind disengage. Not that this happened during the test - I was able to sink a lot of the excess energy into the thought the exam demanded (like empirically solving a 2nd order DE). But when I got back, I couldn't focus on the simple task of writing up a quick response paper. One page, double spaced, easy as pie to do. Couldn't for the life of me. Main thought processes are moving to fast to engage on such a simple task (analogue - slow speed). It's like trying to shift to 2nd while comfortably running in 5th. So I'm pulling some of the energy away through a thought differential - redirect excess thought power into something else. Like taking an analogy and proving it's general completeness.
See, too much caffeine makes the mind start slipping through too much force of thought. And once it slips, it is very hard to re-engage. It just sorta grinds, burning out or slowly cooling down from heat loss from the strain. Like your arms when pressing your hands together for a long time, the mind slowly tires and relaxes the pressure. The trick is to know when you can re-engage and ram force into it again before the engine stalls and you need a nap.

HA! I feel better now. I think I can write that paper now. :) And I only misspelt 5 words, one of which was squiggles :P And no, this took almost no effort to explain. It's not complicated at all if you visualize and cross-connect the concepts. Keep that in mind before putting the analogy down.

Comments

Teaching

Your analogy was very satisfying to read. It is frustrating to have someone introduce an analogy that folds under the slightest scrutiny while one that can be carried to completion is something I find beautiful.

Re: Teaching

W00t!!!1! Success!