The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work. - George Pólya

Life is full of surprises: our approximate condition for the fall of a body through a resisting medium is precisely analogous to the exact condition for the flow of an electric current through a resisting wire [of an induction coil<nowiki>]</nowiki>. ...
<math>m\frac {dv}{dt} = mg - Kv</math>
This is the form most convenient for making an analogy with the "fall", i.e., flow, of an electric current.
...in order from left to right, mass <math>m</math>, rate of change of velocity <math>\frac {dv}{dt}</math>, gravitational force <math>mg</math>, and velocity <math>v</math>. What are the electrical counterparts? ...To press the switch, to allow current to start flowing is the analogue of opening the fingers, to allow the body to start falling. The fall of the body is caused by the force <math>mg</math> due to gravity; the flow of the current is caused by the electromotive force or tension <math>E</math> due to the battery. The falling body has to overcome the frictional resistance of the air; the flowing current has to overcome the electrical resistance of the wire. Air resistance is proportional to the body's velocity <math>v</math>; electrical resistance is proportional to the current <math>i</math>. And consequently rate of change of velocity <math>\frac {dv}{dt}</math> corresponds to rate of change of current <math>\frac {di}{dt}</math>. ...The electromagnetic induction <math>L</math> opposes the change of current... And doesn't the inertia or mass <math>m</math>..? Isn't <math>L</math>, so to speak, an electromagnetic inertia?
<math>L\frac {di}{dt} = E - Ki</math>

The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.

We wish to see... the typical attitude of the scientist who uses mathematics to understand the world around us. ...In the solution of a problem ...there are typically three phases. The first phase is entirely or almost entirely a matter of physics; the third, a matter of mathematics; and the intermediate phase, a transition from physics to mathematics. The first phase is the formulation of the physical hypothesis or conjecture; the second, its translation into equations; the third, the solution of the equations. Each phase calls for a different kind of work and demands a different attitude.