[FRIAM] Learning curves (was, Abduction)

[lrudolph at meganet.net]

Nick wrote, in relevant part,
> This reminds me of the misuse of the "learning curve"
> metaphor.  People speak of a steep learning curve as something to be
> feared.  In fact, people who learn quickly have a steep learning curve.

Behold, complete with ASCII art (so be ready to view this in a monospaced
font, or forever hold your peace), an ancient USENET post of mine to
alt.usage.english, from 1995 (!):

===begin===
Robert L Rosenberg (rros... at osf1.gmu.edu):
>: A learning curve should be the graph of a non-decreasing function (time
>: on the horizontal axis, knowledge of the topic on the vertical axis).  A
>: fast learner would have a generally steeper learning curve than a slow
>: learner.  At least that's the way I've always pictured it.

kci... at cpcug.digex.net (Keith Ivey) writes:
>I agree that this makes sense, but it doesn't seem to correspond with
>the way the phrase is used.  In my experience, something that is hard
>to learn is said to have a steep learning curve.

Rosenberg's explanation not only makes sense, it accords with the
original use by rat-runners and other operant conditioners (cf.,
e.g., _Psychology_ by James D. Laird and Nicholas S. Thompson, p. 164:
"The ... steeper the curve, the faster the animal is learning").
More precisely, *during an interval of time where the curve is
steep, the animal is learning quickly*.

The present use is muddled; as Ivey points out, "something
that is hard to learn is said to have a steep learning curve."
Here's how I unmuddle it (but I don't know what, if anything, is
going on in the heads of most people who use the phrase): by the
Mean Value Theorem, or common intuition, if a (smooth) nondecreasing
function f(t) with f(0)=0 and f(1)=1 is "steep" (has large derivative)
somewhere, then it MUST be "flat" (have small derivative) somewhere
else.  Typical learning curves (I gather from the illustrations in
Laird and Thompson) look either like Figure A or like Figure B:

                            x                                    o
                    x
               x                                                o
           x
         x                                                    o

       x                                                    o
                                                        o
                                                 o
      x                                o

              FIGURE A                          FIGURE B

In the first case, you learn almost everything in a short period of
time near the beginning of the training, then reach a plateau and learn
the rest very slowly.  In the second case, you learn very slowly for a long
time, then take off near the end of the training.

So the question is reduced to another one: which of Figures A and B is
a "steep" curve to the average speaker?

Lee Rudolph

===end===