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I've concluded that chess pedagogy, as commonly practiced for
over 100 years, is almost entirely ineffective.

Articles, books, and lectures usually involve a teacher going
over games, explaining the moves, etc. This sort of presentation
is pleasant for students but I doubt it improves anyone's play!

Instead, here are ten practices I think might work:

1. Play lots of games
Sounds obvious, but most students will have a hard time finding
more than one rated game a week, at least in the US. So it takes
ten years to play 500 games! A student needs 500 games his first
year. Most stories about Western chess talents involve frequent
games with family members at a young age.
There's a trope about the "Soviet school" of instruction and its
"secret methods". I propose the key to Soviet success was largely
high availability of opponents.
Last two days, Adric attended the Berkeley Chess School's winter
camp. I asked the teacher how class time is spent and he said,
"Mostly lecture, going over Fischer games." I don't think Adric
got in more than one complete game each day.
Computer opponents are readily available but they don't engage
the senses to the same degree as human opponents. They also have
trouble playing at an amateur level convincingly. Recent work on
"intrinsic ratings" (Bratko, Regan etc.) could be used to improve
their ability to play below their true strength.

2. Move count
In this exercise, you look at a position diagram and just count
the number of legal moves for one side as quickly as possible.
You can do it on a timer and try to beat previous times (for
positions with the same number of pieces on the board).
A variant is to count captures only.
Custom software could track your average error while gradually
decreasing the time for each trial and increasing the number of
pieces on the board.

3. Learn the correct values of the pieces
The commonly-taught values are inaccurate. I suppose it had to
wait for computer analysis (by Larry Kaufman) to reveal that if
a pawn is worth 1 point, a knight or bishop is worth 3.5, a rook
5.25, a queen 10, and the bishop pair 0.5!
To practice these, look at positions and calculate the material
balance as quickly as possible.
Again, custom software could help. An advanced variant is to
score a suite of positions with an engine and have the student
quickly evaluate them on 3- or 4-point scale (positions can be
chosen to fall into bins matching this scale).

4. Mate in X
Here's an existing pedagogical exercise that's effective: Books
show position diagrams, each with a caption like "White to move
and mate in 3." You have to find a sequence that forces mate in
three moves, visualizing forward from the diagram in your head.
Definitely recommended.

5. Chess Mazes
This is an interesting kind of puzzle invented in 2004 by chess
teacher Bruce Alberston. It's like Mate in X except one side
can't move. All the moves are for the same side in a row.
http://www.amazon.com/Chess-Mazes-Bruce-Alberston/dp/1888690232/

6. Solitaire chess
This interesting chess variant is like peg solitaire but with
chess pieces. It came out in 2010 or so. Very cool. There's a
physical version and an iOS app, both very well done.
http://www.thinkfun.com/solitairechess

7. Board-reversed games
I invented this in 2008. You play normal chess with the board
turned around. You dictate your move to your opponent, and he
moves the pieces. I tried to recruit opponents at my local club
but was not successful.
Many computer chess programs support arbitrary board reversal,
so this can be done without a human opponent.

8. Semi-blindfold games
One side plays without pieces. Easier than total blindfold
because the opponent's pieces help mark the position. I came up
with this recently but it's probably been used before.
A variant is for both players to play blindfolded at a real chess
set. Feeling the pieces engages motor and tactile memory.

9. Push-ups
Due to Ivo Donev. Already implemented in software!
http://www.creativechess.com/Products/Push-Up/push-up.htm

10. Speed learning
Suggested a few years ago by Mike Leahy. Software flashes
positions in rapid succession. During each presentation, the best
move in the position is animated on the board. Leahy suggests it
for memorizing openings, but based on the idea that human masters
essentially memorize millions of motifs, I wonder if it couldn't
be used for all phases of the game...
https://www.youtube.com/watch?v=BuVux1uQVjY

-Carl