> interesting

It worked for me, in violin,
As we know, things should
be as simple as possible but not simpler.

My approach to learning mathematics is
quite different from learning to play violin
but also is based mostly in independent study.

In violin, I got started when I was a math
grad student at Indiana University. Of
course there is a terrific music school there,
and at the time my dorm was next door with
a lot of music students. One day one of the
violin students, darned good, a Stern protege, put
his violin under my left chin, and I was off
and running, took a violin course in the
music school.

After some years of work from Galamian's book
and just self-teaching, I went for
some lessons. Sure, I started with the
"Preludio" of the Bach E major partita.
Of course early on there is a fast shift
from 1st position to 5th; I happened
to get the note after the shift right;
and the teacher exclaimed "You could
play in an orchestra!". Gee, that shift
is the main issue for playing in an
orchestra? I doubt that!

Maybe he thought I was sight reading that
music! Not a chance! I'd worked hours
on each note, note by note, checking
intonation over and over, etc. That
shift I'd likely done 1000 times.

I hadn't really been learning just the
piece but had been using the piece
as an exercise to learn to play violin,
starting from very little!
So, that shift to 5th position,
at two places in that music,
was the first I'd learned. Sure,
I could have used that skill with
that shift elsewhere. I learned
the whole piece; the learning was
a lot of fun.

But, I learned it, self-teaching.

I had a good ugrad math major but didn't
like what the IU math department had me
doing: Of the three courses they started
me with, two of them had just what I'd already
learned in ugrad school, and the third
started with what I'd learned in an NSF
summer program the previous summer.
But there was other material I wanted
to learn. So, IU and I didn't get along,
and I went, got a job, and studied on
evenings and weekends, independently.

That study worked out well and was,
really, except two graduate courses
all I needed for my Ph.D. I did the
research for that independently in my
first summer after the two courses.

So, my approach to learning has worked.

Computer science and programming? I've
taught college and grad courses in it,
but I never really took one.

What I'm saying did work for me.
Of course if you want then you can
try more complicated ways.

For the OP, she was concerned that
her playing sounded mechanical.
Well, once she has the notes on time
and in tune and practices until
they are "in her fingers" well enough
that she can concentrate on the expression,
then she should, just think about
how she wants it to sound.

For convex minimization, once a guy
had a 0-1 integer linear program
with 40,000 constraints and 600,000
variables. He'd tried simulated
annealing without much luck.

I looked at his constraints, and
16 of them were special, and I put
them in the objective function
with Lagrange multipliers.
Then the dual problem was
to maximize a concave function, and
I did the linear programming and
supporting hyperplane approach
I outlined. After about 500
primal-dual iterations in 900
seconds on a 90 MHz PC, linear programming
via the IBM OSL, I found a feasible
solution, guaranteed from the
duality, to be within 0.025%
of optimality. It was fun!
But I do recommend the central
cutting plane idea.