music, math,
  and the human ear
     To me math and music and the human ear
were the keys to my 'getting' mathematics.
     Once I learned about the s-plane, 
and time to frequency domain transformations, 
then it all made a lot of sense to me.
     I could hear how frequencies sound.
I could see the time domain equation. 
I do the transformation and then see
the s-plane interpretation.
     It seems to me that if we teach
frequency domain at the same time
as we teach time domain, then kids
 will just get math a lot better. The 
frequency domain is where everything 
really happens. Time Domain equations
are a creation of the human mind. 
Frequencies can be heard. We can see 
them on an oscilloscope.  We can not see
'time' on a oscope. We know that the
length across the screen is a set period.
But when we see the frequency show up on 
a spectral graph and we hear it. . . we get
real and visual confirmation of mathematical
concepts.
We can map frequencies to a string on a guitar or
on a note on a keyboard. A time domain 
equation is a fantasy of mathematics 
because it exists from + to - infinity. 
Where as a frequency can be heard and we
can see what it does to the air or
to electricity. And so to get a child
into math, tell the child about how 
frequency is a better way to think about
signals than the Cartesian time domain
way of thinking about it.
I rewrote this a little (From when I wrote
it as a blog post many years ago) because
I tend to blog very quickly and don't
spend a lot of time with each passage.
I had said 'Time is a Creation of the
Human Mind' when I meant something more
like 'Cartisian timebased graphs' are a
creation of the human mind.  The time/displacement
graph of a sine wave only show a small part.
The the equation says t from - to + infinity.
IE:  sin(t)
The s-plane graph is much more intuitive.
And then, the z-transform, which is the mathematics
of discretization (ie how digital computers work)
give one great insight into how computers behave,
and the niquist effects.