Pen and paper allows quick revisions of ideas. The computer helps ensure you don't make stupid mistakes on the little details when trying to see the big picture. In my opinion, there are times and places for both tools.
Often I find myself having difficulty with a problem while caught up in using the computer, trying to write a paper in TeX or program some simulation, and when I pry myself away and write it down on a piece of paper, I'm able to solve it in minutes. Similarly, sometimes I'm in the lab with a pen and a pad, can't figure out what I'm doing, and all it takes to solve it is plugging some equation into Mathematica and having it plot a few points out for me. ("Oh, right, it diverges.")
The real way to learn about math is to realize there are parts of it which comes some ways and parts which come others. There's no one way to do it, and probably no one way will work for any one person. It's about understanding; if you don't understand, try something else.