It is wonderful to watch her improvise, which she does twice in the brief interview. It is also wonderful to watch her take the opening theme of the Mahler 5th Symphony and improvise on it. But did you notice that in both improvisations she does the same thing? She starts in minor and ends with a shorter section in major? On a different day, she would do something entirely different, but this day, that is the way her thought went.
What do you do when you improvise? Have I got anything to say about it? I have spent countless hours improvising, most of it when I was young and playing in a band. I was forced to it because I didn't read music back then, and was barely able to pick up my part by ear, so I had to just make it up! In the early years I played mostly electric bass. Later on, on electric six-string, I improvised just as much, because we spent a lot of time jamming on blues and other progressions. 99% of it was just made up on the spot. I can still improvise a bit as I prove to myself occasionally when I am searching for an idea for a composition or working with a student. But here is the thing: improvising is not terribly interesting, at least to me. When you improvise, you mostly rely on things you know how to do: certain progressions, certain textures (like the arpeggios Gabriela Montero is using in the left hand in her first improvisation), certain melodic tricks or gestures. You have a foundation that holds it all together and on top of this you create a decorative filigree that is partly or largely improvised. But all things that your hands have done a thousand times before. A lot of improvisation, therefore, is rehashed stuff that is musically very ordinary. The extraordinary thing is how some musicians can spontaneously produce it.
But as I say, it is rarely interesting, unless you really let yourself go, as I do when I am improvising for a composition. I don't care if I hit some clangers, or go astray or whatever--I just want to see what I turn up that might be interesting. But this kind of improvisation you don't do for others, just yourself. If someone is there, you automatically turn to smooth phrases that come naturally from your hands.
Improvisation is interesting, sure, but not nearly as interesting as it seems! If you want a really good piece of music, with original construction and substance, then you probably should plan it out and write it down. That's what Beethoven and Bach did, Bach most famously in his Musical Offering.
On a visit to Potsdam to see his son, C. P. E. Bach, Bach improvised a fugue on each of six pianos Frederick the Great had just purchased for different rooms in his palace, Sans Souci. The theme, given by Frederick, was awkward and chromatic. However one can easily imagine Bach turning out different fugues in two, three, perhaps even four voices on the theme. Bach was perhaps the greatest improviser of all time as witnessed by the fact that other famous musicians such as Handel, would flee town rather than chance an improvisation duel with the Leipzig master! In any case, after he returned home, Bach prepared a collection of pieces for Frederick which he called A Musical Offering. Crowning the collection was a six-part ricercare on the theme, that not even Bach could have improvised. Here it is, with the score: