I can teach you to improvise

 I could teach you to improvise like this.

First, play a triad CEG on the piano with your left hand.  I will show you the notes if you don't know them. Then you would just play ideas based on that triad, as well as the scale from C to C (all the white nots on the piano) with your right hand. Every phrase should end in a C. There, you are improvising!  

Now, you would be doing this for a while until you felt you could improvise over the C triad. You will notice that you have found certain cadences to end each phrase, like EDC or GEC.  Now, I will ask you to notice how you want naturally to play phrases that answer each other.  You have invented the question answer structure, which you probably already know. You will notice too that you are playing in a certain rhythm, and the phrases will take up the same number of beats. Set a timer and improvise over C for 10 minutes. You will have ideas you like and others you don't. You will get bored and so will have to invent something not so boring, or you will find something you like a lot and delve into that.   

I never said I could teach you to improvise well, or that your ideas would be great or compelling or original. Right now, you just want them to make musical sense, and they do: they end in a logical way, they have a parallel structure. There is logic and symmetry. There is melody, because melody is just movement in the right hand of any type. Notice that you can have simplistic melodies or ones that are slightly not so simplistic. This second step is just noticing what you are doing and how easy it is. You already knew how to improvise before you started. You just didn't know you knew. 

Now, I would show you that this works in D Dorian. Play DFA in the left hand. Play phrases from D to D on the white notes and make your phrases end on D.  Now, you could do the same in A minor.  Play a G triad, and then add the F to the left hand. That is a G7. Improvise over that.  

The next step is to play a chord progression, using the four chords you know.  You can do DGCC, or CADG. All the rest is just adding new chords to what you know, and doing all of this in other keys.