Actually, he's having me play the chords with my left hand and the melody with my right hand. He's having me try a lot of different ways to voice the chords, which is confusing for me because I can never decide which to use. He keeps saying that it just depends on the context and style - which I'm sure is true - but what kind of context and style is it when you're just learning jazz?
Well you tell me...
What led you to want to learn to play jazz in the first place? You must have heard some music you liked, and felt like you wanted to reproduce something in at least that general ballpark, on the piano. What music was it? What kind of jazz do you want to be able to play? What kind of jazz do you LIKE?
In my experience as a teacher, it's the students who had some answers to these questions - who had a desire for and curiosity about a certain kind of sound and style - who made the most progress.
I'm not trying to be obstructive but the fact is that choices of which voicing to use DO come down to what kind of style and sound you are trying to create. There are voicings that sound like Count Basie, voicings that sound like Bill Evans, and voicings that sound like McCoy Tyner. Using the wrong voicing for the context will sound wrong, even though in the right context it would sound right.
There are however some general technical factors that are common to most styles, and clearly in the early stages these need to be covered. They can basically be summarised in three areas: chord sonority; root progression; and voice leading.
What we've been discussing so far is basically chord sonority: what approaches to voicing a chord sound OK and why. We're talking about chords in isolation, not how they lead to each other. So your first lesson there is that for a 7th chord, you really need to play the 3rd and 7th (or at least if you don't, you fundamentally alter the identity of the chord). You don't NEED to play the 5th, or the root (depending on context and whether someone else is playing it).
But you've then got an artistic decision to make about whether you're aiming for a 2-note, 3-note or 4-note sound. Obviously just 2 notes is very thin. 3 note chords are generally more sonorous and easy on the ear. 4 notes nice and chunky.
I'm also curious about how the scheme you're referring to pans out when you DO change chords. Say you start a song with a C7 chord and voice it C-E-Bb (bottom up) in the LH. When the next chord symbol says F7, what do you do? What you've described suggests F-A-Eb (bottom up), but then you're just sticking unconnected root position chords around and not voice leading at all. So what do you do?