With much effort and strain he can play the keys without arm movement.
If your asking him to depress a key with a still arm its not half surprising that you'd end up with flat fingers? we're talking a bout collapsed PIP and DIP joints right? so the only motion that's being used is coming from the MP joint? ...which is really not a problem (infact - YAY! - since you don't want to breed conscious 'curling' [DIP/PIP making a fist motion])
...if you tell a child not to move their arm they'll make it go rigid, and not use it to support the fingers movement.. they'll proceed to grab at the key, kind of pulling back, and their fingers will totally collapse because they have no idea how to balance their arm freely and in the right position.
How old is the child? This may or may not be applicable..
Test it yourself first to ensure you understand whats going on anatomically and what the child will feel.. (that is very important, I guarentee most people will misinterpret this, form you're own idea and opinion about whats going on)
..ALSO, this is not a be all and end all solution, it is merely one possibility that will have to be refined over many years and a great deal of music. I'm sure I'll also spark some technique debate with people saying I'm insane.
We are going to play two notes together, a chord.. say C and D, fingers 2 and 3. Relax the fingers, push forward with the arm. If the fingers are relaxed, all joints will freely move, the fingers will curl under the hand.. The keys will not depress unless you let your arm's weight move down into the keys.
Then you can pull the arm back out, the fingers will extend.. this happens because of their resistance against the key, nothing to do with moving the fingers, just because the finger tip is connected to the key.
Build up a circle of motion with the forearm. So the wrist is slightly higher on the way in and slightly lower on the way out, this ensures a relaxed/supple wrist - and teaches basic phrase shaping. There is no need to be depressing keys yet - this should be fluent, no stiffness before you start worrying about the keys.
Now, as you push in to the keys - very slowly, and starting with a
relaxed natural curve in the fingers - add a 'firmness' (this can be difficult to explain to especially young children) to the fingers, they will remain in the curve position and the keys will depress as the arm moves forward. It need only be very minimal, just enough to ensure the key depresses, no more, and can largely be released once the note sounds. Most importantly, the firmness is not created using a flexion or extension of the fingers similar to what would be used to make or undo a fist - I think (I'm not a physiology professor) it is more to do with the interossei, found in the hand, these are responsible for sideways motion of the fingers.. when both left and right (either side of 1 finger) are used at the same time the result is a firming of the finger, or slight extension type motion if you push it to the exteme - (don't push it to the extreme)
The arm push/pull element, and level of "firmness" will pretty much always need to be reduced once the overall pattern is understood - which is why this is difficult to do with really young kids, because they are not familiar enough with their bodies to understand what you mean.
Once this is achieved you can consider also adding other movements, finger flexion and forearm rotation as examples. Though in this example that will be too much for the student I suspect.
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Once the chord can be played, this process can be used to teach the legato element.. allowing one finger to follow the other, and leads into the use of chord attack and parallel sets to develop technique, alongside pushing and pulling arm circles.
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Lastly, you wouldn't use this level of detail unless its absolutely necessary. You don't want to go sending the student into over thinking things they are already doing.. Or causing a misinterpretation that results in doing something they shouldn't. Exercise with caution.
EDIT:
I will add that this is how one might explain that process to an adult.. you will need to come up with ways to explain and visually demonstrate it that works for a child.