I also never said that he couldn't specify he was "looking at it with Detect Magic", I really don't knwo where you are finding the strings of thoughts that I didn't say.
I have no problems with players telling me they are studying an area or looking for magic or trying to ascertain magical emanations via detect amgic, anything that conveys the idea of what they are doing.
That's not what you wrote earlier. You made a distinction between looking and studying with "Merely walking and looking 'normally' is not the same as 'studying'" (and you emphasized that several times in different ways - but you were *always* using "studying" and always making it a distinction from "looking") and you wrote "I said that you must specify that you are studying the area or object". You made quite the case that "studying" was a very specific term, and that it was decidedly distinguished from "looking". Now you say looking is fine? I'm sorry for reading things as you typed them, rather than as you intended.
Now if they do not specify they are looing in a specific direction to gain information then this is what will happen (I have seen this happen too many times to count). The player wants the benefits of detect magic from the area in front of them but also wants to have the idea that his PC is likewise looking above to ensure nothing is going to fall on them. If you do not require some sort of specification as to what the PC is doing then the player will inevitably choose to have it be the most beneficial at the time, switching what his PC's intentions are from minute to minute (real time not game time) to suit his whim.
Detect Magic operates in a cone. It's 3d. Looking forward as you travel also gives you the floor, ceiling, and walls as well, especially in a tunnel. About all you lack is directly behind you (which, if you're doing this constantly, you've already scanned). Detect Magic even has some amount of object penetration. With a Permanent or Persistent Detect Magic, unless you're counting rounds for one reason or another (e.g., you're in combat, or you're trying very hard to get away from something), it might as well be 360. When you're not in a hallways, it gets repetitive, though (check ahead - any magic? Check behind - any magic? Check left - any magic? Check right - any magic? Check up - any magic? Check down - any magic? No? Okay, we proceed forward sixty feet...). But that leads to something very much like AGC's Illusion wars (also, do note that this particular method leads to an average travel rate of 10 feet per round, even when re-checking things, which is still faster than the party rogue goes when he's searching for traps for every five-foot square; the only time it goes slower than that is when the method detects magic, so the Wizard stops to examine things more closely - which is analogous to the Rogue stopping to disable a trap).
Part of the spell is "studying" but when using the persistant version then the PC must basically say that he is "studying" or "looking" at a specific direction for the "required" duration.
Now if you wish to allow free use of the directional spell that requires "study" to ascertain information and "concentration" to maintain then please feel free to do so. But do not deny that you have rewritten (or ignored) the text that is written in the PHB (and similarly used in other examples).
Outside of combat-rounds, when the action cost doesn't matter? Sure; it saves verbal paperwork, and has no effective impact on the mechanics (at least, not when using a Persistent or Permanent version) beyond rate of travel (about 10 feet per round, on average)
I do think that "study" does not necessarily fall into an action type, but rather it can be done as part of another action. Like Hide is always a part of another action.
Compare to when this started, when you said "I beleive [sic] you are reading the spell incorrectly." in reply to my "It's a Concentration-duration (max 1 min/level) effect, and it's a cantrip. With a caster level-1 wand, concentrating while you walk". That was where you heavily implying the spell requires a full stop.
Likewise, compare to when you said:
I disagree.
Merely walking and looking "normally" is not the same as "studying".
in response to my "If I have a 60-foot range, and travel 30-feet over the course of 1 round, measured from my starting point, I have been studying the range from 30 to 60 feet continuously for that 1 round - which is enough to determine whether or not magic is present in that range, and thus whether or not it's worth stopping for closer examination."
What you're saying now does not appear to sync up with what you were saying then.