My understanding of the new moon is the same as yours. The moon should be between the Earth and the Sun, so you shouldn't be able to see it. My first thought on how this *might* be possible was that I think there is about a 5 degree difference in the moon's orbit and the earth's orbit. So, if the moon was at its farthest separation from the e earth's orbital plane, could it be possible to see any reflected light? I think that 5 degrees, however, would only come out to be equivalent to a change in the moon's relative position by a half day or less, so I don't think that would do it. My guess as to what your student saw was the moon just before or after the new moon, where there is some reflected light, and he saw the "Earthshine." If he does believe he saw the new moon, it's probably worth further investigation with folks who know more about astronomy. Good luck!

I have also seen a new moon as a whole disk. There is a very thin crescent sliver facing the sun, but the rest of the moon also shines faintly, even though it is night there. With binoculars you can even see features on the night side of the moon! The light comes from Earthshine, which is strongest during a new moon, because then the Earth is full. You can verify this yourself using an Earth-moon-sun diagram:

The Earth's phase as seen from the moon will be the opposite of the moon's phase as seen from the Earth. Seeing the moon lit up by a full Earth is tricky, though, since the moon will be between the Earth and the sun (as you said). Look in the twilight right after sunset of just before sunrise, and hope the brightness of the sky does not wash out the night side of the moon.

Well, to be wrong once every 45 years is not so bad after all. But, even though the new moon is not directly illuminated by the sun, there is some light that is reflected off Earth and this very faint light can actually weakly illuminate the moon. It is faint and generally hard to see but because the albedo of earth is relatively high (ice and water) the earthshine can light up the moon. This is aided by the fact that 80%of the lunar surface is covered by a high albedo rock called anorthosite.

On a lunar calendar, on the first day of the month, you can (by definition really) see a very thin crescent moon right around sunset. During the day, the waning crescent would get thinner and thinner as the Moon gets "closer" to the Sun until the time when the Moon and Sun are as close in the sky that they will be: new moon. Then the Moon starts moving ahead of the Sun and you start getting a waxing crescent which you can easily see once the Sun goes down. I've noticed that you can also see what looks like a dim outline of the rest of the moon. I think this comes from light that has been reflected multiple times, assuming it's not just an optical illusion.

Presumably, if things were just right and the new moon occurred at around noon local time, you could have gotten up early that day and seen the last bit of waning crescent too. The local time at which new moon occurs would, of course, would depend on where on the Earth you are when the Moon and Sun cross paths. So the thickness of the crescent you can see at sunset will change on a month to month basis since the time it takes the Moon to complete a cycle is not an even number of days.

Sorry for rambling on. Anyway, I haven't really tried to look for a "new" moon during the day. I guess if you knew exactly where the moon was and could block out the sun you might be able to see something. It would probably be very difficult. It is, of course, quite possible to see the moon in other phases during the day. Maybe you should have everyone in your class figure out when the moon rises and sets each day for a month and go look for it every day (during times when they aren't supposed to be asleep!)

Actually, it is definitely possible to see the 'new' moon, it is illuminated dimly from our perspective by earth shine -- i.e.light reflected from the earth (predominately clouds -- but also blue light from the oceans and red light from the landmasses). I have seen it dimly as well.