When I was in elementary school, I went to a public school for five years and a parochial school for three. The reason given by my Roman Catholic parents for sending me to public school was that for the first two years, I was too small to walk a mile from my house to the parochial school. The parochial school had a bus, but if your house was less than a mile from school, you couldn't ride it. Our house was half a block from the cutoff point. Thus. when I was a skinny five-year-old, I just beat the January cut-off, when you had to be six or start the next year, and started first grade at the public school, which was a block and a half away from our house. During the summer after second grade, when I was seven, I came down with polio and had to miss the first month or two of third grade. The public school said I wouldn't have to repeat third grade, despite having missed a good chunk of it, so I went there for fourth grade, too. In all that time, the parochial kept its ban on short-distance bus rides, and the weather in North Dakota did not warm up.
I'm saying all this because I think I can speak from experience about prayer in school. My public school classmates--and neighbors--were mostly Lutherans of various stripes, some stricter than others, with Presbyterians and Methodists, at least some Plymouth Congregationalists, and one Jewish boy escaped from Germany tossed into the mix. I can vouch for their collective piety, although it was not the same as my own. Every morning in the public school we all stood up and recited the Pledge of Allegiance. Period. No prayers.
Then I went to the parochial school for fifth grade and enjoyed the long "short cuts" my beautiful classmate Judy and I devised--through alleys, over fences--twice a day (we walked home for lunch, too, unless it was more than -15F). We arrived at school, shed our boots and jackets in the cloak room, then stood and recited the Pledge of Allegiance, followed by the Guardian Angel prayer: "Angel of God, my guardian dear, to whom God's Love entrusts me here. Ever this day, be at my side to light, to guard, to rule and guide." Period. Even in the religious parochial school, we had the bare minimum of civic and religious piety. It's not as if we spent an hour in prayer or anything, and it's not as if we knelt down, bowed our heads, or sang hymns, either. The Guardian Angel prayer, to us kids anyway, was more like a rabbit's foot--something to have with you just in case.
As far as behavior being different in the two places--one with added prayer, one without--I can say with great assurance that the place WITHOUT the prayer was somehow better: less nasty, more fun, more open to whatever it is that makes one's small life worth singing about--good teachers, sensible curriculum (in the parochial school, all of our academic grades--say, for five classes--were divided by SIX: the sixth being attendance at daily mass and communion. go figger.) I dunno. I just think that those people who pine for the good old days when our young minds and hearts were blessed by religion are chasing an effing MYTH. The pubs were no better or worse than the Cat-lickers. The kindest and best teachers I had were both Protestant and Catholic. There was no preponderance of beatitude on either front.