Can we use slash in the phrase "over the two periods, 2000/01 and 2001/02"? If not, what is the appropriate punctuation in this case?
No, no one will understand you. Write instead:
over the two periods, January of 2000 and February of 2001,
I'm not entirely sure those are "periods", per se. It seems better to just say the two months:
during the two months of January of 2000 and February of 2001
Perhaps you mean from the start of the first month till the end of the second one; if so, write this:
during the 14-month period extending from January 1st, 2000 through February 28th, 2001
People used to ISO dates might instead write that in this fashion:
during the 14-month period extending from 2000-01-01 through 2001-02-28
Anytime I've run across this being done, it is usually with both years in full and a dash/hyphen between them. Ex. The 2001 - 2002 school year
Actually, I take that back- I see it done without the full second date as well.
Ex. The 2001-02 school year.
If you were to ask me, I would tend to prefer the second way over the first; I just think it looks better. I don't know if the first way is considered "correct" but I do believe the second way is, if not "correct" then at least in acceptable common usage.
I could maybe get with using a hyphen when referring to a period of time- from 2001-04 for example... and using a slash to mean a year-period which starts and stops in the middle of the year. It is also treated as one thing; the 2003/04 school year is said, "the two-thousand three (/) four school year" while 2003-06 would be "Two thousand three TO six"
So I guess my final word is that you may use a slash when referring to a year long period of time that starts in the middle of, say, 2002 and ended in the middle of 2003. "The two-thousand-two (/) three school year" is referring to one year's length of time and therefore the slash is used to join the two school years into one length of time.
Make sense? Hope so! :)