Praise be to Allaah.
It is permissible for a menstruating woman, and also a woman who is bleeding following childbirth (nifaas) to recite Qur'aan without touching it. For more information please see the answer to question no. 2564.
It is permissible for a menstruating woman, and also a woman who is bleeding following childbirth (nifaas) to write Qur'aanic verses, on the condition that she does not touch the letters, because the prohibition only has to do with touching the Mushaf, and writing is not touching.
It says in al-Jawharah al-Nayyirah, which is a Hanafi book (1/31):
It is makrooh for a person who is junub or a menstruating woman to write down Qur'aan if he or she touches the slate or chalk. But if he or she places it on the ground and writes without putting his or her hand on the written letters, it was nothing wrong with that. End quote.
It is permissible for one who is in a state of impurity, or even a dhimmi, to write it without touching it because the prohibition, as stated above, applies to touching it, and this is not touching.
End quote from Kashshaaf al-Qinaa, 1/135
Shaykh Ibn Baaz (may Allah have mercy on him) was asked:
We are female students in the girls' college and we have to memorise some Qur'aan. Sometimes the tests come during one's menstrual period. Is it acceptable to write the soorah on a piece of paper and memorise it or not?
It is permissible for the woman who is menstruating or bleeding following childbirth to recite Qur'aan according to the more correct of the two scholarly opinionsâ€¦ The same applies to the piece of paper on which Qur'aan is written if necessary.
End quote from Majmoo Fataawa Ibn Baaz, 10/209
Shaykh Ibn Uthaymeen (may Allah have mercy on him) was asked: Is it permissible to write some verses on the chalkboard without having wudoo'? What is the ruling on touching the chalkboard on which those verses are written?
It is permissible to write Qur'aan without wudoo' so long as one does not touch it. As for touching the chalkboard on which those verses are written, the Hanbali fuqaha' say it is permissible for a boy to touch the slate on which verses have been written in the places that are free of writing, i.e., so long as his hand does not touch the letters - does the chalkboard come under this heading or not? In my view this is a matter for which there is no definite answer.
End quote from Fataawa Ibn Uthaymeen, 11/214
And Allah knows best.