All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
A menstruating woman is prohibited from reciting or touching the Holy Quran, as stated in the Fatwa No. (357). In addition, being ashamed to tell the recitation teacher doesn`t make a sound excuse that allows you(Questioner) to commit such a sin. Rather, you shouldn't be ashamed to abide by the rulings of Sharia.
We advise you to ask the school principal to exempt you from attending the recitation class when you are in menstruation period to spare you the embarrassment.
In addition, teaching the Holy Quran to women should be entrusted only to a woman.
In conclusion, if a woman had mistakenly recited the Quran during her menstruation, then she must repent, ask for forgiveness and avoid committing that sin again. And Allah knows best.