It turns out the woman is a 37-year-old teacher in West New York.
The child was left on the steps of Weehawken High School last Friday.
John Fauta is the superintendent of neighboring West New York schools.
He confirms that the woman police stopped driving erratically a short time after the baby was abandoned, has been teaching math to special needs children for years.
"She was a good teacher... I've spoken to her supervisor and other who know her and they say this is completely out of character... And there were no signs on a mental issues and so forth," said Fauta.
The woman whose name has not been released is undergoing a psychological evaluation at Jersey City Medical Center.
The infant found only in a diaper, and wrapped in a towel is being treated for injuries including a skull fracture.
But superintendent Fauta says he isn't prepared to draw any conclusions yet.
"I don't want to pass judgment. You never know what the circumstances are."
Authrorities say the woman has two other children, and was out in medical leave in May. She's due to return to teaching in September.