* Searching for similar-sounding names: Try variations of "Bateman" and different spellings.
* Checking Chicago Public Schools archives: They might have records of private schools operating within the city.
* Searching historical city directories: These directories often list businesses and residents, including schools.
* Contacting the Chicago History Museum or other local historical societies: They may have resources or be able to help with your search.
Without more details, it's difficult to determine what happened to this school, as its existence isn't readily confirmed.