I'm reading about all the Ivy League Universities that many of the founding fathers attended and I was wondering how these universities knew if they were qualified or not. For example, Alexander Hamilton attended Columbia University, but he had recently immigrated from a Caribbean island. How did that work? How did Columbia know that he was Ivy League material?
In effect, they had their own version of a standardized test.
The colony colleges [Harvard, Collegiate School (Yale), College of New Jersey (Princeton), King's College (Columbia), University of Pennsylvania, College of Rhode Island (Brown), Dartmouth, College of William & Mary, and Queen's College (Rutgers)] all had some version of an admissions exam that required a student to sit with the college president and a few professors and respond to questions about Greek/Latin, some math, and possibly science.
One thing to keep in mind is that the average age of a Harvard student in 1810 was 15 1/2. College wasn't college in the way we think of it today. So it's less that Columbia knew Hamilton was Ivy League material and more than there was space for him in first year classes and he demonstrated basic proficiency during the admissions interview. Some students enrolled in the colleges after attending a preparatory school like Boston Latin while others got most of their education from private tutors.
This response goes into more detail about the entrance interviews and enrollment criteria but basically, if a young man could answer the questions, pay tuition, and there was room for him... he got in.