At night, Ms. Mariscal had been taking home the teacher’s guide to the third-grade math curriculum, searching online for child-friendly definitions of words like “product” and “multiple.”

At Berkley Maynard, the students credited Ms. DeSantiago with improvement.

“At the beginning, she seemed kind of shy,” said Jeremiah Lewis, 11, during a break between classes one spring morning. “But she learned how to become strict and how to get what she wants.”

Four years into the residency program, principals at Aspire had seen previous graduates flourishing and were eager to hire the new trainees.

Mr. Nutt secured a spot teaching third grade again at Monarch, while Ms. Mariscal accepted a job teaching first grade at East Palo Alto, her alma mater.

There were no openings at Berkley Maynard, so Ms. DeSantiago applied for jobs at other Aspire campuses, and was invited to give a sample lesson at Golden State College Preparatory Academy, a school for sixth through 12th graders in Oakland.

When she asked the class for names, one boy gave her an alias. Channeling all that Ms. Saberton had taught her, she looked him in the eye. “I’d really like you to give me your name,” she said firmly. This time, he complied.

The rest of the lesson went smoothly, as she explained the mathematical concept of scale factor using a childhood photograph of herself with her sister. The students were hooked by her mix of personal and practical. At the end of class, she handed out a short quiz, and 22 of the 24 students made the correct calculations.

A few days later, a job offer arrived.