The Department of Education and the teachers union disagree on many things, but they agree on one basic truth: Good teachers are vital.

“It’s this amazing connection when that light bulb goes off between student and teacher that is not replicated anywhere else in life,” the teachers union president, Randi Weingarten, said. “Particularly people who have overcome adversity or made something of themselves, they’ll talk about some teacher or several teachers who really made a difference in their life. It really is interesting, this special responsibility and authority and role that a teacher has in a student’s life. Even Joel Klein talks about it.”

Indeed.

At an event last week, Schools Chancellor Klein said, “Teachers changed my entire life.” The graduate of Queens public schools spoke of teachers’ having encouraged him to “reach for the stars.”

“They didn’t dumb things down because my parents hadn’t gone to college,” Mr. Klein said. His favorite teacher at William Cullen Bryant High School was his physics teacher, Sidney Harris.

As New Yorkers prepare to elect the next mayor, they might wonder who shaped the minds of the men and the woman whose names will be on the ballot in September and perhaps in November – and how the candidates’ experiences in school might affect the way they would govern the city’s public school system. The New York Sun spoke to all of Mayor Bloomberg’s rivals about their favorite teachers.

The two candidates who were born to teachers, Rep. Anthony Weiner of Queens and the former City Council minority leader, Thomas Ognibene, named their mothers as their favorite educators.

Mr. Weiner said he never had his mother as a classroom teacher, but he said when he failed math in his second year at Brooklyn Technical High School, his mother, whom he called a “disciplinarian math teacher,” made sure it didn’t happen again.

“When I finally swallowed my pride and asked my mother for help, I realized how lucky the city was to have someone like her,” Mr. Weiner said. “She really turned me around.”

When he was growing up, he said, his friends who had his mother as a teacher would “grudgingly” call her tough but good.

“That’s the kind of teacher who ends up being the best,” Mr. Weiner, who has called in his campaign for putting more resources toward attracting and retaining quality teachers, said.

The congressman said he is constantly stopped and thanked by former students of his mother, and he predicted: “If everybody who had my mom for math votes for me, I think I’m going to be okay.”

Mr. Ognibene never had his mother as a teacher, either, but said she was “brilliant” in that role.

“She would get frustrated with me sometimes. I wasn’t always the best learner, but I think my mother had a level of patience that was unparalleled,” he said. “She just absolutely was a patient woman, and I think that’s what made her a great teacher.”

Mr. Ognibene, a Republican of Queens, said adults who once had his mother as a teacher still run across streets to hug her and “thank her for saving their lives.”

Other candidates praised teachers with whom they had close personal relationships, or teachers who taught them a special skill or way of learning.

The City Council speaker, Gifford Miller, who attended St. Bernard’s private school at Manhattan, said his favorite teacher was David Kingwood, who taught him English in eighth and ninth grades. He said Mr. Kingwood taught him a love of reading.

An English teacher also shaped the life of the Manhattan borough president, C. Virginia Fields. She said her favorite teacher was Daisey Winston, who taught her high-school English at Birmingham, Ala.

“She was very special to me in many ways,” Ms. Fields said. “Not only was she my high-school English teacher, but she was my neighbor in my community. She was my Sunday-school teacher in my church. She was my summer-school teacher. Even before I reached high school, I had been exposed to her in other teaching situations.”

The borough president said Mrs. Winston was so special because of the time she invested in young people, to make sure they had grasped concepts – whether they were religious concepts from Sunday-school class or literary concepts from English class.

“The time given. The commitment. The compassion,” Ms. Fields said. “Teachers who care – she definitely exemplified that in many, many ways.”

The “lightbulb moment” also came in English class for an investment banker, Steven Shaw, who, like Mr. Ognibene, is challenging Mr. Bloomberg for the Republican nomination. He said his favorite teacher was his ninth-grade English teacher, Rita Jean Rappaport, who taught him to diagram sentences.

“She was probably unpopular at the time because she focused very much on structure, very much on the basics of grammar and of language, but the emphasis that she had on basic skills has enabled me to learn how to write properly, learn how to compose sentences properly, and to do all those sorts of good things,” Mr. Shaw said. “So, that structure has helped me throughout my entire life.”

He acknowledged that diagramming sentences, “especially when you get into the advanced levels of it,” can get “a little trivial,” but he said it has shaped his view of education.

“I think it’s important to have a focus on the basics,” he said. “Give kids the basic skills, rather than focusing on things such as theme schools and other things like that. … Get kids taught in all those disciplines first, and then we can worry about other things after that.”

For the Democratic front-runner, Fernando Ferrer, inspiration happened later in life, when he was an undergraduate at New York University, a school he recently called “one of America’s great universities.” His favorite teacher was his philosophy professor, Hiram McClendon.

“He was a Southerner who went to Harvard. He had a very interesting, hybrid way of speaking,” Mr. Ferrer said. “He was a spectacularly wonderful man, insightful, and he gave me an appreciation of things I could never repay him for.”

The current mayor could not be reached for comment on the subject of his favorite teacher.