LOS ANGELES -- Jim Horsman never thought much about a career in public schools, but in 1984 he saw a newspaper advertisement begging for mathematics and science teachers in Los Angeles and decided to try it.

Like thousands of other specialists nationwide with little or no training as teachers, Horsman was sucked into a great occupational vacuum, given a somewhat disjointed introduction to the difficulties of classroom work and left on his own.

In his case, the gamble seemed to pay off. His keen interest in mathematics and his aggressive, somewhat idiosyncratic approach to learning produced one of the most successful calculus programs in the city at Manual Arts High, a low-income neighborhood school not known for academic excellence.

Many of his students, and the few teachers he said he respects, called him a success, a symbol of the advantages of bringing rough-edged experts into the system.

Now, other students and teachers are saying quite the opposite, calling Horsman an insensitive racist whose classroom achievements have come at the expense of the self-esteem of Hispanic and black students. One black student has accused him formally of saying that "the reason blacks don't get ahead is that they are all ignorant," a charge he vehemently denies. His critics say other troubling incidents are on his record.

Despite having 17 students pass the difficult Advanced Placement (AP) calculus examination in May, Horsman has become the target of a Los Angeles Unified School District investigation and been told he may be transferred from the school and away from the program he spent four years building.

Horsman's case hones to a sharp edge a lively debate in American education about how teachers should be trained and accredited and whether the system is in trouble because teachers do not know their subject matter or how to teach it. Supporters of emergency credential programs such as the one that lured Horsman say they attract knowledgeable scholars, but critics say expertise is useless if clumsily applied.

Opinion is divided at Manual Arts, a pink stucco school of 2,500 students in a gang-ridden area of south central Los Angeles.

Arian Tescum, the student whose accusation launched the investigation of Horsman, declined to comment but said she remains in the calculus class. Kim Johnson, who dropped out of Horsman's class two weeks ago, said Horsman tried to intimidate the students by suggesting any accusations against him would only hurt them. Horsman said she misinterpreted his remark.

There were also complaints that he ordered latecomers at a summer class to do pushups and that one tardy girl broke her arm. The girl was a well-conditioned basketball player, Horsman said, adding he was stunned and sorry about the incident.

Some faculty leaders challenge even the worth of Horsman's classroom record, which few other teachers of disadvantaged students can match. "At any particular school," said Joshua Pechthalt, a social studies teacher and teachers' union representative, "even in poverty-stricken communities, there are students at the high end of the skill level who see college as a possibility and are willing to bust their behinds to get to that goal. And to get those students to achieve is not particularly difficult."

Horsman's AP class, Pechthalt said, is not as important as efforts to "teach basic reading and writing" to students "at the lower end of the scale." He charged that Horsman "has demeaned students and made them feel that they will not succeed."

Virginia Ford, head of the school's magnet and Advanced Placement programs and one of the few faculty members to whom Horsman speaks, strongly disagreed with the critics.

Her AP program began in 1981 with just seven participants, but last spring she had 150 students taking difficult national examinations for college credit in chemistry, economics, history, English, Spanish and calculus. "All of these kids are at risk," she said of the student body, which is about 70 percent Hispanic and 30 percent black. "Every single one of these kids needs help and needs support."

Many of Horsman's students have defended him. Jermaine Clare, a black senior applying to Massachusetts Institute of Technology, called Horsman "a great teacher" and said, "I've never heard him say anything negative about blacks."

"If it wasn't for him," said Eduardo Garcia, a senior, "I wouldn't be where I am right now." Garcia takes a special calculus course at USC and is applying to colleges with strong math and engineering programs.

All sides agree that Horsman, 35, a slim, 6-foot-3, socially awkward Ichabod Crane, often speaks without thinking and does not care much about other teachers. He is also a fervent anti-union Libertarian who crossed teacher picket lines in 1987 and 1989.

"He told me once that he was going to teach me how to teach math," said Phyllis Williams, a Manual Arts math teacher who is also a representative for the teachers union, United Teachers-Los Angeles.

Horsman fell into high school teaching after five years as an instructor in math and economics at the University of Southern California, where he still teaches a special summer course for his most promising high school students. He cheerfully conceded that initially he was "one of the worst teachers you ever saw" and did not learn much from the district's training program for those hired with emergency credentials.

Lecturers at the Thursday training sessions faced a room full of inattentive neophyte teachers "grading papers and polishing their nails," Horsman said. One day, he said, he challenged a lecturer who, reading from a set of papers, said, "If you believe in something, it will come true."

"So if I wanted to be a frog," Horsman recalled asking, "I could become a frog?" The lecturer glanced at his papers in bewilderment, Horsman recounted, and added, "That's what it says."

Ben Lujan, the district's director of teacher recruitment and selection, said studies show positive results from the training program for 250 uncredentialed teachers hired each year, and even Horsman acknowledged that his technique has improved with time.

He is known for beginning each class period with a quiz and taking his students through two calculus texts and hundreds of sample AP questions. Another student, Lucila Pereira, said Horsman showed "infinite patience" and taught her more math in six weeks "than I had learned in the past few years of my life."

Whether such testimonials will save him remains unclear. Marv Starer, the principal, has declined to comment on his investigation other than to say he is concerned about anything that might inflame racial tension at his school.

Horsman said he will try to do better. "Somebody once told me that most teachers are just terrible," he said, "so if you improve by just 10 percent every year, in a few years you'll be the best teacher in the school."