It's a Wednesday morning, and Zenaida Tan is warming her students up with a little exercise in "Monster Math."

That's Tan's name for math problems with monstrously big numbers. While most third-graders are learning to multiply two digits by two digits, Tan makes her class practice with 10 digits by two -- just to show them it's not so different.

On this spring day, her students pick apart the problem on the board -- 7,850,437,826 x 56 -- with the enthusiasm of game show contestants, shouting out answers before Tan can ask a question. When she accidentally blocks their view, several stand up with their notebooks and walk across the room to get a better look.

The answer comes minutes later in a singsong unison: "Four hundred and thirty-nine billion, six hundred and twenty-four million...."

Congratulations, Tan tells them, for solving it con ganas. That's Spanish for "with gusto," a phrase she picked up from watching "Stand and Deliver," a favorite film of hers about the late Jaime Escalante, the remarkably successful math teacher at Garfield High School in East Los Angeles.

The Los Angeles Unified School District has hundreds of Jaime Escalantes -- teachers who preside over remarkable successes, year after year, often against incredible odds, according to a Times analysis. But nobody is making a film about them.

Most are like Zenaida Tan, working in obscurity. No one asks them their secrets. Most of the time, no one even says, "Good job." Often even their own colleagues and principals don't know who they are.

As part of an effort to shed light on the work of L.A. teachers, The Times on Sunday is releasing a database of roughly 6,000 third- through fifth-grade teachers, ranked by their effectiveness in raising students' scores on standardized tests of math and English over a seven-year period.

The findings are based on an approach called value-added analysis, which is designed to allow fair comparisons of teachers whose students have widely varying backgrounds. Although controversial, the method increasingly has been adopted across the nation to measure the progress students make under different instructors.

L.A. Unified has had the underlying data for years but has chosen not to analyze it in this way, partly in anticipation of union opposition. After The Times' initial report this month showed wide disparities among elementary school teachers, even in the same schools, the district moved to use value-added analysis to guide teacher training and began discussions with the teachers union about incorporating data on student progress into teacher evaluations.

The results of The Times' analysis are not a complete measure of a teacher by any means, but they offer one way to see whether an instructor is helping -- or hindering -- children in grasping what the state says they should know.

The Times found that the 100 most effective teachers were scattered across the city, from Pacoima to Gardena, Woodland Hills to Bell. They varied widely in race, age, years of experience and education level. They taught students who were wealthy and poor, gifted and struggling.

In visits to several of their classrooms, reporters found their teaching styles and personalities to differ significantly. They were quiet and animated, smiling and stern. Some stuck to the basics, while others veered far from the district's often-rigid curriculum. Those interviewed said repeatedly that being effective at raising students' performance does not mean simply "teaching to the test," as critics of value-added analysis say they fear.

What's clear from the data is that these teachers have an immediate and profound effect on how much children learn. On average, their students leapt 12 percentile points on tests of English, from the 58th to the 70th. In math, the gains were more stark: a 17 percentile point jump, from 58th to 75th. All in a single year.

The idea of publicly rating teachers by name has generated enormous controversy among educators and experts across the country. The debate has focused on whether the method is sound and the publicity is fair to those with low rankings.

Often lost in that discussion are the benefits of singling out those who consistently succeed.

U.S. Education Secretary Arne Duncan said as much in a speech last week, denouncing a culture in public education that has long been averse to talking about success stories.

"The fact is, rather than shining a light on effective teachers, our education system hides them," he said.

Unsolved mystery

Experts have long known that highly effective teachers can overcome the challenges students face both inside and outside of school. But why they are so successful -- and whether their skills can be passed along to others -- remains largely a mystery.

Most of the things districts track about teachers -- their age, years of experience, education and credentials -- do not appear to matter much, at least when it comes to raising students' performance on tests.

What does matter? Is it chemistry, technique, dedication, rigor? Might it be a thousand smaller, almost invisible things, depending on the subject and type of students?

Hundreds of books purport to answer those questions, but no clear consensus has emerged. And few of the competing theories have been rigorously tested, said Thomas Kane, a leading education researcher at Harvard University.

"It's very difficult for an individual teacher to distinguish between the valuable suggestions and the snake oil," he said.

That's in large part because there is no agreement on how to identify the best teachers. It's something Kane and other education researchers have spent much of the last decade trying to sort out.

In a seminal study in 2008, Kane and a colleague set out to experimentally test the reliability of the value-added approach, which assesses a teacher's effectiveness by measuring the year-to-year gains of each student on standardized tests.

Among other things, some researchers had been concerned about the wide variation in value-added results for individual teachers from year to year, the potential for error in the findings and the possibility that the results would be skewed by how students were assigned to classrooms.

In Kane's experiment, conducted at Los Angeles Unified with administrators' permission, 156 district teachers who volunteered for the project were randomly assigned to classrooms. Kane and his colleague tried to predict, using value-added analysis, how students would do under those teachers. The projections were then compared with the students' actual results.

The conclusion: Value-added analysis was a strong predictor of how much a teacher would help students improve on standardized tests. The approach also controlled well for differences among students, the study found.

With \$45 million from the Bill and Melinda Gates Foundation, Kane and other researchers are now following 3,000 teachers in six school districts to see if other types of evaluation -- including sophisticated classroom observations, surveys of teachers and reviews of student work -- are also good measures of teacher performance.

In the meantime, Kane said that, although it is not perfect, "there is currently not a better measure of teacher effectiveness than the value-added approach."

Identifying the most successful teachers is merely a first step. The next is to study them closely and find ways to pass their techniques on to others.

It is no simple undertaking.

Some of the top teachers in The Times' analysis said in interviews that they weren't sure exactly what made them effective, or were skeptical that whatever it was could be distilled and passed on.

Diane Hollenbach, who recently retired from Broadous Elementary School in Pacoima, was the most effective elementary school teacher of the roughly 6,000 analyzed by The Times.

"The ones that love their students and love their job do well," she said. "You can't bottle that, and you can't teach it."

Jilla Sardashti, who taught last year at Parmelee Avenue Elementary School in the Florence-Firestone neighborhood, said she teaches critical thinking skills from the first day of school.

"These kids are as smart as any other kids in the district," said Sardashti, whose students are mostly poor and Latino and often still learning English. "I'm really good at figuring out what they need, and I provide them with experience to know about the world around them."

Hollie Bloch, who retired in July from Balboa Gifted Magnet in Northridge after teaching in the district 39 years, said that challenging students -- especially high-achieving ones -- was essential.

"I teach Shakespeare to children," she said. "If the teacher's expectations are high, and you have control of the classroom, those kids should do well."

Said Aldo Pinto, a 32-year-old teacher at Gridley Street Elementary School in San Fernando: "The biggest challenge is getting them to buy into the fact that school is important."

He does that by telling students his own story as the son of Mexican immigrants.

Pinto, like most other teachers interviewed, said his good results had not been recognized.

"No one is ever really singled out, neither good nor bad," said Pinto. "The culture of the union is: Everyone is the same. You can't single out anyone for doing badly. So as a result, we don't point out the good either."

"When I worked at a bank, I was employee of the month," he added. "For LAUSD, for some reason, it's not a good thing to do."

Extraordinary gains

Morningside Elementary in San Fernando is in many ways an average school for the district.

Like students at Parmelee, its pupils are largely poor and still struggling with English. Its test scores are below the state goal but in the middle of the pack for L.A. Unified.

In this very ordinary school, year after year, Tan quietly accomplishes extraordinary things.

In the 2008-09 school year, four of Tan's students started below grade level in math. By the end, they were all advanced. In English, nine of her students started below grade level. All but two ended the year at grade level or higher.

Tan is 62 but looks to be in her 40s. An immigrant like many of her students, she understands what they face. She is still self-conscious about her strong accent from her native Philippines, which she left at 27.

When not teaching, she is a marathon runner, with the wiry frame to show for it. Last spring, she finished Boston's in 4 hours, 20 minutes.

Inside the classroom, she sets a sprinter's pace, at times zipping around her students' desks in an athletic shirt and shorts.

Tan is not reading from a district playbook or drilling her students in how to take tests. She says she has little patience for the district's rigid curriculum and at times ignores it. That gets her into trouble on occasion with district administrators, who urge teachers to stay on the same pace.

Tan brims with innovative ways to reach limited-English students, handle discipline problems and keep the kids engaged. "I do a lot of singing, games," she said. "It doesn't look like a lesson."

But no one asks for her advice. She says her fellow teachers at Morningside consider her strict, even mean. She tends to keep to herself.

"Nobody tells me that I'm a strong teacher," she says.

That's OK by her, she adds. Year after year, she watches her students make enormous progress and feels a quiet sense of satisfaction.

Two generations

Tan's students made strong gains in each of the seven years analyzed by The Times. Indeed, she seems to have been pulling students up for a generation.

About twenty years ago, Karina Reyna and her twin sister both had Tan as their first-grade teacher, an experience Reyna still remembers vividly.

The girls had been born in Mexico and entered the U.S. illegally with their parents, neither of whom had graduated from high school. The family lived in a working-class area in San Fernando, where Reyna's father installed carpets.

By first grade, Reyna said, she still didn't speak English. Ms. Tan was determined to change that.

"I really didn't like her," Reyna recalled. "I remember crying every day."

Tan pushed Reyna relentlessly, accepting nothing but her best work. Reyna's English improved, but when she continued to struggle in math, Tan stayed after school to help her catch up.

"Now I recognize it wasn't mean, it was strict," Reyna said. "She was pushing me to do what I was capable of. Maybe she even saw something I didn't see."

Reyna and her sister Daniella stayed in touch with Tan over the years. Tan attended their 15th birthday party and years later Daniella's 2006 graduation from Cal State Northridge, which both sisters attended after becoming U.S. citizens.

"I don't think I'll ever forget her," said Reyna, who works for an insurance company and plans to finish her college degree soon. "Without her, there wouldn't have been somebody saying, 'You have to finish school; you have to go to college.' "

When Reyna learned that her daughter, Jazmin, had been assigned to Tan, she was convinced the girl would thrive.

"I told her, 'She's really strict,' " Reyna recalled. " 'You're going to be pushed, but it's going to be good for you.' "

She was right.

Jazmin entered Tan's class in 2007 above grade level in math. By the end of the school year, she had vaulted 74 points to 600 on the state test -- a perfect score.

Having just finished fifth grade, Jazmin was recently accepted into a gifted-magnet middle school. Reyna expects her to graduate from college and go into medicine. She hopes her son, a kindergartener, will also be assigned to Tan.

"She pushes kids to be their best," Reyna said.

Only two options

Tan measures her success in stories like these.

But by the LAUSD's measure, Tam simply "meets standard performance," as virtually all district teachers do -- evaluators' only other option is "below standard performance." On a recent evaluation, her principal, Oliver Ramirez, checked off all the appropriate boxes, Tan said -- then noted that she had been late to pick up her students from recess three times.

"I threw it away because I got upset," Tan said. "Why don't you focus on my teaching?! Why don't you focus on where my students are?"

Ramirez said he wants to give more recognition to his excellent teachers, but with no objective measure to rely on, he's concerned about ruffling feathers.

"What about the teachers who feel they should have been recognized?" he said. "There'll be a whole mess. The district knows this would open up a can of worms."

"That's why it doesn't happen."

jason.felch@latimes.com

Times staff writer Doug Smith contributed to this report.