When Lisa Suben took a job last year as the fifth-grade math teacher at the AIM Academy in Southeast D.C., she was told her lessons had already been prepared for her. AIM was the second charter school founded for low-income D.C. students by KIPP, the Knowledge Is Power Program. KIPP had gained a national reputation for math instruction. The KIPP leaders in D.C. had good reason to think, as they told Suben, that "we have math pretty much figured out."

Suben, 23 at the time, still thought she could do better. She told her supervisors she was going to produce her own fifth-grade math curriculum. A year later, her students achieved the largest one-year math score jump ever seen at a KIPP school (or any other school that I know of), from the 16th to the 77th percentile.

Given KIPP's history, this is quite an event. The two KIPP founders, Mike Feinberg and Dave Levin, were trained by Harriett Ball, one of the most imaginative and successful math teachers in the country, when they were struggling teachers in Houston. Feinberg and Levin started their own fifth-through-eighth grade middle schools in Houston and the South Bronx, each of them taking the key role of fifth grade math teacher so they could introduce their newest students to math the KIPP way. Susan Schaeffler made sure she, too, was the fifth-grade math teacher when she started the first KIPP school in D.C., the KEY Academy. That first KEY class was, on average, at the 34th percentile in math when they started fifth grade in 2001 and rose to the 92nd percentile for those who graduated from the KEY eighth grade in 2005.

Knowing of that record, you would think that Suben would have thanked Schaeffler, now director of all three KIPP schools in the city, and AIM's principal Khala Johnson, for making life so easy for her and used the materials they gave her. KEY had the best math scores in the city. How could anyone improve on that?

But one of the secrets of KIPP's success in attracting the brightest young teachers and raising achievement for low-income children throughout the country is its insistence on letting good teachers decide how they are going to teach. KIPP principals, such as Johnson, have the power to hire promising young people such as Suben and let them follow their best instincts, as long as the results -- quality of student work, level of student classroom responses, improvement in standardized test scores -- justify the teacher's confidence in her approach.

Johnson and Schaeffler were variously startled, amused and intrigued by Suben's determination to do math her way. They say they are also very pleased with the results, which justify both the hiring of Suben and the KIPP insistence on lively engagement of every child in class.

Suben turned 25 last month. She grew up spending the school year in Rochester, where her mother worked as a property manager, and the summer in D.C., where her father worked for the World Bank and started a music store. She was always a good math student. After she graduated from George Washington University in 2003 with a degree in economics, she joined Teach For America -- which places recent college graduates in low-income schools -- and she was assigned to be an eighth-grade math teacher in Opelousas, La. When she completed her two-year commitment she came back to D.C. to be closer to her family. She was a natural hire for KIPP, which finds many of its teachers among successful survivors of Teach For America's sink-or-swim approach to classroom training.

In D.C. KIPP has been using the Saxon math series, a no-frills approach that often works well with students whose parents never went to college. Suben said she did not have anything against Saxon. She still has copies of Saxon books and a rival program, Everyday Math, in her classroom. But she thought all the textbooks she had seen had flaws.

"I've found that most traditional textbooks oversimplify and isolate concepts, and yet, are still too difficult for non-readers to use. They don't generally push students to think, but offer repetitive, and boring, practice," she said. She started writing each lesson nightly. This was a remarkable feat of youthful energy when you consider that KIPP teachers work 10 hours a day, and Suben was putting in another three hours each night at home composing the next day's lesson on her Dell laptop.

Suben said: "My primary goal as a teacher is to help my students understand the reasoning behind math rules and procedures. I have several core beliefs about this: (1) Understanding is constructed by the learner, not passively received from the teacher. (2) Understanding is built by making connections between as many strands of knowledge as possible. (3) Understanding is galvanized through communication. (4) Understanding is only valuable when you reflect on it and question it."

The core of her method is the workbook she produced last year on the fly. It "lets students build their own notes and create their own examples. It is incredibly active learning," she said. They were encouraged to write down the meaning of important terms and strategies they used that worked with certain kinds of problems.

"I certainly refer to traditional textbooks for ideas and guidance as I write," Suben said. "My sequence and pace are set by a long-term plan that I have designed to catch the students up on second-, third- and fourth-grade material as well as introduce every single D.C. public schools fifth-grade standard by testing time. I model my word problems after the eighth-grade text that I used in Louisiana because those problems require the level of understanding that I am looking for. I focus on non-traditional problems so that students are forced to think."