Midha has always loved math — it came easily. She suffered some doubts about her abilities in 10th grade, thanks to an algebra teacher who did the standard “chalk and talk” at the blackboard. But during freshman year at Wesleyan University, she took calculus with a professor who considered mathematics to be an art. “Everything he said was so profound,” she recalled. “I was like, ‘Oh, my god. This is the coolest thing ever.’” And that was that. She became a math major — and also a music major; she plays classical guitar.

Midha decided to become a teacher while still in college, after giving lessons in music, math, Spanish and web design at a summer session for middle schoolers. She applied for the fast-track New York City Teaching Fellows program and got her start at a school in Brooklyn straight out of college, earning her master’s degree in teaching at the same time. She taught for four years, and then took a two-year detour, leaving the country and working in marketing and public relations at a fashion house in India. When she returned home, she realized she missed the kids. “Then I found East Side, and I was like, ‘This is it.’ This is a special place.”

East Side Community High School opened in 1992, part of the second wave of a movement of small alternative city high schools that dates to the 1970s. It is part of the New York Performance Standards Consortium, a group of 28 schools across the state that oppose one-size-fits-all standardized testing in favor of performance-based assessment.

A recent case study on the school, titled “Learning by Heart,” describes East Side as a place that succeeds due to its combined priorities of academic rigor and personal relationships with students. Students came to appreciate just how special East Side was in 2012, when a custodian noticed that the school’s 90-year-old brick façade was detaching from its steel frame, requiring an emergency evacuation and a five-month relocation to an impersonal school with windowless classrooms that resembled a high-security bunker. At East Side, by contrast, there is a lot of love. “The teachers love our kids, the kids love us,” said Midha, who is like another parent for some students. The school serves grades six through 12 and draws a diverse mix of students, most of whom live on the Lower East Side. More than half of the student body is Hispanic and about a quarter is African-American; about a quarter receives special education. But 100 percent are governed by inquiry as the lever of learning.

On day two of the struggle problem, Midha, dressed in slacks and a T-shirt and New Balance runners that allow her to keep in perpetual motion, provides her students with only select pieces of information and instruction.

“I’m going to give you one hint for your struggle problem,” she says. “Here is the one hint I’m going to give you: The goal is to find the total height of the building. The full height. The total height.

“Remember: Where was the angle of elevation measured from? The eye. So when you are drawing and calculating, remember that. Your job is to calculate the total height of the building. … Remember: There is a reason we measured the eye height. There is a reason we measured the eye height.” Repetition, and more repetition, is key for penetrating the adolescent brain.

Midha also provides a bonus hint of sorts, pointing out to her students that they measured the eye height in inches and the building distance in miles, and that the worksheet asks for the height of the building in feet. With that, she leaves her students to their own devices —“Good luck!”

She circles the room, surveying the progress, asking simply: “Does it make any sense? … Why doesn’t it make any sense?” Despite her hints, the relevance of the eye height is proving elusive, and the inch-feet-miles conversions are confusing. She reiterates her tips one-on-one with the groups, and then lets them loose again, declaring: “I’m going to walk away now … ”

“Nooo, come back!” Fabian pleads, flipping his pencil in frustration and dinging himself in the eye.

Midha and her fellow East Side math teachers recently read 5 Practices for Orchestrating Productive Mathematics Discussions, by Margaret Schwan Smith and Mary Kay Stein. They dissected and discussed it chapter by chapter at their Friday meetings. “It’s about the idea of increasing rigor, increasing the ratio, and the conversations that you have with kids,” Midha said. The “ratio” refers to the amount of discovery and learning the students do on their own versus the amount of explicit play-by-play instruction they receive from the teacher. “Basically, it’s the thinking ratio,” Midha said. “The kids need to be doing the thinking, not the teacher.”

Inspiring kids to think, battling their inclination toward intellectual inertia, requires finely tuned orchestration. The book’s five practices consist of anticipating how students will try to solve problems, monitoring their work as they struggle and progress, selecting various students’ approaches to share with the class, sequencing the shared ideas in a logical way, and connecting it all to the overall goals of the curriculum. As Midha did the rounds from one group to the next, these factors were clearly in play — she gently and deliberately coaxed along the conversations, nudging the kids toward a breakthrough, where it all, finally, made sense. “It is pretty powerful when they do have that aha moment,” she said.

“It’s kind of like, not a weight off of my shoulders, but a weight off of my shoulders in my head,” said Savannah, 16, who used to hate math. “Suddenly it’s like you took a medication or something, you relax and the struggle’s gone and you realize the truth — you realize the truth behind the struggle.”

FAIL

On Friday, May 27, in an affluent suburb west of Boston, Aaron Mathieu introduces a lab experiment to his advanced placement biology students at Acton-Boxborough Regional High School. Mathieu, 42, has a boyish appearance and a voice that sounds alternately serious and laid back. His students scatter into groups, chatting boisterously. Mathieu reins them in with a reminder to prep their test tubes. Then he relaxes, calling the students cherubs, an affectionate term he picked up from his high school physics teacher. (He likes it because, unlike “guys,” it “offends everyone equally,” he said.) Later that day, in another class, he recollects puns about a recent career field trip to learn what eye doctors do (“the visit was very eye-opening”). His students poke fun at their teacher, calling him a nerd.

Six years ago, Mathieu decided to try a risky project in his honors biology class. The students would collect bugs from the school’s grounds, grind them up and extract DNA. Then they would try to isolate DNA from a specific type of bacteria living inside the insects’ reproductive tracts.

The experiment was complex, and Mathieu and two fellow teachers had spent a couple of years mulling it over and trying to decide whether to attempt it. They were drawn to the project because it combined molecular biology concepts and techniques with the real practice of science. If the experiment was successful, the students had the potential to make novel discoveries — to identify new species carrying that particular microbe, known as Wolbachia pipientis, and to uncover new strains of the bacteria. But the project could turn out to be a spectacular failure. “We were worried about how we would feel as teachers and how the students would feel if they spent five days doing all these novel experimental steps” — extracting DNA, copying sequences, separating them and visualizing the data — only to look down and see nothing, Mathieu said.

Most of the existing lab projects in the biology curriculum were designed like a recipe in a beginner’s cookbook — with predictable results that almost always came out as planned. But Mathieu wanted to expose his students to the true nature of science — its sense of exploration, the highs of discovery and the challenges of failure. To do that, he knew that he had to remove the safety net. “In order to innovate, we have to take risks and put kids into situations where they take risks,” he said. Spending a week on a lab that doesn’t work risks squandering limited time and other resources and discouraging the students. “We realized that everyone, teachers and students alike, would learn a lot by attempting the project, even if we did not successfully generate any data,” Mathieu said. “To me, the only waste of time is if the students didn’t learn from the experience.”

The first attempt at the Wolbachia lab ended just as Mathieu and his colleagues had feared — after a week of pipetting, centrifuging and careful note taking, all but one group of students had a blank gel. The students were distraught, focusing only on their missteps and the lack of a clear answer when the project was complete. Mathieu, however, used that response to shape his teaching philosophy. “I took that as a sign they are not learning enough about the process of science,” he said. “It meant we needed to embed the process of science throughout the curriculum.”