All but a handful of ninth- and 10th-graders at Pathways in Technology Early College High School have an ambitious summer goal: to pass the Regents exam in geometry before school starts in September.

To that end, they are enrolled in a six-week long summer enrichment class meant to get them up to speed on the information technology-themed school’s academic expectations and prepare them to take the state’s geometry exam this month. Classes are long — two to four hours each morning — and involve a mix of group projects, drills, homework, and writing assignments.

GothamSchools spent the morning in one marathon math class two weeks before the Aug. 16 exam. As the students worked in pairs on projects, four teachers hovered above, sometimes chiming in with explanations of geometry concepts and sometimes reigning students in when they wandered off-task.

After class, the lead teacher, Jamilah Seifullah, explained how she kept track of the students and what she wanted them to learn. As when we chronicled Ryan Hall’s math class in May, we’ve included Seifullah’s commentary in block quotes beneath our observations.

Seifullah, who taught geometry to a small cohort of advanced math students last spring in the school’s first year, took turns directing the class with Rachel Jamison, an English teacher who is pitching in with math instruction this summer. Jamison is also offering English lessons, but not for credit and during a shorter class period. With the Regents exam approaching, she and Seifullah agreed to combine the classes for longer math sessions, but weave in tasks that build literacy skills.

**10 a.m. ** Already, 32 P-TECH students had been working in pairs on a major assignment for almost an hour. Sitting at round tables in groups of five or six, each pair was using a computer to put the finishing touches on presentations on various geometry concepts, such as surface area and the isosceles triangle theorem, they would later present to their classmates.

Each presentation included of formulas, definitions, and practice problems. After each presentation, a pair of students at each table moved to a different table to present their projects to other classmates. Students had been teaching lessons to each other over the past week, and on the day we visited, five pairs were presenting.

Seifullah said she picked the concepts to assign based primarily on the Regents exam required for graduation. I’ve looked at the most important topics in geometry, as far as the Regents are concerned, Seifullah said. I assigned those topics to students and they then worked with me and Ms. Jamison to develop a powerpoint lesson plan. The plans needed to have independent practice, guided practice, as well as homework.

**10:20 a.m.** As Seifullah flitted among groups, she paused over one student who was looking up geometry terms on Yahoo! Answers, a website most teachers consider unreliable. Seifullah explained that in her assignment instructions, she directed the class to seek help on more reliable sites, such as the Regents Prep web site maintained by an upstate school district or another site called Math is Fun.

“Every time I come over here you’re on a site that I haven’t recommended,” she said. “I could never verify the information on all the sites out there.”

Most of the students were typing out notes into slides on the computers or writing on notebook paper, but several were listening to music with headphones plugged into the computers. And at the table behind them, a student was reading a comic book.

[There were] only two or three that I really saw going off task, Seifullah said. They’re moving around so often that they really don’t have an opportunity to. If they’re working on something independently, some of them ask if they can listen to music. Most of them stay on task because it is part of their grade, and most of them do think it’s important to teach the lesson. They don’t want to get up in front of their peers and not be prepared.

**10:30 a.m.** Seifullah told all the students except those who were presenting to rearrange themselves at the tables and return their computers to the class’s cart. Each table would have two lesson presenters, she explained, and the other students would listen to each lesson for 20 to 40 minutes before hearing from a new set of classmates.

“Get ready to take notes,” Jamison told the students. “This is your classwork credit. Don’t just sit and stare.”

Seifullah said the students learn more when she asks them to teach each other. This creates that supportive environment, where they are supporting each other to do the best job that they can, she said. When they put in even the minimal amount of time to learn what they need to learn to teach, they learn more than if they just sat in the classroom listening to a teacher talk the whole time. When I’ve give quizzes after lessons that previous students have taught, the students who taught the lesson have gotten almost every question right on it.

**10:37 a.m.** With their audience in place, ninth-graders Justine Maximilien and Ettienne Durand began passing out worksheets with multiple-choice drill problems called a “Do Now.” Most of the problems had to do with determining and changing the angle measurements of various isosceles triangles, or triangles with two equal sides. The other students copied the definitions and formulas on each slide and then shared their worksheet answers.

Durand said it has been easier to stay focused during the lengthy math lessons than he expected when he accepted his slot at P-TECH, which was his third-choice high school. He said he was intrigued by school’s information technology focus but didn’t initially realize summer classes would be a requirement.

“I didn’t know we were going to do geometry, but I’m okay with it,” he said.

**10:53 a.m.** At a table across the room, 10th-grader Brandon Scott shared a PowerPoint slide about how to calculate the volumes of different three-dimensional shapes. The “aim” of that lesson, another slide read, was to learn “what are the different formulas for volume,” and answer the question “What do you think volume means?”

Yaayaa Whaley, a math teacher who had been hired just two days earlier, listened patiently at the table. During a lull, she explained the difference between a rectangular and triangular prism to the group.

Seifullah later walked over to Scott’s table and passed around blocks shaped like various prisms and cylinders. She also took out a paper towel roll and cut it into small circles.

“A cylinder is a shape made up of lots of circles, stacked together,” she told the group, using the prop to illustrate the concept.

I like to give them a real world example, she later explained. She used other props to make similar points throughout the morning. For example, at other tables she offered the groups hollow cylinders into which they poured bags of uncooked rice to illustrate their different volumes.

**11:13 a.m.** After the students collected their notes and moved en masse from table to table, and the presentations began again — though it took up to 10 minutes for some groups to get back to work.

As Seifullah and the other teachers watched the presentations, they used grading rubrics to mark the presentations on creativity, the accuracy of the slides, students’ communication skills, and audience engagement.

**12:36 p.m.** Jamison put up the final task of the day on the room’s SmartBoard: to complete, in writing, three sentences: “In today’s lesson(s), I understood…” and “I could teach someone to…” and “I was productive when I…”

“This is your ticket out of here,” she said, to groans from some of the students. One student refused to write at first, and Jamison led him into the classroom doorway for a private talk.

Jamison said the writing exercises are essential as P-TECH works to integrate writing across its math and science-heavy curriculum, but they are usually not welcomed by the students. They are not writing enough, she said. This is just another medium for them to explain the process they went through today. As you saw, many of them really have a hard time with that. Many of them were resistant to it. Seifullah echoed her concerns. In technology, you’re expected to read all kinds of documents and eventually to write them, she said. They’re being prepared for a career in [information technology], so they have to be the kinds of students who can read, and prepare, and write code.