Jason and Sandy Roberts, parents in Pasadena, Calif., have produced to my astonishment the nation’s most accelerated math program.

Advanced Placement Calculus BC is taken by fewer than 5 percent of U.S. high school seniors. At the Pasadena Unified School District Math Academy it is an eighth-grade course. High school students in the program, who include the Robertses’ oldest child, devour college courses far beyond calculus and are building a complex online strategy game called Space Empires.

I have found only one other U.S. program, at the University of Minnesota, that teaches calculus to eighth-graders or younger.

I thought the ravages of the pandemic would set the Robertses back, but I was wrong. “We were determined not to lose a step,” Jason Roberts told me. “The instructors were uncertain about how to proceed given the district’s guidance to throttle things down, but we were given an exemption as a special program and were allowed to push forward.”

Students in the PUSD Math Academy who complete AP Calculus BC in the eighth grade move on to these university-level subjects in high school: multivariable calculus, linear algebra, differential equations, abstract algebra, probability, statistics, combinatorics, number theory, graph theory, real analysis, topology and complex analysis. The district secured University of California approval for those unusual high school for-credit courses.

The Pasadena public schools, where most students are from low-income families, have had a mediocre reputation for decades. But in recent years its administrators have created strong AP and International Baccalaureate programs with high levels of participation. It has established magnet schools and dual-language immersion classes.

When the Robertses started a lunch-hour class in an elementary school in 2013, Superintendent Brian McDonald, a former accountant and math teacher, visited and asked them to help him create a pilot program based on their model. He had the district hire more advanced math teachers. In the 2020-21 school year, 154 students participated in the Math Academy.

Districts usually shudder at the idea of deep parent involvement in curriculum. A group of families with ties to the NASA Jet Propulsion Laboratory tried to get a more affluent district near Pasadena interested in similar acceleration but were turned down. Officials said their proposal raised “concerns regarding its developmental appropriateness for 13- and 14-year olds.”

Most students in the Math Academy are what schools call “gifted,” but few other districts advance such students this way. One parent said he knew his daughter was bored in school but did not understand how bored until he saw how happy she was in the academy.

The Robertses met when they were undergraduates at the University of Chicago. Sandy was interested in marketing and eventually became a nonprofit fundraiser. Jason applied himself to technology start-ups and worked on the original real-time system at Uber. His interest in accelerating math instruction dated back to when he taught himself calculus in the ninth grade out of a book.

Their program started when they coached their oldest child’s fourth-grade math field day team. That happened twice a week from December through the end of the school year. The following year they had a three-days-a-week enrichment class for 13 fifth-graders. Then they had algebra, geometry and trigonometry classes in middle school. In September 2017, I watched the program’s first eighth-graders — six of them — dive into AP Calculus BC. Five of the six passed the AP exam the following May.

As part of a plan to construct intelligent agents for the Space Empires strategy game, Jason Roberts said, the high school students are building “all the core elements from scratch, support vector machines, neural networks, etc. . . . They will be forced to adopt disciplined software engineering practices that typically aren’t learned until one becomes a professional and has to code something more than a one-week project.”

Pasadena’s accelerated system will preserve the online learning tools it developed during the pandemic. “Some of our classes are quite large, like 20-plus, so having all or even most of the kids at the whiteboard doesn’t really work,” Roberts said, “and keeping everyone on task is always a challenge with this age group. Also, we’ve found that nothing outperforms the personalized learning capability of the online system, which now adapts to each student’s specific strengths and weaknesses.”

“While the kids who are extroverted and competitive absolutely love the games and board work, the introverted and uncompetitive prefer to quietly learn at their own pace without a lot of interaction with other kids or even attention from the instructor,” Roberts said.

The program had 80 students in a six-week summer camp this year. It also started a self-paced course for 35 students ranging from seventh-graders through rising 10th-graders. The program has had instructors meeting fifth-graders once a week in schools with many low-income families to cultivate a love of math and prepare more students for advanced work.

American experts are glum about what will happen in schools this fall. They see mostly obstacles to improvement. But people like the Robertses are ready to raise expectations. This might be a good time to give more of them a chance.