In St. Petersburg, Florida, seven-year-old Mary Adler, a mathematical genius, lives with her uncle and de facto guardian, Frank. Her best friend is her 43-year-old neighbor, Roberta Taylor. Frank, a former philosophy professor now boat mechanic, feels strongly that Mary should attend a normal elementary school so she can have a normal childhood.

In first grade, Mary shows remarkable mathematical talent, which impresses her teacher. It emerges that Mary’s mother, Diane, had been a promising mathematician, dedicated to the Navier–Stokes problem (one of the unsolved Millennium Prize Problems) before committing suicide when Mary was six months old. Mary has lived with Frank ever since.

Despite Mary’s initial disdain for average children her own age and her boredom with their classwork, she begins to bond with them when she brings her one-eyed cat, Fred, for show-and-tell. Later, she defends a classmate from a bully on the school bus. After the incident, the principal, having discovered Mary’s math talent, encourages Frank to send Mary to a private school for gifted children, offering the opportunity of a scholarship. Frank turns it down, based on his family’s experiences with similar schools.

The principal contacts Frank’s estranged mother and Mary’s maternal grandmother, Evelyn. Evelyn, a former mathematician herself, believes that people with capabilities such as Mary’s have an obligation to use their talents to further help society and therefore feels that Mary should be specially tutored in preparation for a life devoted to mathematics, much as Diane was. But Frank is adamant that his sister would want Mary to be in a normal school and have the childhood she did not have.

Evelyn sues Frank for full custody. While she is on the stand in court, it is revealed that not only did Evelyn homeschool Diane, she kept her socially isolated, so that she could be completely focused on mathematics. When Diane and her teenage boyfriend ran away to a ski resort, Evelyn filed a lawsuit and threatened to charge him with kidnapping, forcing him to cut ties with Diane. She attempted suicide for the first time shortly after, something Evelyn argues was an isolated incident. When Frank takes the stand, he admits working at a low-paying job without health insurance. His lawyer, worried that the judge will side with Evelyn due to her financial resources, convinces Frank to take a deal orchestrated by Evelyn’s lawyer.

Mary is placed in foster care, where she can attend the private school Evelyn wants until her 12th birthday, when she will be able to decide where she wants to live. The foster parents live near Frank, and he is entitled to scheduled visits. Mary is devastated and refuses to see Frank when he tries to visit. Thanks to a tip from Mary’s teacher, Frank rescues Mary’s cat Fred from the pound moments before he is about to be put down, along with two other cats. Frank realizes that Evelyn, who is allergic to cats, is overseeing Mary’s education in the guest house of the foster home.

Frank goes to the foster home and reconciles with Mary. He informs Evelyn that Diane had completed the Navier–Stokes problem, but left instructions for Frank not to publish the equation until after Evelyn’s death, revealing Diane’s deep resentment towards her mother. Frank offers Evelyn the opportunity to publish Diane’s work if she drops her custody case, to which she reluctantly agrees. Mary is placed back in Frank’s custody, living a normal public-school childhood while taking college-level courses in her spare time.