A man sees what no one else can, and we call him a genius. A man sees what no one else does, and we call him crazy.

This Oscar-winner for Best Picture is a movie about a man who was both, the true story of genius John Forbes Nash, Jr., who revolutionized mathematics and then became mentally ill. More than 40 years later, as he edged back into sanity, his contribution was recognized by some people in Sweden. They awarded him the Nobel Prize.

As the movie opens, it is just after World War II, and a group of bright young mathematicians are arriving at Princeton. They are proud because “mathematicians won the war,” and they are eager to make up for lost time. Nash (Russell Crowe) stands out. He is tactless, he does not go to class, and he does not produce anything publishable. A teacher once told him that he had a double helping of brains but half a helping of heart. It is not that he does not seem to care about norms of social behavior and academic performance – he does not even seem to notice them.

Then Nash has an idea, an anti-Darwinian notion that proves that more success for more people is achieved through cooperation than through competition. The elegance of his proof is a stunning achievement, and he is rewarded with an important position and allowed to select the classmates he wants as his colleagues. He even meets a beautiful student (Jennifer Connelly) who enjoys his directness and appreciates his “beautiful mind.” Nash is successful, saving the day when only he can see the pattern in a string of numbers from an intercepted Soviet message.

But then Nash begins to see patterns where there are none, and he is hospitalized. His powers of logic and focus and the love of his wife help him to reconnect to reality, and after decades of effort, he is able to teach at Princeton.

There is a heartbreaking moment near the end when Nash is leaving a classroom and a man he does not know approaches him to ask him something. Before answering, Nash turns to a student to ask whether she sees the man, too, because he is still not sure which people he sees approaching him actually exist. He has simply adapted to his delusions, by requiring proof, in classic mathematical or at least empirical terms.

This is an extraordinary story, and it has been made into an extraordinary movie. Crowe is, as always, simply magnificent in a role that would provide irresistible temptation for showboating for most actors. There are superb performances by everyone in the cast, including Connelly (an Oscar-winner for Best Supporting Actress), Paul Bettney, Ed Harris, Christopher Plummer, Judd Hirsch, and a dozen others. What is really special here is the way that screenwriter Akiva Goldman and director Ron Howard have found a way to present both Nash’s genius and his mental illness in such compelling, cinematic, and accessible terms. Both in essence become characters in the story as we go inside his head and wonder with Nash what to believe. This is what makes the movie more than a disease-of-the-week special with color-by- numbers “heartwarming” moments of triumph over adversity. This is what makes the movie itself a true work of art.

Parents should know that the material might be very upsetting for kids, or for anyone who has relatives with mental illness or who knows very little about it. There are some strong scenes of family tension and peril, including a child in jeopardy, scuffles, and potential domestic abuse. There are graphic scenes of shock therapy and self-destructive behavior. A character is in peril involving shooting. There is also some crude language with sexual references.

Families who see this movie should talk about mental illness, about how people with mental illness need to be treated, and about what is different now in the way we treat the mentally ill from the days depicted in the movie. Families who want to know more should check the website for the National Alliance for the Mentally Ill.

Families who enjoy this movie will also enjoy “Apollo 13″ and “Parenthood,” also directed by Ron Howard. They might also like to read the book, by Sylvia Nasar, or Surely You’re Joking, Mr. Feynman!, a very engaging book about a brilliant physicist. Families who like board games should try Go, the Chinese board game played by the mathematicians in the movie. The rules are fairly simple, but the strategies are endlessly complex.

NOTE: DVD extras include all kinds of extra goodies, including an entire separate disk featuring footage of the real John Nash and more information about his work.