The death of actor Maggie Smith at age eighty-nine has led to an outpouring of appreciation for her great career. Though hers was primarily a theatrical life centered in London, she also made a notable impact on American film and television, especially late in life, playing formidable old women with sharp eyes and acid tongues in the TV series Downton Abbey and the Harry Potter film franchise.

Those performances brought her belated international fame and considerable fortune. “Harry Potter is my pension,” she once observed.

She’d had early success as a stage actor, moving smoothly from her childhood education at the Oxford School for Girls to performing at the Oxford Playhouse and then getting hired for professional roles. By the early 1960s, she was starring in the National Theatre as Desdemona opposite Laurence Olivier’s Othello and swiftly became a leading light of the British stage. Her own summation of her life was typically pithy: “One went to school, one wanted to act, one started to act, one’s still acting.”

But though she made a number of films, including California Suite, Travels with My Aunt, Murder by Death, Sister Act, A Room with a View, Gosford Park, and The Lonely Passion of Judith Hearn, to take the measure of Maggie Smith’s brilliance, you don’t have to look any further that her incisive Academy Award–winning performance as the title character in The Prime of Miss Jean Brodie (1969), directed by Ronald Neame. That’s the one to see if you’d like to watch something dazzling in Smith’s honor. Among other wonderful things, this complex performance represents a damning portrait of naive “progressive” politics that’s the most unsung aspect of a highly praised film.

With a script by Jay Presson Allen (Marnie, Cabaret, The Verdict), which was based in the 1961 novel by Muriel Spark, The Prime of Miss Jean Brodie is about a charismatic schoolteacher at a traditional all-girls school called Marcia Blaine in 1930s Edinburgh. Miss Brodie’s modern flair and liberated ideas make her the idol of her favorite students and the target of the icy disapproval of the conservative headmistress, Miss Mackay (Celia Johnson).

Miss Brodie’s dedication to teaching is genuine, if typically high-flown and increasingly misguided, and she expresses it in her Scottish accent in a line that becomes a haunting refrain: “Little gels, I am in the business of putting old heads on young shoulders, and all my pupils are the crème de la crème. Give me a girl at an impressionable age, and she is mine for life.”

Confident to the point of arrogance about her secure position as tenured faculty and certain of the devotion of her students, Miss Brodie continues in her unorthodox teaching methods and daring love life, which are sometimes uncomfortably intertwined. One of Miss Brodie’s lessons involves having her students open their history books in case Miss Mackay or some faculty member should come into the classroom unexpectedly, and then narrating with dramatic intonations and gestures the story of her first love involving a soldier who was killed in World War I.

“Hugh fell like an autumn leaf,” she laments as one of her more susceptible students sobs aloud.

Miss Brodie’s field trips with her favorites, known as “the Brodie set,” have a tendency to cross paths with the men she’s romantically involved with. Her former lover is Teddy Lloyd (Robert Stephens, Smith’s husband), the art teacher at Marcia Blaine. He’s a married painter of no great success but considerable sexual bravado, and he continues to pursue her obsessively. Her current lover is the mild-mannered music teacher, Mr Lowther (Gordon Jackson), who wants to marry her.

The students are fascinated by what little they can observe and what a lot they can imagine about Miss Brodie’s love life, and two of them, Sandy (Pamela Franklin) and Monica (Shirley Steedman), are caught writing a love letter purportedly from Miss Brodie to Mr Lowther that’s full of naive sexual crudities. Miss Mackay, hoping to catch Miss Brodie in a serious enough transgression to get rid of her, seizes her opportunity to use the letter as evidence that Miss Brodie’s students are becoming alarmingly precocious because of her bad influence. But Miss Brodie rises to the occasion and easily foils her with smiling calm, noting that the letter is obviously the work of twelve-year-olds who are naturally becoming curious about sex and relying on their imaginations to generate their childish effort.

At first, Miss Brodie seems like a progressive heroine offering admirable resistance to a fossilized authoritarian educational structure, embodied in the cold, scowling Miss Mackay. Her airy, ironic remark to Teddy Lloyd that perhaps he and his wife, expecting their eighth child, haven’t heard about modern advances in birth control is one of many she makes early in the film that are both funny and exhilarating in the way they seem to make Miss Brodie a recklessly daring champion of women’s rights in general and of the welfare of the girls in her charge in particular.

Smith gives a dazzling performance, full of verve, style, and humor, swanning around and tossing the ends of her long scarves behind her, followed by a gaggle of adoring acolytes. Her erect posture with the head thrown back accentuates her racehorse build, tall and marvelously slim. If you haven’t ever seen the sexy Maggie Smith, get ready for it.

Miss Brodie’s extraordinary self-assurance only makes her more glamorous when she assures her girls, “I am dedicated to you in my prime.” And it’s so rare and thrilling to see a film portrait of a woman so independent, commanding, and self-directed that it’s quite an emotional workout to have to recognize her glaring flaws, her fundamental wrongheadedness, and ultimately the danger she represents. Miss Brodie’s overconfident tendency to hold forth to her students on any and all subjects, seeking to mold them in ways that will extend far beyond their school years, becomes increasingly alarming as the girls move through their teenage years toward graduation. She tells Sandy that her habit of “peering” at people and intently studying their motivations and behavior indicate that she’s destined to become a spy. And she predicts that Jenny (Diane Grayson), the most beautiful of the Brodie set, will inevitably become Teddy Lloyd’s next muse and lover.

But instead, underage Sandy has an affair with Teddy Lloyd in order to defy Miss Brodie’s cruelly limited vision of her: “I won’t be your spy, Miss Brodie.”

Always enthralled with the new and forceful in modern life, and seeming to have no grasp of politics, Miss Brodie becomes an ardent fan of Benito Mussolini and Francisco Franco. She urges activism on her gullible students, arguing that they should model themselves on Franco’s supposedly self-sacrificing Nationalist forces fighting the Spanish Civil War. The most vulnerable of the Brodie set, the gentle naif Mary McGregor (Jane Carr), obediently runs away to join her brother fighting in Spain, not understanding that he’s fighting with the Republican forces against Franco. She’s ultimately killed on the train that’s taking her to the “wrong side” of the war. When Sandy finally rebels and confronts Miss Brodie on her culpability in this tragic event, she reveals that Mary didn’t die a heroine as Miss Brodie claims — “She died a fool!”

After this, Sandy is determined to take down Miss Brodie, whose personal life and career both unravel with disturbing speed. Though, as Sandy says, somebody had to “put a stop” to Miss Brodie and the damage she’s doing, it’s nevertheless heartrending when her “prime” appears to be over. And Smith’s faltering, fading delivery is painfully memorable: “I had thought it would last until I was at least fifty.”

The break in her voice between “at least” and “fifty” is inspired.

But perhaps her most memorable line is Smith’s whipcrack delivery of the word “assassin!” It’s shrieked repeatedly at Sandy’s back as she walks out of Marcia Blaine after having delivered to Miss Mackay all the evidence she’ll need to end Miss Brodie’s teaching career. Sandy walks out as a new graduate, a young adult, and seemingly as shattered a woman as Miss Brodie, who’s been left behind in the empty halls of Marcia Blaine. Over Sandy’s pale, tragic face in close-up, we hear again Miss Brodie’s line, now softened in regretful memory but also foreboding in what it indicates about Sandy’s fate: “Give me a girl at an impressionable age, and she is mine for life.”

Miss Brodie is a daunting role that requires an actor of tremendous command and control to play her. (Vanessa Redgrave played the part in the original British stage production, and Zoe Caldwell won a Tony for her performance in the American stage version.) And that’s an indication of what Maggie Smith brought to the parts she played: an apparently effortless ability to dominate scenes whenever she appeared, and to bring such decisive skill to her performances that even the slightest of them got etched into the memory. We have a shortage of formidable women actors in films, absolutely, and we couldn’t have done without Maggie Smith. But sadly, now we’ll have to.