Innocent Prodigy

A young, but incredibly bright child whose high intellect comes with rather large doses of naivete. He seems incredibly deep at times, but he has moments where his innocence reminds the audience (and the other characters) that he's still a child.

Truth in Television, since real-life prodigies typically have emotional maturity better fitting their actual age than their intellectual level.

Compare Adorably Precocious Child. See also Ping-Pong Naivete.

Anime and Manga

 * Ten-year-old high-schooler Chiyo from Azumanga Daioh always keeps acing test after test while failing miserably to understand mature stuff any teenage schoolgirl would already know by heart, down to understanding nothing of Nyamo's drunken sex-ed lesson.
 * Negi of Mahou Sensei Negima. He's beginning to grow out of the naivete though.
 * Shinra Sakaki from CMB, So Touma form Q.E.D.. Hell, Motohirou Katou loves this trope.
 * Hiiragi from Hanamaru Kindergarten is very intelligent and knowledgeable for her age but still enjoys children's activities such as going down slides, and playing dress-up.
 * While whether Hiroki Sawada of Detective Conan: Phantom of Baker Street wanted to reset Japan is debatable, the ten-year-old MIT graduate does want to play with the other kids.
 * Rebecca Miyamoto from Pani Poni Dash! is an MIT graduate and a teacher at age ten. Doesn't stop her from throwing tantrums or hiding behind things whenever she gets scared or upset.
 * Alice from Aria, though her more childish moments are quite few and far between.
 * Diamond from Pokémon Special. He likes to eat and watch robot anime (singing its theme tune in an attempt to bolster his confidence when he's scared), but he is shown to be very thoughtful and sensitive at times- he's associated with Mesprit, the legendary Pokemon of emotion, and delivers heartfelt speeches that convince legendaries to fight alongside him.
 * Edward Elric and, to a lesser extent, his little brother Alphonse USED to be this in Fullmetal Alchemist, before all of the Break the Cutie kicks in.
 * Sister Maria from Boku wa Tomodachi ga Sukunai shot her way through the education system and ended up as a high school teacher at 10 year old. Also, she ended up as an insufferable, foul-mouthed Bratty Half-Pint...until she met Yozora.

Literature

 * Charles Wallace Murray from A Wrinkle in Time and its sequels is an example of this trope.
 * For that matter, Huckleberry Finn, making this Older Than Radio.
 * Discworld has a rodent version. Dangerous Beans is brilliant but by most of the other character's standards rather naive. However, the book clearly sympathizes with his idealistic views.
 * Elantris: Daorn and Kaise have elements of this. Brandon Sanderson says this snuck in as a mild criticism of Orson Scott Card's depictions of hyperintelligent children in Ender's Game.
 * Temeraire is a dragon who manages to be this.
 * Mark Clifton's "Star Bright" has Star Holmes and Robert Howell, ultra-intelligent toddlers who invent a way to teleport themselves through time. They use this ability to play hide and seek. Star's father also tells us that Star's intelligence doesn't prevent her from enjoying dolls, fairytales and playing grownup.

Live Action TV

 * Micah Sanders from Heroes. The audience first sees him fixing his laptop as he tells his mom, "The memory board's bad," and he can talk to machines. Yet Micah reads comic books religiously and uses them as a guide, making him something of a miniature Ascended Fanboy, and he idolizes his father.
 * Shane at the start of Weeds shows aspects of this but has already lost a lot of innocence (part of the Backstory is that his 40-year-old father died of a heart attack right in front of him) and grows out of the last of it over the course of the first couple seasons.
 * Lex from Flight 29 Down.
 * Lost: The young who was definitely extremely intelligent and introspective and possessed the prerequisite dash of naivete, but very quickly lost his innocence due to his, which eventually culminated to his  and.
 * Jill Petterson in the Are You Afraid of the Dark? episode "The Tale of the Final Wish" qualifies, with her superstitions and obsessions with fairy tales, as opposed to the other kids her age. Until her wishes come true in a bad way.
 * Franklin from My Wife and Kids. He is a genius, outsmarting pretty much the entire cast. But at the "where do babies come from" department, he still believes that pregnancy comes from plant seeds. Justified, since he was purposefully left out of Sex Ed classes.

Newspaper Comics

 * Peanuts: Linus Van Pelt, Charlie Brown's best friend. He has great insight into some situations and a very high intellect such as being able to precisely quote any passage of Biblical scripture, but he also believed in the Great Pumpkin, which brought him ridicule despite his intelligence. And he hates to be separated from his beloved blue blanket.
 * As an infant it was even worse. As Charlie Brown struggled to balance one card on top of two others to start a house of cards, Linus used the rest of the cards to make a Gothic cathedral.
 * Calvin of Calvin and Hobbes waxes philosophical about the nature of the universe...to his stuffed tiger. While pretending to be a space cadet. Or a superhero. Or a jungle explorer. Or a private eye. Or a dinosaur. Lampshaded by Calvin's mother in a punchline, "How can one kid be so smart and yet so dumb at the same time?"
 * Mafalda's greatest concerns are: the terrible ordeal of the starving kids in Biafra, the permanent tensions between USA and the Soviet Union, the armed conflicts in the Third World, and Mom's terribly hideous soup.

Video Games

 * Tails of Sonic the Hedgehog fame. Most depictions stick solely with one extreme or the other for the most part (eg. Adventures of Sonic the Hedgehog and Sonic the Hedgehog Tails are extremely childlike and not to bright while the games and Sonic X counterparts is arguably the most sensible and soundly intelligent member of the team), though he is a rather valid example of this trope in Sonic the Hedgehog The Movie, as well as in recent games where his childish traits are trickling back in.
 * Vivi from Final Fantasy IX.
 * Eiko, aged six, meanwhile, has been living on her own as Team Mom to a bunch of moogles. While she's clearly an incredible child (not to mention her white magic), she's very naive, due to her age and lack of interaction with adults.

Web Comics

 * In The Inexplicable Adventures of Bob, all three of the Peanut Butter Monsters have aspects of this, since they're born geniuses but have little to no life experience yet.
 * Gin from Jackies Fridge is four or five, acts more like a ten-year-old, and comes out with insights like this.
 * Carrie from Everyday Heroes, sometimes with just right question at the right time.

Western Animation
""Huh, this toy contains small parts. But why would they include small parts in a toy for someone my age...? Unless I'm supposed to eat them! Of course, it all adds up!""
 * The Simpsons: Lisa Simpson, while Wise Beyond Her Years and one of the few Springfield citizens who isn't stupid, incompetent or just plain lazy, loves her Malibu Stacy doll more than anything else (except, perhaps, activism) and likes immature cartoons like "The Itchy and Scratchy Show." She also wants a pony. On some occasions she'll happily join Bart's destructive antics (indoors bike jousting? Sure!).
 * Ed, Edd 'n' Eddy: Edd is depicted as a genius whose maturity makes him the Only Sane Man. Yet the nature of some of the show's more risque jokes often completely escapes him.
 * Time Squad: Otto is has a fairly complete and detailed knowledge of all of history. And yet he still plays with crayons and coloring books.
 * The title characters from Phineas and Ferb, both of whom are creative and technical geniuses, but use their talents more for their own amusement than anything else. They also manage to regularly figure out the solutions to other characters' problems, while still maintaining an oddly childlike sense of naivete.
 * At least Once an Episode, somebody will ask Phineas if he and Ferb are too young to be doing whatever they're doing. Phineas always responds: "Yes, yes we are."
 * Stewie from Family Guy is the Evil Counterpart to this trope. He's a Wicked Cultured Evil Genius despite being barely a toddler and often almost as naive as you'd expect a child his age to be.


 * Froggo from Histeria! spends time providing the occasional historic narration and poetry reading when he isn't busy having lunch, playing video games, or working on his inventions.
 * Arguably Dib from Invader Zim. Usually a Straight Man spouting Cassandra Truths, there are nevertheless times when he shows a childlike optimism, especially around adults who seem to treat him nicely (such as Dwicky and, at times, his father).
 * Ike Brovlovsky from South Park has had a relationship with a teacher and was involved in a Government Conspiracy to elect Barack Obama. And yet, he's still in kindergarden and confuses Milan for a Disney film.
 * Kit Cloudkicker of Tale Spin a lot of times shows immense knowledge of flying and sometimes displays a lot more common sense than adult peers such as Baloo. However, he often displays a rather bratty overconfidence, and will not accept that he is still too young to actually fly himself.
 * Amberley of The Dreamstone has moments of this Depending on the Writer, she is mature enough to hold a stable job and manages to keep her head out the clouds a lot more than Rufus and even the Dream Maker at times, but has a very childish temper which sometimes leads her to go headfirst into dangerous situations.