To amend Chekov's maxim: If an axe is nailed to the wall of a third-grade classroom in the first act, it must draw blood by the third. Hey, at least An Invisible Sign observes the unities. But why the weapon is introduced into the school in the first place and why it's allowed to stay there for the majority of the film are just two of the numerous imponderables thrown up by Marilyn Agrelo's head-scratching folly.
Based on Aimee Bender's bestselling novel, the film is a puzzling mix of sentiment, black humor, and loads and loads of nonsensical behavior. Granted, An Invisible Sign is framed by a dark fairy tale (which invites us to view the subsequent film as such?) and deals with one certifiably crazy character and another who is at least a borderline case, but the entire film seems to take place in a world where every human action utterly defies any kind of discernible logic. Basically, An Invisible Sign runs according to its own sense of "quirk," a quality that Agrelo and her screenwriters inflect away from the calculated "cool" of other Amerindies, but their particular flights of somber fancy are no more palatable—and given their context within the film, arguably less so—than the tired Juno boilerplate.
"Life is much harder than math" is the lesson learned by twentysomething depressive Mona Gray (Jessica Alba, in an odd, inexpressive performance). Struggling to live an adult life many years after the nervous breakdown that left her beloved father suddenly insane, Mona lives a self-punishing, isolated existence. Her only interests are the math equations she sees everywhere in nature (visualized by Agrelo via CGI pop-ups) and which she mystically ties in with a belief in her dad's possible recovery. But when a job as a local math teacher falls into her lap, she takes it, along the way befriending a young girl with a dying mom and finding a potential romantic partner in an equally quirky teacher, Ben (Chris Messina).
This must be some school: While the arguably mentally ill Mona conducts show-and-tell sessions where students bring in objects both household and natural that resemble numbers (hence the axe), Ben has his pupils talk either nicely or meanly to the plants in their homes and record the subsequent results on growth, while the dizzy principal looks the other way. This is clearly supposed to be deliciously offbeat, but the fact that the film takes place in a world that doesn't feel at all fantastic and that it wants to deal seriously with weighty subject matter makes these quirky details—and many, many others—far more puzzling than enjoyable.
Eventually, through the help of another borderline insane number-obsessed character, Mona starts to come to terms with her math addiction as well as her father's illness. Her arithmetic teacher when she was a kid, Mr. Jones (J.K. Simmons), now runs a hardware store, but still wears a chain around his neck every day with a different number (the higher the digit, the better his mood). The ex-pedagogue's principle role is to explain to the young girl the importance of actually living life and not viewing it passively through the scrim of mathematics. Fair enough, but since comparatively few people's particular insanity is to see the world as consisting primarily of a combination of numbers, the film's lesson is far more limited than Agrelo and company would seem to intend.