So as we are gearing up for Chana's bat mitzva, as per my Rabbi's advice I am making sure she is capable of davening her basic chiyuv: shemona esrei. (Birchas hatorah she can do because she says them before we do chumash every day.) Her preference is to daven the maariv shemona esrei.
Since she is going to sleepaway camp this summer, it came up that she doesn't know how to daven all the different things the other girls daven. She knows this since she went to daycamp this year. I told her I'd be happy to help her learn any of the other parts of davening. She declined. Then she brought up aleinu. "The girls all sing something after shemona esrei!" I said I'd be happy to sing it with her until she learns it. She said she'd rather not. But she kept complaining about it: "And they move during it. And they stand up. And they bow down and I have no idea when!"
So I sang the part during the bowing: "V'anachnu KOR'IIIIIIM, u'mishtachavim, u'modim, lifnei MElech, malchei hamelachim, hakaDOSH baruuuuch hu." I sang it in the classic tune. After I sang it, she said, "Oh. You bow down when it talks about bowing to Hashem. That makes sense."
So the speaking Hebrew is extremely useful as far as unschooling goes.