"Help me understand about unschooling. I know kids learn to talk and walk with no overt teaching but they learn by example, I think these are more of a biological drive in the human being. I am not sure learning Pi falls under the same category."
If you want to look at it from a "biological imperative" standpoint, I think you can definitely lump learning about pi into it. I think human beings are programmed to learn to walk and talk, but I think another thing that sets us apart from most of the rest of the animal kingdom is that we are programmed to learn from others. Humans have memes as well as genes: bodies of knowledge that are passed along by the culture that surrounds us. And I'm not talking about school-ish learning here. I'm talking about the sort of learning that in the distant past let hunter-gatherer kids know which plants near where they lived were safe to eat, how to fashion a weapon, how to build a shelter. We are hard-wired for learning. These days the things children are driven to learn are less of the "building a shelter" type and more of the "swapping graphics cards" or "calculating interest charges on a loan" type. We are hard-wired to learn whatever is necessary in our particular environment for becoming a productive, capable member of our society.
Unschooling doesn't mean no formal learning: it just means no uninvited teaching. Much of an unschooler's learning may be informal, but if they want formal structure to their learning that's totally cool. My unschooled 11-year-old is currently getting up every morning to sit down with a high school math textbook and do work with pencil and paper to master it. She has decided that higher math is likely to be useful to her, so she wants to learn it. My 14-year-old son who until recently had nothing you could call handwriting gradually discovered that there were a few occasions in real life when it was helpful to be able to write neatly and efficiently with a pen. Since real life wasn't giving him enough practice to get good at it, he set to work making himself practice on a daily basis, and now has a neat legible written script.
Proving learning to an overseeing body hasn't been a problem for us. My kids are learning like crazy, and progress is well-nigh inevitable. It isn't necessarily linear and steady, but over the long term, like a school term or two, there's always stuff I can point to as evidence.
I've also not found that my kids need much if any prodding to challenge themselves. Occasionally (rarely) they have needed some substantial support from me in following through on their desire to challenge themselves. But for the most part when they know that they are fully in charge of their own learning their ambition and motivation rises up and propel them forward. In fact they often challenge themselves far more than I would ever have thought of expecting of them.