Now two writers of note have tackled this question. Dinesh D'Souza has written an excellent article for Forbes in which he demonstrates how the future president spent his youth in places where anti-colonialism was the prevailing ethos. He says of Obama:
Here is a man who spent his formative years--the first 17 years of his life--off the American mainland, in Hawaii, Indonesia and Pakistan, with multiple subsequent journeys to Africa.
D'Souza spent his own youth in Mumbai (aka Bombay) and knows whereof he speaks. He also makes much of the fact that one of BHO's autobiographical books is entitled Dreams from My Father, instead of Dreams of My Father. This strongly suggests that his Muslim African father was a source of his inspiration and values.
The other article, which makes favorable reference to the D'Souza piece, is by Thomas Lifson, editor of American Thinker. Lifson writes of the Indonesian expatriate experience of Obama (then Barry Soetoro) whose family were treated as inferiors by the highly subsidized (white) expatriate Mobil Oil executive families, the firm for which his stepfather worked as a low-paid local.
Lifson had a related experience in post-war Japan and shares with us what that may have felt like to young Barack. Lifson entitles his article "The Grudge," which give you an idea of how it felt to him. We know Obama left Indonesia and went to live with his grandparents.
In Noonan's parlance, we may not "get" Obama because our formative years were very different than his. As we try to get a handle on what is a Barack Obama, people who have had expatriate or cross-cultural experiences may turn out to have more empathy with, and be better able to explain, our President.