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Turning a problem on its head

Reader question:

What does "turn a problem on its head" mean? Please give examples.

My comments:

When you turn something on its head, you twist it, turn it sideways or upside-down. To turn something on its head literarily is to alter it, to contradict it. In a dictionary (Longman) explanation, to turn something on its head is "to consider a statement or idea in the opposite way from the way it was intended."

My favorite example of turning a problem on its head runs something like the following (I have a lot of practice at this):

Owen (not his real name) told me the other day that he has little interest in studying English because his teacher back in school many years ago never liked him. "That teacher," he said, "was never encouraging. I think I never developed an enthusiasm for English because of that teacher."

Oh dear! All teachers beware! This is how pupils pay you back (just kidding).

I don't have your teacher around, Owen, to give his side of the story. I'll give you the benefit of the doubt. But even supposing what you said is true, I can't help turning your problem on its head a little before giving it back to you: Your teacher's not your problem. Keep him out of it.

In all likelihood, your teacher could say that he was not discouraging. Or he could say that he was less than encouraging because you had never shown any interest. He could, you see, point his fingers right back at you.