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A kushiyah is a challenging question. It comes from the word kasheh, meaning hard or difficult. Kushiyot are the bread and butter of the Talmud.
If a rabbis position on a specific halachah can withstand any kushiyot it faces, then his view is legitimate; if, on the other hand, there is a case in which the position is not defensible, then the rabbi is stuck with the kushiyah and his opinion is considered unproven.
The late Nechama Leibowitz emphasised the difference between a kushiyah and a question. The latter is a request for information, as in: In how many days did God create the world?
A kushiyah is the result of a sense of difficulty or problem. Hence, Leibowitzs famous question, Mah kasheh lRashi?, What is the kushiyah driving [the great biblical and talmudic commentator] to interpret a verse as he does? Rashi was not in the business of simply supplying extra information; rather his commentary was always in response to a perceived difficulty in the text.
On Pesach, the children ask the Four Kushiyot, also known as the Mah Nishtanah. The kushiyah is a crucial educational tool. When a teacher or parent hears the point at which learning becomes kasheh, difficult, he is able to guide the student or child beyond it to the next level of understanding.