I am not sure when I learned about Fibs, however, a simple Google search will bring up several bits of information such as the New York Times article,
by Motoko Rich and The Fib Review.
In general, a mathematical fib is a poem where the number of syllables in each line corresponds to the Fibonacci sequence. For example, the first line contains one syllable word. The second line also contains a one syllable word. Line three contains two syllables which can be made from one word or more. Line four contains three syllables, line five contains eight syllables, line six contains 13 and so the sequence continues.
Here is a student example from this year.....
Notice the number of syllables in each line of the fib follows the Fibonacci sequence of 1, 1, 2, 3, 5, 8, and 13. This student could have continued but writing a poem in this sequence is not as easy as it may seem. Try for yourself:)
Here is a series of other Fibs written by my 2013-2014 seventh graders.