Grand master of the Inoue school of dance - Yachiyo Inoue V talks about the meaning of folding fans on the day of the Kotohajme. Yachiyo Inoue is a National Living Treasure but she keeps on practicing her skills everyday. She also teaches every maiko and geiko in Gion Kobu herself. On 13th December they visit her house and gift her mochi cakes as a symbol of thanksgiving and respect. In return, all maiko and geiko receive new fans that they will use within the following year.
Source - Winter Comes to The Geisha