This book is an attempt to look at mathematics from a new and somewhat unusual point of view. I have started to systematically record and analyze from a mathematical point of view various difficulties experiencing by children in their early learning of mathematics. I hope that my approach will eventually allow me to gain a better understanding of how we—not only children, but adults, too—do mathematics. This explains the title of the book: metamathematics is mathematics applied to study of mathematics. I chase shadows: I am trying to identify and clearly describe hidden structures of elementary mathematics which may intrigue, puzzle, and—like shadows in the night—sometimes scare an inquisitive child.