So we've found the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ..., but how does that lead to the "golden spiral"?  Here's how:

1) draw 2 squares next to each other, each 1 unit by 1 unit

2) draw another square and place it on top, to fit neatly it will need to be 2 units square

3) draw the next square next to the "1" and "2" squares, what size will this need to be?

4) draw the "5" square underneath and keep going until you reach the edge of your paper

5) mark a point at diagonally opposite corners of each square then join up these points to make a nice smooth curve.

6) The resulting curve is none other than the golden spiral, formed from "Fibonacci-sized" squares