Suppose that it takes 0.6 seconds for a mass on a spring to move from its highest position to its lowest position. what is the period of the spring?

The mass on the spring is bouncing. 

We would call it a wave-like motion, except that it all stays in the same place.  But, just like a wave, moving from the highest position to the lowest position
is one-half of a full wiggle. 

(The other half consists of moving from the lowest position back up to the
highest position, where it started from. 

So, half of the wave-like motion takes 0.6 seconds. 

A full cycle of the wave motion ... the actual period of the bounce,
is double that much time . . .
                                                   1.2 seconds.