unpurpose(ful)ly incorrect

For the first time today, I learned that purposefully and purposely are not, in fact, the same word.
. . . And, of course, I did not learn this until I after I (mis)corrected a student's writing on my white board.

So, after reading this helpful guide and discussing the topic with my brother, Joel has determined that

purposefully is for a purpose 

-and-

purposely is with a purpose.