Roosevelt on the front. The back has Providence, R.I. Standard Emblem. This is the last item in that cigar box I took a photo of. I've posted many of them but I could post about a hundred more. Back to coins now.
Interesting piece. There is a similar one selling on Ebay for $20. The NY Historical Society has something similar, dated 1920, but it appears to be a larger medal? The Manhattan Historic Sites Archive has a copy of the association's Charter and By-laws. Similar piece here. Another one for sale with some additional info here. There appears to be quite a bit of information on this pin and its associated organization out there. I'm not sure if it's suffrage related or not, though.