Daisy, I think what you describe might be seed pods. Look at some of the photos on google, they have a cup shaped bottom which holds seeds with a flattish pointed cap.
https://www.google.co.uk/searc...
On the other hand, some plants do appear to make clusters of small leaves at the end of a stem which would root when laid on the ground, look at the photos posted by jojoe on this old thread ..
The thread "rooting portulaca???" in
Purslane and Portulaca forum
This photo shows seed pods alongside the clustered leaves on the end of a stem. Could this be the leaves you were thinking of, which would perhaps become detached from the main plant with a small amount of stem attached and that is what would take root?
http://garden.org/pics/2012-08...
