Can I start some new African daisies...

Yes, you can start new plants from cuttings. Snip off a shoot, trim off some of the leaves, and stick it in a pot of moist compost. Place the pot out of direct sunlight. Water enough to prevent the compost from drying out. The cuttings should root in a few weeks. To check whether roots have formed, see whether roots are protruding from the drainage holes in the bottom of the pot, or give the cutting a gentle tug to feel whether it is anchored in the pot. A few shoots may wither and die before rooting takes place, so take several cuttings from the same plant.