Thanks for the tips Skoobysnax!
I've put the cuttings in a larger container (a plastic bottle cut to size) filled until right below the leaves.
I was a bit hesitant to make any cuttings at all, but one of my 2 plants didn't make it through the winter so I wanted to make sure I got some fresh plants going while I still can.
My mother plant is growing tall and slanky, only the tops of the stems are growing large leaves, the other leaves only get to a few cms before turning brown and dying. Could this be because it is rootbound? I guess I should repot it soon to give it some more space.
I don't water my plant often as I've heard it is easy to get root rot that way (in fact, I believe that's how my other plant died).
I wait until the leaves start to droop slightly, and then give them around a liter of water (with a bit of fertilizer in the spring/summer) and let the pot drain. I do mist the leaves a few times a day so they can absorb a bit of water that way.
Also, is the leaf browning due to overexposure to direct sunlight?
In spring and early summer my plant always looks super healthy with gorgeous large silky deep green leaves, but as summer progresses I get more and more browning. I live in a rather cold climate, and on a windowsill with moderate sunlight and temperature it does far worse than in full light in my veranda (where the humidity and temperature is also quite a bit higher).