@Horntoad
Jay, I took some photos this afternoon. Maybe they will help.
This is how I have it currently. It's in a good sized container filled with heavy red clay and the smallest leaf is the joint that is planted in the clay. The container with clay is submerged in a plastic cattle tub. The yellow cattle tub is about the size of a big round galvanized wash tub.
Here I just lifted it out of the cattle tub and drained some surface water off. That dark leaf stem leads to the joint in the clay, anchoring it. The wiggle tails like dirty water, so I need to water other plants with it.
I tried to get a photo of the growth. It's essentially running like Bermuda grass. Runs, makes a joint with roots and a single leaf; then another joint and another. At the end you can see the small new leaf already with roots but also already sending out a new growing point.
Obviously I don't understand what it's going to do. Just make one long rope indefinitely? Will a joint eventually put out more leaves and a bloom I have visions of a 1/8 mile long runner circling around and around in the stock trough

and still growing. Sure isn't growing like the Nympheas. Can you tell from the photos if it's American Lotus (Nelumbo lutea) from these photos. To me, it doesn't look like the photos you gave of Nuphar lutea.