yeah, the leaves die back. In cold winter country like yours and mine, elephant ear is just dug and stored for the winter.
Gently lift them from the soil, keeping as much roots as you can. Cut the foliage off, rinse the dirt off the bulb/ tuber, pat it dry. Leave them in a warm dark dry place with good air circulation for a week or so to be sure they are thoroughly dry. Then wrap them in paper and store them in a cool dark place until the soil has warmed up in spring, re-plant.
I guess it is possible to keep them alive and growing as houseplants for the winter, but seems like more trouble than it is worth. Especially since the leaves of yours are already dying back, I would just store as above.