Answer: As you probably know, when the tops of potato plants flower it usually indicates that the tubers are beginning to form. This is a critical time for watering, so keep the soil moist. After flowering is the time you can harvest 'new' potatoes. The longer the plants are alive, the larger the tubers will become. Normally when the tops yellow and die, the tubers are mature. It doesn't sound as though your potatoes went through the entire process. When you plant potatoes, use certified disease-free stock, and keep the plants watered well. Leave only a small portion of the growing vines exposed to encourage additional root development. Once the plants blossom, stop mounding up the soil. Sometimes too much fertilizer, especially nitrogen, will encourage the growth of the foliage at the expense of the tubers.
Seed potatoes are pieces of actual potatoes that contain one or more "eyes" (sprouting points). Most gardeners I know cut them up to get more starters. Make sure they each have at least 1, preferably 2 eyes. Plant seed potatoes in deep, well-drained but moist soil. Dig a one foot deep trench and place the seed pieces about 12" apart in the trench and cover them lightly with soil. As the sprouts emerge, bury them with soil. Do this 2-3 times until the soil is mounded around the emerging sprout. The buried part of the plant will develop roots and provide plenty of space in the soil for forming the potato tubers.
Potatoes are succeptible to many fungal and some bacterial diseases and perhaps that was a factor. To reduce the potential for a problem, rotate them to a different bed each year. I hope some of this info helps!
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