Answer: The water chestnut used in Chinese cooking is Eleocharis dulcis. The part we eat is sometimes called a corm and sometimes called a tuber -- but in layman's terms we eat the root part. It looks like a reed and grows as a bog plant, meaning in standing water, and needs full sun. Most of the world supply is grown in Asia and more recently in Australia. You might find the following information interesting http://www.boofarm.com/earthcare/Wchestnuts.htm but basically it can be grown in any container which holds water, the corms are harvested in late fall after the leaves have died back.
Do not confuse this plant with the noxious water weed Trapa natans which is also commonly known as "water chestnut".
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