Answer: Paper whites and other varieties specifically intended for forcing can be started the way you have described; the bulbs are left as is and the pan is placed in a bright location. Keep the water level at just below the bottom of the bulbs. They will root into the water, and on about three weeks, the bulbs should flower. After that, the bulbs are discarded. These types of daffodil are not winter hardy in cold winter areas such as yours. If those are the instructions that came with your bulb, then that is how they should be treated.
However, most types of daffodils are winter hardy and would not be forced as described above. Instead, they would be planted in a pot of soil, the pot would be chilled for several months to allow the bulbs to root and come into growth, then the pots would be moved to a cool and very bright location where the flowers would open after several weeks. The bulbs could then be planted in the garden where the foliage would be allowed to grow, mature, and ripen and the bulbs would be left in place to bloom the following year.
March is very late for starting them into pots and beginning a chilling period, but you could try that, although I would not be too optimistic about the long term results. The soil outside will be warming up shortly so there isn't time to chill them naturally in the ground.
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