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The worst way to brew gourmet coffee is to do it the way your parents and grandparents used to do it - by leaving it to boil and stew on the stove for hours on end. You want the coffee to brew for 4-6 minutes only and then either drink it right away or keep it in an insulated carafe. Keeping it hot over a hotplate may be convenient, but it really spoils the more subtle flavors of your coffee.
Two ways to avoid 'stewing' your coffee is to either use a French Press or use a coffee brewer that drips the coffee into an insulated carafe, like the Melitta Mill & Brew Coffee Brewer.