Divinity is sensitive to moisture and oil, so I always use metal pans, bowls, and spoons. I wipe all utensils and pans/bowls well with a mixture of 1 part vinegar/1 part water to remove all oil traces. Also- make sure your ingredients are all at room temperature, especially the egg whites.
Prepare cookie sheet by buttering well or spraying with cooking oil.
In large, heavy saucepan, combine sugar, Karo syrup, and water. Heat over medium flame, stirring constantly until sugar dissolves. Cover pot and cook for 3 minutes to allow sugar to wash down from sides of pan. Uncover and continue to cook, without stirring, until it reaches 250 degrees.
While sugar is cooking, beat egg whites in large metal bowl with mixer until stiff peaks form. When sugar syrup reaches 250 degrees, pour 1/2 sugar syrup in thin stream over egg whites, beating constantly.
Return remaining sugar syrup to heat, and cook until 270 degrees. Pour syrup over egg white mixture in a thin stream while beating constantly. Continue to beat at high speed until mixture is thick and shiny. Add vanilla or almond extract and beat well.
Drop by spoonsful onto prepared pan. Immediately press pecan half or cherry half into top of each piece. Allow to cool and set, and store in air-tight container.
NOTE- if you would like to color your divinity, you may add a few drops of food coloring with the vanilla extract. Be careful not to add more than 3 or 4 drops, it will thin the candy too much and cause it not to set.