When a child is done chomping on her gum, see what she can design with it on an index card with a toothpick. Make it a timed contest with her friends, and have the group vote on whose sticky sculpture is the best. Display them for a bit before throwing them away. For a more advanced craft project, a child can attempt a more complex work of sculpture using only gumballs or "bricks" of unchewed gum.
For older children, bubble gum balls make a fun addition to a list of available materials for a DIY lamp. Not only will students learn about the construction of a common household item, but they can decorate their lamps in a potentially pleasing and artistic way. The children can glue gum balls to the base of a lamp, or construct a wooden box for a base and glue the gum balls to it. Gum wrappers could be decoupaged onto the shade, which could be made of an upturned gumball bucket.
If you're willing to devote some time to an art project, bubble gum might be the medium for you to create a striking portrait or landscape. Several artists have garnered attention from layering bubble gum of different colors and shades into pieces of art. Some public art displays have included exaggerated areas of bubble-gum graffiti; in others, the artist has flattened wads of bubble gum found in public spaces and painted them in a variety of ways. Have the children try something similar, or have them make a mosaic out of unchewed gum squares.
Maybe you don't want to work with a material that's been in your mouth. It's understandable. Keep the wrappers on and let children use bubble gum as one of several materials in a themed collage project. For instance, gumballs or wrapped gum can make an apt addition to a list of materials to use in making a collage about childhood. Or your child can create a bouquet of bubble gum using a flowerpot, floral foam, fake grass, stem wire, decorative paper and ribbons. Attach the wrapped bubble gum to the stem wire to form flower buds.