The ERIC Clearinghouse on Disabilities and Gifted Education identifies some of the characteristics of gifted children. These include their intense curiosity, their tendency to ask a lot of questions, their ability to learn quickly and remember things without much trouble, and their tendency to think about the world differently than their classmates. Identifying gifted children and engaging them in activities that suit their mental facilities will keep them engaged in the learning process, rather than bored.
Using opportunities to incorporate creative thinking and activities into everyday lessons is one way to engage gifted children. Using "what if" questions to look at a different way of exploring a subject is one way of doing this. For instance, a study on rainforests could be an opportunity to ask a child to imagine what it would be like to be a lizard in the rainforest. This would allow the child to gather facts about the lizards and present them in different ways.
Project EAGLE, a program for gifted kindergarten children, has come up with projects that involve poetry and painting. One activity is based on a poem by J.D. Evans "In Search of the Xanthus," about the search for the imaginary creature. The poem is printed on a booklet that has large-size type and large line drawings that allow the children to trace and color the drawings. Similarly, there is a second booklet with a poem, also by J.D. Evans, about another creature from outer space. This too allows for similar painting activity.
There are also various theme-based projects that you could engage gifted children in at the kindergarten level. One such project involves hexagons, which are six-sided figures. After distributing some hexagon-shaped figures, ask the children to name them and define their properties. Ask the children to find other combinations of shapes that could form a hexagon. For instance, six triangles together. There are a number of other similar activities that the children could engage in on each theme.