Children at this age are developing essential prereading and math skills. In a box, assemble magnetic letters and numbers along with a cookie sheet for them to manipulate them on. According to Healthy Children in "Hand and Finger Skills of Your Preschooler," 3- and 4-year-olds are starting to develop skills that allow them to grasp a pencil and start forming letters and numbers independently. Pencils and writing paper in the box gives them the opportunity to practice these skills.
Preschoolers enjoy creating, and a box with many supplies allows them to use their small motor skills. Glue sticks, paper, self-adhesive foam shapes and crayons can be used independently by a preschooler. Old magazines and greeting cards that they can cut pictures out of are appreciated. Throw in odds and ends, such as ribbons, chenille wires, beads and any other crafty supplies, to keep the box fresh and interesting.
Modeling dough can be purchased inexpensively or made at home using a recipe found on the Internet. Preschoolers can spend a lot of time manipulating the dough, creating items such as snowmen, snakes, bears, or anything else in which they are interested. Along with the dough, parents can include items in the box like cookie cutters, small rollers, cars to drive through the dough and other small tools that can be purchased to use with the dough.
Children seem to have an innate interest in animals, so parents can create an activity box to foster this learning. Small plastic animals, along with accessories like fences, trees or barns, can be manipulated so preschoolers can use their imagination. Include some books about different animals in the box that preschoolers can look at independently or with an adult. If a child favors a certain type of animal, such as horses, encourage that interest and place things in the box that help the child learn more about them.
According to "Sensory Integration Activities," sensory development is crucial in the preschool years. Many children this age enjoy sensory activities that not only help calm and focus some children, but help them learn about the world about them. In one activity box, parents can place a sensory material, such as loose sand, rice or beans, along with spoons, cups and bowls that children can use to scoop and pour the material.
Preschoolers not only enjoy listening to music, but participating in making it as well. Fill a box with assorted CDs that a child can choose from, along with some instruments the child can play while he marches around. If traditional instruments are not available, place beans in a plastic container to shake as a maraca, bang some spoons together or tap the bottom of a pot.