What does Big Bird have in common with a hornbill, or Smokey Bear with an aardvark?
They teach you something. But to find out what, you may have to visit the National Museum of African Art in Washington this summer - or at least visit your local library.
However, a good deal of the research you might do in the library has already been well done as part of the museum's ''Thinking With Animals'' exhibition, which offers all sorts of animal-inspired objects from various parts of black Africa. East Africa is not included.
As John E. Reinhardt, acting director of the museum, comments in his preface to the excellent booklet accompanying the show, the problem of what to select from the rich variety of available African art is that ''animals in African art are a Beethoven symphony awaiting the next production of a fine orchestra and inspired maestro.''
This time the interpretation brings out the relationship of animals to people , and the lessons people may learn from close observation of the animals among which they dwell. The exhibition is organized into four sections focused on different aspects of animals as viewed by Africans: animals whose behavior serves as an ideal example for humans, animals that represent status and power relationships, animals thought to possess abilities like those of spirits, and animals who mediate between men and gods or spirits.
Many of the objects are masks; some are utensils, simple pieces of furniture, or doors; a few are gold weights, puppets, or sculptures; and one is a shrine. Labels describe the uses of each, along with photographs of the objects in use.
But for a more lively sense of the effect of the masks as worn, one would have had to go to the Renwick Gallery in Washington this spring, when ''Celebration: A World of Art and Ritual'' presented videotapes and color-slide shows of ceremonies in which the meaning of the headdresses and masks and drums could be more directly experienced by the viewer. It is a pity the two exhibitions could not have run simultaneously to illustrate the full artistry of African communication.
Still, there is more than enough lore to keep the visitor happily engrossed for days. Take the chameleon - with its ability to change its skin color; its human-like habit of carrying its young on its back; its anthropomorphic paws; its independently rotating eyeballs; its prehensile tail; its lightning-swift, long, sticky tongue; and its nocturnal activity. A remarkable creature by any standard. To the Africans it suggests transformation and spiritual contact, as indicated on an ancestor mask and on jewelry in the exhibition.
The hornbill appears many times in combination with other wildlife as well as on its own in the category of animals as ideal example. The birds' family life sets an example cooperativeness and skill of a kind. The males are good hunters; the females good builders. They dispose of agricultural pests efficiently. Humans would do well to follow suit, obviously. The most unusual rendition of the hornbill motif is that of Senufo women's hairstyles.
Antelopes long ago taught men how to farm, according to myth, so they are very important to the Bambara people of Mali. In one headdress, or mask, the antelope's openwork mane is said to represent the sun's course across the sky during the agricultural year. The horns illustrate millet growth. The lower section of the sculpture refers to planting and germination.
The aardvark and pangolin, both good diggers, may be combined with the antelope to remind the farmer of the ability to dig with strength and speed. This combination promotes the idea of cultivation, the importance of agricultural techniques, and the transformation of seed into plant.
Other animals - the leopard, the buffalo, and the elephant, for instance - imply earthly or supernatural power. Some colorful Dahomey banners that use these motifs show that the storytelling quilts made by black women in America have a long and honorable ancestry, perhaps heretofore unsuspected and unappreciated. And then there are sacrifice animals, such as the mudfish shown in sculpture. They are either used as direct substitutes, or as economic equivalents of the real thing, bearing in mind the artist's proper payment.
The only problem with this well-designed, well-presented exhibition is that it seems to be aimed primarily at adults. The concept of ''thinking with animals'' is such a natural for children that one wishes at least some of the materials in the show were presented with them in mind.
Meanwhile, we make do with Sesame Street, the Muppets, Winnie-the-Pooh - and Spider Man, whose animal image came from Africa originally, though the exhibition never mentions him. Maybe the youngsters already know what the museum is trying to tell their parents, and one should let Spider Man take care of himself.