As the others have said, art therapy can be used to gauge a child’s development in hand/eye coordination and perception. There are certain stages children tend to go through when they draw.
From infancy to about two or two and a half years old, a child’s drawing are usually scribbling - mostly vertical and horizontal lines back and forth. As the child gets a little older, she begins scribbling in a circular pattern.
From there, children usually begin drawing thing in their immediate environment - family. Early drawings tend to be just the head with limbs attached. Once the torso is introduced, there is indeed a period where the belly button is fantastically important. Ears tend to be drawn before noses, interestingly enough.
The size of the subjects tends to be very canonical - Dad the largest, Mom next in size, brothers and sisters smaller.
At some point, (around four or five) children usually begin drawing the outside world. Almost always, the sky and grass begin as bands of color at the very top and very bottom. Usually within a few months, the child begins to color the entire above-ground area as sky.
Perspective is extremely limited as showing detail of the people involved are much more important. Transparency of objects is common because children haven’t learned how to draw foreground, background, and intervening layers. So, to show a drawing of a person holding a hot dog, the child draws the person first, with their arm sticking out and their fingers extended. Then they draw the hotdog on top of the fingers. When drawing a family gathered around a table or say, picnic blanket, the child will often flatten the table, draw the farther members upright and draw the nearer ones upside down - as if you were looking at them from above instead of from the side.
Children tend to draw flowers as either tulips (straight stem, cup with three or four points at the top) or daisies (straight stem, petals surrounding a center, always seen full on). Trees tend to be a straight brown trunk topped by a green leaf circle. The sun is almost always drawn in the upper lefthand corner. Houses have a door in the middle, a window on either side of the door, and a chimney with smoke coming out of it. Sometimes the windows have curtains.
Most children start to leave these conventions behind by about 11 or 12 years old. Development can usually be gauged within a year’s variation. Problems are indicated only when there is a substantial change in the way the art is produced or when nearly all of the art produced is markedly and consistently different from the norm. Even then, it’s only an indication.
<snipped couple of paragraphs on Art Brut, Naive Art, and Innocent Art. Fascinating, but off topic.>
There are some other things that were mentioned in my classes that I want to verify before presenting them as absolute:
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children’s drawings are so archetypal that standard representations of house/tree/flowers appear even in non-Western cultures.
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several artists and art psychologists were given copies of work by Adolph Hitler without being told it was his work. They critiqued his work as being very detached and cold and as treating the human figure as an object instead of a person.