As a medical doctor, Montessori believed the brain goes through a series of “sensitive periods” for cognitive development. For example, the distinction of color, size, shape; math concepts, writing, etc. Also, just as something within the child prompts them to work at sitting up, crawling, walking, etc., so it is with cognitive skills.
This something shows itself through interest. That is, interest prompts the child to work diligently on perfecting a new skill they’re ready to develop.
So, just as with motor development, a child demonstrates interest in activities designed to develop a skill they’re ready to perfect. To accommodate this, she had a variety of special material/equipment made to hone particular skills and concepts.
Therefore, in a Montessori classroom, the directress/teacher demonstrates how to use a particular piece of equipment. Then she places it where the child has access to it. But she does not require, or even ask the child to use it.