As a medical doctor, Montessori believed the brain goes through a series of what she called “sensitive periods” for cognitive development (the distinction of color, size, shape; math concepts; writing, etc.).  Also, just as it’s something within the child that prompts them to work at sitting up, crawling, walking, etc., so it is with cognitive skills.

This “something” that prompts them to work diligently on perfecting a new skill can be described as “interest.” 

So just as with motor development, she believed that during each sensitive period for cognitive development, a child would naturally be interested in activities that sharpen a particular skill.  So she had a variety of special materials/equipment made to hone particular skills and concepts.

So a hallmark of the Montessori approach is allowing a child to choose which they are interested in/drawn toward working with. The directress/teacher simply demonstrates how a particular piece of equipment is handled, then places it where the child has access to it, and allows the child to decide when to use it.