What do you think accounts for the continued popularity of the montessori approach, over 100 years after its inception in italy?

The Montessori approach provides an unstructured learning, where Children can explore the topics they are most interested in. 

It has such advantages as: 

Lack of stress
Teachers can pay attention to every child
Children can follow their talents

Often, the learning is deeper - the children really understand how things work rather than memorize it by heart - because they learn out of curiosity. 

Additionally, children learn to be active in their education and in life, to take lead and initiative rather than only to follow the lead of others. 

Hope this will !