I am a huge advocate of the Montessori method. It is really hard to explain, but I strongly encourage you to go "observe": in a classroom. This is common practice in a real Montessori school so you shouldn't feel strange about it.
The focus of montessori is, instead of making all the kids do the same work at the same time, embracing the unique developmental stage of the child to allow them to be successful and really master each level of work. Teh goal is to develop intrinsic satisfaction in learning and a sense of being "proud of self" rather than relying on making mom and dad proud.
The work itself relies a lot on manipulatives. If you go to a classroom ask for a demonstration of a "lesson." I swear I didn't understand base 10 math really well until I saw it demonstrated on Montessori beads!!
Montessori is also a mixed age classroom where the kids stay with one teacher for 3 years (ages 3-6 including kindergarten). This is so great for teaching the kids to work together, respect each other, teach each other, and grow to be leaders.
I don't think that Montessori creates really advanced learners - people who send their kids their to make them gifted and put them far ahead in school will be disappointed, I think. But they will get kids who love school and who are inquisitive - and who have a much deeper conceptual understanding of math and reading concepts that will prepare them to be great learners.
Last, and maybe most important, Montessori really focuses on educating the "whole child." Montessori focuses on independent thinking, responsibility to self and community/environment and respect and "grace and courtesy.".
My older daughter was in a Montessori primary classroom through kindergarten and is now in a public first grade. She did adjust slowly, but for her it was more a social issue of being in a big class with new kids, not an academic problem - except she was way ahead in math. I wish we could have left her at her Montessori school for elementary school beacuse those kids do amazing things. But it was too expensive. Our younger daughter starts next year.
We alays joke that we notice when we ar out with a group of Montessori friends.the differences - one table at a restaurant will have kids literally wakling on the tables and our kids are waiitng patiently for their food and are having conversations with each other and the adults. The expeience, with parental reinforcement, really seems to lead to mature and self-assured kids in general.
I would struggle to find a con, but I guess you have to support the approach and set similar expectations at home. The kids I have seen not do well are the ones whose parents turn on the tv during dinner and don't read with their kids... that sort of thing, but those kids are in trouble anyway in my opinon. And yes, it is expensive, at least where I live - $7k per year..