Just want to point out this sounds more like a private->public pros and cons list. I’m not doubting it at all, just not sure if it’s due to the Montessori method. Most Montessori’s are or are operate similar to the private setting.
While the atmosphere wasn't toxic like I hear people say about public school, few of my peers were interested in learning, excelled academically, etc.
A non-negligible portion of the kids in my school were the ones kicked out of the public school for grades or troublemaking. I think they did better at my school, but that population affected the academic experience. My brother went to another nearby religious school and it was the same.
My school was caught in a loop of poor funding -> sub-par teachers -> less enrollment -> less funding.
I expect a school focused specifically on self-actualization and skill-building to have better results than an arbitrary private school.
If you just throw kids together and expect the syllabus/curriculum to prevail, you'll quickly find the Lord of the Flies elements of public school social dynamics come into play and for a child often are more important than academics. At that point, simply being "private" has no advantage.
Based on the experience given above as well as my own experience, I'm going to guess the poster was talking about Catholic parochial schools. Catholic parochial schools were the original public schools in North America and still operate as such today, except obviously no public funding in the United States (they do in some provinces of Canada).
Either way, they'll accept basically anyone. However, they do consistently give better results anyway, even adjusting for income and social status. Actually, the effect is most pronounced in the lowest incomes. The richer you are, the less difference parochial v public makes (of course going to super cushy private schools puts you at an advantage).
In math the kids definitely leveraged the binomial/trinomial tools, beads and other tactile methods to have a better understanding of the concepts of mult/div/exponentials/roots long before they would have approached in the standard US arithmetic methods. They were also doing algebra and geometry by the end of Montessori.
Montessori also spends a lot of time working on holistic views with regards to science and history and then give the kids the freedom to do their own research and projects on the topics. The normal school models tend to take more of a cause-effect + memorize the dates method as that's what tracks the testing. It's not generally till university that they then go back to something that usually requires critical thinking.
On critical thinking there's an emphasis there in Montessori on involving the kids in asking more questions and then building a model to analyze and defend their position. Public school is optimizing for learn by rote and it's a culture shock going to Uni where they back to something that's utilizing critical analysis.
My older child is in IB at the moment and they're using much of this as it's taught in more of Uni style.
I also like that philosophy is compulsory, I feel that the humanities are under served in our current school systems.