My kids are not school age yet, and I am not sure on if I will home school or not. But I do think its possible to get good socialization exposure while homeschooling. There is the neighborhood kids, you have sports and clubs kids can join, religious groups.

Plus not all homeschooling is just a student staying at home all day. Some people "homeschooling" I know are groups of parents getting together to educate their children together in small groups of ~5 kids to share the responsibility, and hiring a tutor to fill in the gaps. Monday they go John's house, his mom has a philosophy degree and teaches them. tuesday they go to Janes house, her dad is a Mathematician and teaches them. etc.

Everytime I see these kinds of arguments, it sounds like someone desperately trying to argue that a park playground is almost as entertaining for kids as an amusement park. Your example of 5 kids socializing with each other is definitely better than 1 kid at home. It’s also definitely worse than learning to socialize in a school of 500 kids each day. This is undeniable unless you have an argument of how a pool of 500 kids would somehow have less diversity of personality, thought, languages, physical features, intelligence, etc.

I used to work at a YMCA, and the local homeschool group asked us to do a PE class, which I taught.

I had the kids doing swimming, rock climbing, and all kinds of traditional PE games.

I worked with "normal" kids most of the time, and I will say the homeschool kids stuck out. They're more awkward around kids their age, but far less awkward around adults. They know how to speak and act, in large part. And they were disproportionately ahead of their peers academically--though I think that's probably a selection bias for the parents seeking out homeschool PE classes.

This was in the early 2000s, before Facebook. I'm sure the avenues to connect have only grown with social media.

15+ years ago, that might have been the case. Now, you might find some friends in the 3-8 year old range, but then the kids just...don't do things anymore. In both suburban neighborhoods I've lived in the past 10 years, there are basically zero middle school or high school kids doing anything except playing video games and messing around on their phones from the comfort of home. School is quite literally the only social interaction most of these kids get aside from their parents, and if they didn't go to school, they'd just spend more time playing video games or on their phones.

Outside of the coasts or university towns, there aren't any "mathematicians" with kids just waiting around to form homeschooling groups with you.

It's going to depend greatly on your geo location and socioeconomic circumstances, but a homeschooled kid who interacts a lot in the neighbourhood (big "if", IME; those kids all have a lot of school friends) is still going to miss out on broader social, cultural, racial and financial exposure. Example: my white, middle-class kids have a lot of people exactly like them in community groups and sports clubs, but lots of eastern european & asian immigrants in their school classes. This is super-important in elementary school when they're far less aware and insular about interacting with people who are "different" IMO

You don't need a degree in math to teach children age-appropriate math topics. Teachers don't become teachers just because they have a degree in that subject, they have been taught the methods on how to teach. Having prior knowledge of the subject is almost irrelevant. Teaching is really just applying solid methods on how to build knowledge from the most basic concepts as well as having the patience in dealing with humans who are not fully formed in their emotions.

There is no such thing as “the neighborhood kids” anymore. Having any kind of social circle for your children is going to require your facilitation and effort… a lot of it. It’ll be extra hard without the common bond of shared activity.

Not knocking what sounds like your choice to homeschool, just sharing something that has changed from my youth.

John Jane Mary set up is incredibly idealized. In a big city I have not been able to find anyone willing to commit to anything except one off play dates in a museum which has nothing to do with actual education.

Having a degree in philosophy or mathematics or whatever does not automatically make someone a good teacher. Teaching - particularly with young children - is a skill that is almost orthogonal to subject knowledge.

“Good socializaron experience” is the exact opposite of “religious groups”. Said as someone who went to a private Christian school for 7 years.

It sounds like school with extra steps.

Part of "socializing" is observing that one's parents aren't the absolute authority in the world. Parents sometimes butt heads with teachers, coaches etc. No home schooling scenario can provide this experience. I think it leads to enhanced levels of narcissism in both students and parents.