On the other hand, the individual family is arguably too low-level to do all of it. The base of human knowledge is simply too broad even for two well-educated parents to teach their children completely on their own. My knowledge of calculus, dimly recalled from high school classes many years ago, just can’t cut it; I’m going to need help teaching my children that.
"the base of human knowledge" - wait, as in University? I mean Cardinal Newman in Catholic Idea of a University says the meaning of a University is to provide a teaching of human knowledge in all and every field. Yea, if your children are in for University, do give them a chance to study calculus too. And to discuss how it translates into real mathematics. For logarithms I have translated "10-logarithm of 2 is the power of ten, if raised to which tens multiplied by themselves produce two" into "10-logarithm of two is the ratio where a power of ten and a power of two approach each other at the proportionally most possible (they will never be identical)".
Generally speaking all children need to learn about God. If parents do not know their catechism, there are catechists provided by parish. If parents do know it, they or their children are eligible as catechists, if the post becomes vacant, meanwhile they can teach their children without overburdening catechists. And an older sibling may very well teach it to younger siblings.
A farmer's boy was usually taught all there was to know about farming (i e traditional way) by his parents at their farm. A cobbler's boy usually learned what he needed to know about cobbling by a colleague of his father, ditto for a goldsmith's boy or a blacksmith's. My greatgrandfathers on ma's side were the two types of smith, my grandfather was first cobbler's apprentice before changing trade (ending up as a distiller). He did have an appetite for knowledge, but he got it mostly from reading - and I was his apprentice in reading.
Louis XVI had this hunch that both smiths and cobblers might usefully know something about drawing, so he started opening drawing schools, only getting around to the one in Paris, for free, for poor children. This is one thing that earned him a reputation of wastefulness: masters wanted to reserve drawing lessons for the apprentices and journeymen they wanted to be masters, not for every one, whom they wanted to stay journeyman (employee).
But if you are only dimly aware of calculus, it means you hhave not needed it in your trade, which means your children, if they get same or closely similar trade won't need it either.
A girl was typically taught wifing and mothering, sometimes - optionnally - she got a wide reading as well. Is that such a bad base for human knowledge?
I appreciate the drawing lessons (I never learned to draw well) but calculus as an excuse for your children absolutely needing a school? Seriously?
But more to the point: if parents are not quite sufficient as school, they may still be quite sufficient as their own children's "school board". The fact that some parents, indeed most, won't chose calculus for their children as an argument against that? Seriously!
First version of same comment on same quote (that quote ending by his inadequacy for teaching calculus)
Assuming your children need to know it. If your memories are dim, chances are you have had no need of it. If your children have similar interests, chances are they will have similar professions and no need of it either.
On a farm, formerly, parents taught children what there was to be known about farming. If they missed Leibnitz’ differential calculus, big deal. In towns, one cobbler would send his son to another to be taught the trade (it was my grandpa’s first attempt at a profession), one goldsmith to another goldsmith (which was his father’s trade, though he stayed the level of journeyman, never becoming a master).
Louis XVI rightly thought that goldsmiths and cobblers could both need drawing, which is why he wanted to start drawing schools for free for poor apprentices. In Paris one opened. But calculus? Seriously?
I left him with a comment I told him not to publish, chosing between my two ones