Back in the mid-to-late 20th century, there was a movement to teach mathematics rather that way, though I don't think it ever got as far as teaching small child

Back in the mid-to-late 20th century, there was a movement to teach mathematics rather that way, though I don't think it ever got as far as teaching small children the Peano axioms. The "New Mathematics", they called it. I remember encountering talk of sets and functions in UK primary school circa 1975. It never worked very well in practice. I expect that when you're fortunate enough to have unusually able teachers teaching unusually able pupils it might do OK -- but then, in that situation, anything will do OK. (Tom Lehrer fans will know about this from his song "New Math", with "It's so simple, so very simple, that only a child can do it" in the refrain.)