He refers to Arnold Zwicky's distinction between the Normal, the dialect or variety of dialects we learn from childhood by listening to or speaking with other people, and the Formal, the dialect we learn through reading, writing, and schooling. Can is Normal; may is Formal. And it is a mistake to think that Formal is somehow more correct than Normal.
But wait. As spoken and written American English have become steadily more conversational over the past century, can/may has eroded. Here's Bryan Garner: "Although only an insufferable precisian would insist on observing the distinction in informal speech or writing ... it's often advisable to distinguish between the two." But there's more. Educated speakers regularly say "can't I" rather than "mayn't I," "can't" rather than "may not." And "because may is a more polite way of asking for permission, a fussy insistence on using it can give the writing a prissy tone."
Lane Green moves from can/may to the deeper issue of how badly classrooms have handled language instruction. "When children are suddenly told that what they know their parents and nearly everyone else says, and what they have been saying all their lives thus far, is 'wrong,' there is a disconnect between the child's native competence and the new idea of an invisible but Platonically correct language out there. ..."
The way they are taught grammar leads to humiliation, and they learn that "grammar is a set of rules for torturing your natural sentences into an unnatural form that will satisfy a teacher."
We wind up with adults who are twitchy about the way they use their own language, apprehensive about being embarrassed. And the ones who were given no formal instruction in grammar after the defects of the traditional approach were recognized are no better off. That's why often when I am introduced to someone and say that I am an editor, they say, "I guess I'd better watch my language," and I have to suppress the impulse to murmur, "Too late."
Until we get books on language for students and the general reader informed by linguistics rather than ill-informed pedantry, the best we can try to do is to insist that the Formal is something to learn for particular purposes and the Normal is just swell.
*An explanation for readers who said "Huh?": Many of us who are still above the turf were taught in childhood that can expresses ability, may permission or authorization. Thus a child asking "Can I?" gets the fish eye from a teacher and must recast it as "May I."
**Published in 2018. (I've fallen behind.)