here is this anarchist's take:
there would be no school "system". learning would happen primarily through the experiences and activities of life itself. since people would be much more directly engaged in the activities that sustain and fulfill them them, the basics - things like growing and preparing food, making clothes and shelters/dwellings, working with water, constructing the things they need and want - would be learned fairly young.
as people (of all ages) discover the things that particularly interest them, they can focus on those areas, sometimes (often?) with others that share the same or similar interests. [edited to add] of course folks sharing a given interest will have different levels of expertise and experience in those areas; those with more advanced skills would naturally and organically impart some of their knowledge to those with less skills who are open to it. [end edit] if they want to do that in a more "formal" setting, cool, have at it. those involved and interested are the ones that would make it happen. there would be no institutions, and the idea of "going to school" would not make sense - living life would serve the function of "learning" far better than that outdated, authoritarian, capitalist concept.
the school system has as its primary goal - at least at the level where it fits very neatly into the network of institutions that dominate life - to churn out "productive citizens". this anarchist wants a world where neither of those words have any meaning.