It's a bit like bike-riding - easy to do (once you know how), but hard to teach. In fact, it's quite easy to teach badly because one thing is certain: commas are difficult. It takes children literally years to master the comma, and a lot are still struggling with it well into KS3 - in fact at the end of KS2 they cause more trouble than any other punctuation mark.
(This interesting fact emerges from a case study by a linguistics undergraduate, which shows what a useful subject linguistics is. The essay is available at www.phon.ucl.ac.uk homedickoccomore.htm) So how do you explain commas? The standard advice is to use a comma when you would draw a breath or pause. But that really doesn't help much for all sorts of well-known and obvious reasons. How many breaths do you draw when reading a list like "apples, bananas, oranges and pears"? In any case, lung capacity is hardly relevant to punctuation; pity the poor asthmatics in your class.
Another idea that some punctuation pundits sometimes push is that commas have something to do with subordinate clauses. That's certainly true in German, but not in English.
It's true that some subordinate clauses tend to have a comma if they're at the start of the main clause: If it rains, the picnic will be in the gym.
But this comma actually has nothing to do with subordinate clauses - it would still be there if we replaced the subordinate clause by a prepositional phrase: In the event of rain, the picnic will be in the gym.
Moreover, some initial subordinate clauses really shouldn't have a comma: What I read made me shiver. (Not: What I read, made me shiver.) Better, perhaps, to focus on meaning, and what will help the reader to work out the sentence-level grammar on the way to the meaning.
A comma may be helpful if it separates a complete sentence from an optional modifier (an adverbial in grammar speak) - for example a comma after "if it rains" or "in the event of rain".
Take the adverbial away and you're left with a full sentence ("The picnic will be in the gym."). But a comma right in the middle of the clause proper, between subject and verb, is downright unhelpful because neither "what I read" nor "made me shiver" makes a complete sentence.
Of course there's more to it than that, but you can't teach commas without talking about grammar.
Richard Hudson is professor of linguistics at University College, London Geoff Barton is headteacher of King Edward VI School, Bury St Edmunds, Suffolk