Thinking out loud...

So...

If one of my students writes "One plus one equals three"(1) how should I respond? Or alternatively, "Colourless green ideas sleep furiously"(2)...

Obviously both sentences are syntactically correct, and I'd obviously be quite pleased with that. But is it fair to let them walk away thinking this is correct English? Is it correct English?

(1) is not good mathematics, I guess. But it'd be fine in a sentence like "It's possible to design a mathematics where 1+1=3", which I feel sure is the kind of things young mathematicians say to each other when they're a bit bored. So it might actually be excellent mathematics - it's just that it's not common mathematics; that is, it's not part of the common misconceptions about what constitutes mathematics. Alternatively, it might arise from a mistaken idea of what any one of "one", "plus", "equals" and "three" means. Or again, it might be intended as some kind of figure of speech. From all that, it appears that it may not be "wrong", and even if it is, it's not actually possible to say what the mistake is.

I can only comment 'Usually we write "1+1=2" or "1+1<3 p="">
As far as Chomsky's famous saw (2) goes, it amuses me that time has rendered it fairly easy to read this as "The unimaginative ideas of the environmental movement are ignored by the powerful despite acrimonious debate." I might deem it unlikely that one of my students has perpetrated such a monstrous wordplay, but it's a bit dismissive. A whole bunch of them are smarter than I am. Perhaps they've read Chomsky. Perhaps they enjoy puns.

These are extremes, sure. Extremes help cast ideas in relief, which can be a useful clarifying technique. Most of the sentences I correct are not, in fact, syntactically correct, so it looks like an easy business to document the errors. But actually, correcting the errors means assuming that I know what the meaning is that they wanted to express; by analogy, if I offer only "1+1=2" above, I've ignored the possibility that the required correction was "1+1<3 p="">
I'm thinking about these things because I'm thinking about software that provides feedback to students, to increase the volume of feedback they can receive from non-trivial texts (textbook drills are 99% trivial, and about equally useless). Purely rules-based software doesn't work - I guess it could provide a large list of broken rules, but it's not clear how helpful that would be - and software isn't that good at "meaning". No doubt it will get there, but it looks like quite a tricky journey. Right now, it relies largely on statistical evaluations of collocation, which it do quite successfully based on correct input - but we're talking about incorrect input.