I hesitate to comment on this, but any person teaching non-natives not to use good or great as an adverb should always give the caveat that in spoken non-formal English it may be more appropriate to use the grammatically incorrect good-as-adjective. Firstly because they're going to hear it perhaps more than the grammatically correct version, and secondly because if they want to sound natural in certain contexts it may be more natural to use it.
I think it is good to mention both alternatives, but I've never considered it inappropriate to use correct grammar. People still can understand each other, and it's okay to have a different style. I actually admire those non-native English speakers who use more correct and distinct English than the average American. It reminds me that, if I am not careful, my own English can become careless and sloppy, and at times my meaning unclear.
It really depends on the language and the context. In Indonesia the grammatically 'correct' form of the language is only used in formal settings, and even then not always. Anyone who insisted on speaking that way would immediately mark themselves as a foreigner.
More generally, you're going to speak like the people around you, whether it's your native language or a learned one.
See http://articles.chicagotribune.com/2011-04-06/features/ct-tribu-words-work-good-20110406_1_verb-languages-grammar-girl-s-quick for the explanation of why using good like this is correct grammar.
Edit: To all the naysayers out there, keep in mind that language is a socially defined thing and changes over time. You don't have to accept the change, but it would seem unlikely that you can stop it.
The article says that using "good" with the linking verb "to be" is grammatical, and I agree. In that situation, you're using "good" as an adjective and linking it to a noun. Nonetheless, "going good" is not grammatically correct. "To go" can't function as a linking verb. I can "be" good, but I can't "go" good. Sorry, but my teeth are bared on this one. The distinction between "good", the adjective, and "well", the adverb is a useful one for clear communication.
Added note: at the end, the article says "do good" is grammatically correct. I agree with that only when "good" is used as a noun (i.e. "I did good" = "I did a good deed"). Otherwise it's incorrect. And you can imagine a German in a job interview looking bad for using it the second way.
I think you meant that "going good" is not grammatically correct. The fact that not everyone uses correct grammar doesn't make it incorrect to be correct. Or correct to be incorrect. But sometimes, of course, you just want to be cool. I think Duo should primarily teach correct English. You are then free to adapt that to how your associates talk, if you so wish. You want your job to be going well, but perhaps your social life is going good.
I should second dgroves1522. Both "being good" and "doing good" have their acceptable (even if not 100% grammatically kosher) uses. In the contexts in which they are used, they typically have different meanings, comparing to the same sentences using "well" instead -- just like in the example liked to by ZacharyBraun. Compare "- How are you? -I'm well, thanks!" and "-Would you like another beer? -No, thanks, I'm good!" Using "well" instead of "good" would render the second exchange meaningless.
As for the statement that "To go" can't function as a linking verb, that is not entirely correct either. Someone in this thread has already mentioned "going crazy", and all of us refrigerate food to prevent it from going bad.
Hmmm. My uses of "being good" and "doing good" are 100% kosher, and saying "I'm good" in the context of wanting another beer is grammatical (both because it is using "good" as an adjective and because it's idiomatic). As for using "to go" as a linking verb, I agree with you, but it's only used as a sort of linking verb in idiomatic expressions.
Which begs the question, why doesn't "going good" count as idiomatic? This is where I admit defeat. With a lot of people across America, "going good" is correct usage in the context of a jovial greeting. I'm biased towards academic, northern English. In my defense, though, "going good" would sound bad in a situation where a German in an English-speaking country would really need it, like a job interview or a business deal or something. Still, I change my mind. http://instruct.westvalley.edu/lafave/DFW_present_tense.html
Except people say this all the time. It's a valid expression in English.
Sure, if you want to teach foreigners to speak English "real good" - then, by all means, accept it as a correct answer. (And while we are at it, let's also allow for an unlimited number of "like" and "totally" in every sentence - after all some people talk like that all the time.) If the goal is to help the learners speak really well, on the other hand...
This is an interesting discussion about "going well" or "going good". Isn't it interesting that the "rule" says that "going well" is correct even though "going good" is commonly used? Personally, I think that grammar rules should be flexible and allow for modern usage, dialects, and innovations, so I am more of a descriptive linguist. I know, though, that there are many people out there that are in the other camp; they are prescriptive linguists. Here's a discussion on the subject: http://faculty.washington.edu/lauramcg/courses/ling200/ppt/lect2_intro2.pdf
Like every language, German has borrowed words from other languages. I've read that when German takes on a foreign language they tend to lean closer to the native pronunciation so French borrowed words will still sound a bit French.
Of course that's just a tendency so of course not everything follows those rules.