In the midst of a paragraph, one of my students write, 'They taught us a way of cleaning.' I felt it sounded exponentially better to say, 'They taught us a new method of cleaning,' or 'new way'. Why is that? Is there a grammatical principle?
Grammatically, they taught us a way of cleaning is fine, so your desire to add an adjective is a personal preference.
The next question is why do you have that preference? It is probably a pragmatic thing. You want a reason for them being taught a way of cleaning because it is obviously not a random act. Is it because it is a new way of cleaning, or a better way, or a faster way, or an easier way? In technical terms, the sentence violates Grice's maxim of quantity by not giving enough information.
As a native English speaker in North America, it does perhaps sound a tad odd, but that's a hard call to make without reading it in context. Several thoughts:
- if you want to keep the word "way", I would use the infinitive: "way to clean"
- if you like "method", I would use "of": "method of cleaning"
- the most natural to me, without any other context, is "they taught us how to clean"
Edit/response to comment: I emphasize native because I feel it's important to point out I'm not an ELL. If I were responding to a question about another language, I would highlight that I am NOT a native speaker and to take my response with a huge grain of salt, as my ear may not be good.
I'm sorry, I can't give reasons. In my experience, what I said stands. I suspect the things that bother you reflect the difference between our two languages. (Tongue in cheek there, but they really do seem that way.)
The sentence is fine as it stands. I believe this is a more common form of expression in the UK than in the US. I think of it as a slight understatement. In the US one would usually say, "That's a good idea". In the UK you often hear, "That's an idea".
I've heard from people from the U.K., "You know, it might be an idea if you ...."
In the US I almost always hear, "It might be a good idea...."
I think it sounds just a tad less pretentious not to add the word "better" or even the word "good" and let people decide for themselves if one's way is really better or even good. Actually, it might indicate a certain confidence that it is.
Edit: Forgot to address "new". "New" is similar to my "better" example which is more common, and new usually implies better.