|This page in a nutshell: That phrase doesn't mean what you think it means.|
Well-meaning Wikipedians, fully aware of our core content policies and our guideline about fringe theories and covering them only with due weight, sometimes innocently suggest that we "teach the controversy" when presented with a real-world conflict in reliable sources.
"Teach the controversy" is not a general statement to apply to remaining neutral about a subject of social or research conflict. It’s the catchphrase of a specific group of anti-evolution lobbyists, the Discovery Institute, to promote injecting creationism into the curricula of American public schools.
Even if it weren't a misappropriated slogan of religious pseudo-science, the idea is off-base anyway, for multiple reasons.
Wikipedia does, indeed, have a duty to accurately reflect what the reliable sources are telling us, including when real-life experts are in sharp disagreement. This is not Wikipedia "teaching" anything – Wikipedia is not a textbook, guide, tutorial, handbook, or how-to of any kind.
Nor is every such conflict in the sourcing a "controversy"; the assumption that there is one is considered original research. Often it's simply a factor of having looked at only two sources, one of them more current or more authoritative than the other.
If there is in fact a genuine controversy in the field in question, which we might write about as a controversy, sources from and about that field will tell us that this is the case. If they do not, then simply note the conflict ("According to.... However, according to...."), and bring it up on the article's talk page. Chances are, some other editors know where to get more information, and a consensus discussion can determine the weight that particular sources should be afforded. If there's an accuracy dispute between scholars, it is described by Wikipedia without taking part in the dispute, or manufacturing a controversy.