One of this age's great crank ideas, that the U.S. is a "republic" and not a "democracy," is gaining so much ground that people in Michigan are trying to rewrite textbooks to get rid of the term "democracy." And the discussion is such a mess that a New York Times article about the fight manages to get it wrong.
The truth is actually simple: For all practical purposes, and in most contexts, "republic" and "democracy" are synonyms. The big difference is that the first comes from Latin and the latter from Greek. To say that the U.S. is a republic, not a democracy, is like claiming to eat beef and pork but not cows and pigs.
The debate may seem like hair-splitting, but it is important in the same way all assaults on knowledge are important — it's part of the never-ending fight against attempted partisan intervention into education, whether it's denying evolution or pretending the Civil War wasn't about slavery. But it's most important because opposing the idea of democracy can be a step toward opposing the reality of democracy, at a time when voting and other structures of formal equality are at risk.
In the Michigan case, the eradication of the word "democracy" is being pushed by conservatives, who want the K-12 social studies curriculum to "be based on a close, originalist reading of the United States' founding documents," according to the Times. Why they care isn't really clear. Perhaps it's in service to ultimate policy goals, but it may just be yet another form of identity politics, setting off True Conservatives from everyone else. At one point, conservatives were equally obsessed with being for "liberty" and not "freedom." Now it's this one.
The fight over language long predates the U.S. itself. Here's how the confusion started: For centuries, European (and eventually American) proponents of popular government looked to the ancient Roman republic for inspiration. It's not that they actually wanted to create a new Rome, but they viewed that era as a symbol of long-lasting non-autocratic government. So the American revolutionaries called what they were creating a republic, even though it really had nothing to do with the actual institutions of Rome. There was no equivalent of Roman consuls or tribunes or a Roman-style Senate. The Founding Fathers did, however, take the names of famous Romans as pen names, and they called the upper chamber of Congress a Senate, and there's a Tiber Creek in Washington.
Since Rome was popular, they used Latin words. So James Madison in Federalist 10 described the good version of popular rule, which employs a "scheme of representation," as a republic, whereas bad government — "democracy" — doesn't have the same feature. In fact, the classical world never really had the concept of representation. Madison's vocabulary, which he shared with other 18th century proponents of popular rule, was just a mild form of propaganda. That's all that "a close, originalist reading" will discover. None of this had to do with specific institutions or forms of government. Just word choice.
In the 19th century, Athens became popular, and so politicians such as Andrew Jackson and Abraham Lincoln started calling popular rule "democracy." As with Rome, they weren't actually imitating Athens; they just were associating their goals with a popular civilization.
What Americans in the 18th, 19th, 20th and 21st centuries all have in common is that at no time has anything other than some form of democracy/republic gained even a small amount of adherents. Not only has there never been a serious effort to install a monarchy, but there's been no interest in any kind of formal aristocracy, or any national established church that would have political authority. Even supposedly "undemocratic" portions of the Constitution, such as the original Senate (chosen by state legislatures), or the courts, or the Electoral College, are just forms of popular rule, not an alternative to it.
A constant throughout that history are struggles over exactly who would count as "the people." And that's where the words we use for U.S. constitutional government start to matter.
When Madison said the U.S. was a republic and not a democracy he meant (in today's vocabulary) that it was a representative democracy, not a direct democracy. Given that all modern democracies employ a "scheme of representation," that's an unimportant distinction today. But questions about what counts as a democracy – what it means to have popular government – are highly important, and contested, right now.
The biggest of those concern who counts as citizens, and ensuring that those who are included as part of "the people" have full citizenship. That's obviously been a long debate, with the franchise sometimes getting expanded (in several constitutional amendments, in the Voting Rights Act, and in other ways) and sometimes contracted (after Reconstruction and during the Progressive era in particular, and at many other times). And advocates for making it easier for everyone to vote and for ensuring political equality argue that democracy requires those measures.
So to say that the U.S. is not a democracy is to at least imply a position on the actual questions about democracy that people debate today. And not just a mild position, such as an argument that democracy does not require, say, early voting or strict one-person, one-vote rules. Instead, to say that the U.S. is not a democracy is to declare the entire debate irrelevant and wrong-headed, even though all but a small fringe in the U.S. believe that it's a democracy.
I don't really know whether the "republic-not-a-democracy" folks are sincerely just misguided pedants or if they are actively trying to use a confusion in the language to place restrictions on voting rights. Either way, they're just plain wrong. In 21st century America, democracy and republic should be used interchangeably.