In assessing a Miranda related 5th Amendment issue, what constitutes a waiver?

The Miranda rights are basically the government saying, "Hey, person we suspect of criminal activity, we want to be super clear with you that if you want to not say a word to us and force us to work harder to convict you of a crime, that is your right to do so. But just know that the second you open your mouth and start talking to us, there's a decent chance that you'll say something stupid that we will absolutely use against you in court to convict you. So, again, you probably want to shut up, and if you don't want to shut up, you probably want to have an attorney advise you what to say so you don't say something stupid. Or you can ignore this advice, but don't say we didn't warn you, because we are warning you, which is why this is the standard thing we say when arresting people."

With that in mind, a detainee who understands this risk but still decides to talk is allowed to do so and have what they say be later used against them so long as they actually understand the risk. In other words, for a detainee to actually waive their Miranda rights, they must be able to actually appreciate the consequence of their waiver. If there are any facts that show confusion or ignorance on behalf of the detainee, then their waiver may actually be invalid. That being said, practically speaking, the government's burden to prove that the detainee's waiver was valid is only a preponderance of evidence, meaning so long as they can show that they read the Miranda warnings to the detainee prior to the detainee choosing to answer questions, the court is likely to uphold the waiver as valid. Be sure to argue it both ways on an exam.