Howdy jweschbuy,
I see you're looking for help with using reminders on your iPhone, so let's see how we can help.
For confirmation, are you noticing if this is occurring when the reminder appears on the Home Screen, or is it occurring on the Lock Screen? In addition to this, are you seeing that the notification is viewed within the Notification Center by swiping down or is the alert popping up directly on your screen?
If there is an update available, you'll want to install that to see if it clears up what is occurring. Remember to create a backup of your information prior to installing the latest iOS version if there is an update available. For help with these steps, use the following resources:
How to back up your iPhone, iPad, and iPod touch
Update your iPhone, iPad, or iPod touch
If you see that you're up to date, or if updating doesn't resolve this, it would be best to Contact Apple Support to have this looked into in more detail.
Cheers!