Physics's *don't* know that only negative charges move. We can create ion currents on demand in many environments. We *do* know that the current flowing *in a metal wire* is negative charges in motion.

As for how to determine that, you do a [Hall effect][1] measurement. 

The measurement works by subjecting a current in a relatively wide bar to a magnetic field perpendicular to both the current and the width of the bar and then measuring the potential difference across the width. In this era of turn-key precision voltage measurements is easy enough to do in a high school laboratory if the students can follow the underlying arguments.


  [1]: https://en.wikipedia.org/wiki/Hall_effect