Taken literally, what you want doesn't work:
If I was a student at your school, I'd bring my own device, put it between the computer and your LAN, and sniff all traffic. (Or, if you haven't locked down the Windows PC, do it on the PC itself). As soon as your computer tries to connect to my Windows PC behind my device, I'll see the traffic, and I'll know both your MAC address and your IP. And there's nothing you can do about it.
But that won't make your computer insecure, without proper credentials, I won't be able to access your computer in any way. All I know is some computer communicated with mine.
If you just mean "the students shouldn't see my device in the dropdown menu in Internet Explorer" (why? Maybe telling them you are monitoring them will make them behave, whereas hiding that from them probably won't...), then possibly that can be done with some Windows configuration, though I don't know how. If you want to access their stuff, Windows has to know that your machine is there, so you can't hide it from the OS - all you may be able to do is to hide it from the user.
Thank you, but my computer is already set to disable Network discovery. What else am I doing wrong? I can still configure my firewall to block pinging, though. – ditheredtransparency – 2017-02-03T19:48:09.483