Apparently he has told his class that this system is "foolproof"
Well, obviously it's not "foolproof". HereDepending on your capabilities, there are someplenty of ways to cheat.
Your professor has a point in that your standard wireless network card won't simply support a simultaneous connection to multiple different APs, thus preventing your from using that particular interface for an Internet connection. (Although with some tinkering you could possibly alternate between networks without letting the professor's AP take notice by tweaking your driver to omit the layer-2 management frames that are supposed to notify the AP of your intent to dis-/reassociate.)
However, there are also easy workarounds:
While your standard network card won't support a connection to two different Wifis at once, you can just buildBuild in a second network adapter (or plug in an external USB one, once the professor leaves) to connect to a network with Internet access. You can easily do this without interrupting your existing connection.
Connect with your phone or another device instead of your real computer. On that device you can setconfigure the broadcasted MAC address to match the one of your computer which might. This could fool theyour professor but probably wouldn't withstand a forensic investigation of the traffic.
Use Bluetooth. Most laptops have built-in BT adapters and, so you cancould just tunnel your traffic via BT to a hidden device that itself is connected to the internet.
Get creative. There are plenty of ways to bridge an apparent air gap. You might use your sound card to transmit data in a small range (or even your hard drive for that matter) - but then again you could also spend that time studying for the exam.
One effective way to stop these trickscountermeasure might be capturing every student's screen during the exam, but personally I find that very intrusive. Ultimately, if the professor allows students to use their own computers there will always be some way to prepare the devices to cheat.