I don't really understand what you're planing to do so situation A & B follows:
A. If you wish to keep track of visitor during one session, from page to page, like "welcome John", "hi John, still there", "sure John...", etc... it really doesn't matter if you use cookie or not.
You just say "session_start()" in your code and it starts with cookies or not depending on the server configuration (variable session_use_cookies, i believe, you can phpinfo() your server to know how it's configured) and also depending on the client's configuration: if he doensn't accept cookies, php will automatically add a PHPSSESID=kshadf1234897fdhascnaskd2 in every link the page sent to the client (even images!!!!).
When you call session_start() in the next page (or the same if the client links there) php will load every variable registered on that session as a global variable.
B. If you wish to keep track of who voted allrady (in order to get one vote per person only) you will need a very different thing: permanentcookies.
The cookie remains in the client's computer even after the session is over and you will identify (by wrinting some code) the client next time around.
If you believe that more than one student will share the same computer, this is no secure way to do it, because cookie remains in the computer not in ther persons body.... so you will need some sort of passowrd or activation key in order to ensure one vote per student....