I would say your professor is wrong and right. I am actually working with the college of engineering web dev team and we do everything in PHP. I don't think he is really correct though because from what it sounds like, hrefs are more than satisfactory for what you are doing. We primarily use switch case statements to pass POST statements via AJAX to themselves and then handle them through independent cases. An example of this would be like
if (isset($_POST['whatever'])) {
switch($_POST['whatever']) {
case "1":
//I can now do something if the POST variable passed as "whatever" was deemed '1"
break;
case "2":
//I can do something if the POST variable pass as "whatever" was deemed "2"
break;
etc etc...
So two main advantages of this is one it makes it really simple because you don't have to juggle millions of href sites. I'm sure for your application it might not be a big deal but when you really start doing large projects where each website is fairly similar to one another, it is nice to know you can count on php's dynamic framework. Another advantage is that people don't see your code as much ^_^ If I looked in the source and saw a ton of href links, I might be able to know a little more about your file tree structure and that's never good (there are ways to workaround this with redirects and using PHP/javascript but yeah...)
OVERALL though he IS your professor but just to let you know from what it sounds like you are trying to achieve, hrefs are totally fine. I know a lot of professional sites like Yahoo or even Stackoverflow use hrefs to link their sites (just look to the left at the Related Questions. they are all linked using hrefs).
You said the computer course wants you to create an entire PHP web system and just to let you know generally computer courses are for education rather than actual practicality. I think you are doing the right thing by asking questions, but for the sake of the class, your professor is probably encouraging you to learn PHP despite it being not the best way (this is always subjective though). Perhaps you can find a way to find a middle ground where you can develop in PHP but still use hrefs? Hope this helps.