I don’t think the Ed.D. usually takes less time, but it doesn’t get the respect of the Ph.D, certainly not among academics. Most Ph.D.s take a minimum of 4 years beyond the bachelor’s degree. It is very difficult to predict how long a degree will take, because while you can definitely finish your doctoral coursework in two years, the dissertation may rely on a lot of things beyond your immediate control (like your advisor’s demands). As far as students finishing faster at better schools, the answer is no, because the expectations of them are also greater. They may be (on occasion) brighter, but they are also learning from more demanding faculty, who want more from them.

In a Ph.d program in math the first two years are often dedicated to doing basic graduate coursework and passing comprehensive exams. How long this takes depends upon your preparation. Well prepared students with some graduate credits behind them may be able to get this done in one year, and then start on the dissertation. Others may take much longer.

It is also hard to tell how long it takes to write a dissertation. If you have done an REU (Research experience for undergraduates) or an undergraduate thesis, and have a Ph.d topic in mind, things can go faster. What often takes time for the Ph.d is that you have to spend some time getting to the forefront of your field. This means mastering research papers, taking specialized courses and seminars, and attending conferences. The more you already know about the subject, the faster things will go.

Having said that, the general goal is 4 years to get the math Ph.d, and many people take 5. In many programs the minimum residency requirement is 3 years (counting some time for an MS, if you already had one), but for most people that might not be enough time.

If you are making good progress on your dissertation, some univerisities will let you leave campus and finish writing in absentia. This may be a way for you to stay enrolled in the program, and still make progress towards the degree, even though you have to be somewhere else (they will still want to see you making progress, though).

Also, it is not unheard of for people to start a Ph.d program, get to the ABD (all but dissertation) level, take a leave to go do something else, and then come back to complete it. Many places give you 7 years or sometimes more in total to complete the program.

The time you take to do the Ph.d does not really depend on how good a school you attend. Some of the best, in fact have extra requirements, like the ability to read math in one or two foreign languages.

The Ed.D will not demand as much mathematical content from you. But it may still take a long time to complete, as you still have to do a research dissertation.