We just started back at school on Monday and I am taking a graduate algebra course. I have taken undergrad Algebra I and Algebra II. Of course I expected that the graduate Algebra I would be much more in depth than the undergrad course, given that it is meant to prepare a student for the comps. So, I went in to this class thinking that, in undergrad Algebra we covered X so in this class, we are going to cover k*X, k > 1, since we should be able to do more in depth stuff for each topic, given that the students (ostensibly) are familiar with algebra. I still think that the k*X thing will hold true, but this doesn't seem to be the major difference. The major difference seems to be not the material that is covered but the WAY in which it is covered. For example, in the undergrad course, we began by defining groups. Then we talked about a bunch of different properties of groups, then we did some stuff with sub groups, then we did some stuff with quotient groups and THEN we did stuff with homomorphisms. In the graduate course, we defined groups and then homomorphisms on the first day of class. Now, it isn't that we covered all the sub-group, quotient group and other stuff in the first day, we just did the homomorphism stuff first. At first, I just thought that we were going to do things in a different order, but not for any particular reason. But then the prof. explained to us that real mathematicians want to know how objects relate as soon as they know what those objects are. In fact, he said what he did on the first day of class was to define the category of groups (I have no idea what this means, but an explanation is coming). Apparently, once this has been done, we HAVE to define sub-groups a certain way and we HAVE to define quotient groups a certain way. So, this is really intriguing, and I feel like I am doing "actual" math now, even though everything in class has basically been a re-cap of what I know (so far). It is just the way we are looking at things is much, much different than what I am used to, and it feels more rigorous and "mathy". Also, the attitude of the prof. in the grad course is much different than in the undergrad courses. I can't really explain it as well as I feel it, but it is like the prof sees us more as future colleagues than just students. So, not only does he teach us the technical stuff, but he also teaches us about the culture of mathematicians. There is a lot of "the amateur way is to do this ... but we want to be mathematicians, so we do it this way ..." So, it is quite difficult for me to explain exactly what I mean, but my question is to those who have (are in working on) graduate degrees in math (well, it really doesn't matter if it is math, it could probably be anything). Do you remember things like I have described? Did you like it as much as I did? Like me, did it make you want to do math even more?