> He's talking about a fairly advanced math topic, abstract algebra. That's actually a remedial-level topic for MO, which focuses on research mathematics. It's

> He's talking about a fairly advanced math topic, abstract algebra. That's actually a remedial-level topic for MO, which focuses on research mathematics. It's meant to be an easy example that everyone can relate to in this discussion, like talking about how children learn how to add. You'll see they say, "simple example". I'm not trying to be elitist, just trying to explain MO. As an aside, I found the conclusion of this simple example about how normal subgroups are about being kernels of homomorphisms hard to relate to, how would you know that without even knowing what a homomorphism is, which needs knowing what a group is? These a-ha moments that come after learning the material and make the learner assume that the teacher is an absolute idiot by not starting from the a-ha moment are very frustrating. It's kind of impossible to start from a-ha without the learner first bumping into all the dark corners and hard work that light the way to the a-ha. I myself have had topics ranging from (simplicial, singular) homology to the completeness of the reals and elliptic curves explained in so many ways and nothing ever made sense to me until I sat down and struggled through all the explanations offered to me. I don't think there's a way to convey insights to a mind that hasn't struggled towards those insights. Here I am going to quote Michael D Alder, from the introduction to an old edition of some lecture notes of his that I can no longer find online: I try to start every course of lectures with an overview or outline of the material in the course. Before I do that, a (true) cautionary tale. While I was an undergraduate doing Mathematics, I told my Calculus lecturer, a celebrated mathematician called Ian Porteous: “I have been getting the strong sensation in your course of being dragged at high speed down a narrow track in a jungle. There have been dimly sensed paths off to the sides but we have been galloping after you and have no time to explore these possibilities.” I went on: “What I would like, is to be taken up to a high place and given a view over the country through which we have been and that where we are headed, an overview of the subject.” He replied with an amused grin, “Wouldn’t we all.” He elaborated the point, which was that in order to have an overview of a subject in Mathematics, you have to crawl all over it on your hands and knees, and then you move next door and do the same with another mathematical subject, and then, if the subjects are related, you may get a higher level view of the two bits you have sorted out in detail. And then you can do some more detailed understanding of some other bits and link those, and maybe get a higher order insight linking the links. But a higher order view from a height without doing the detailed work is not possible. The reason it is not possible is that you are looking at ideas. You can only develop a higher level idea by understanding the base level. We could tell you the words for the higher level ideas, but they wouldn’t mean anything. For this reason, course outlines in Mathematics are intelligible only after you have done the course, not before. This is extremely maddening, particularly to philosophers, journalists, post-modernists and others with similar intellectual handicaps, but that’s the way things are. It follows that explaining what the course is about in a general way is a waste of time and can’t really be done, but I shall do it anyway. Edit: I just serendipitously found a passage of his in a very similar vein. Read section 5.1, "Cultural Anthropology": ftp://www.biophysics.uwa.edu.au/pub/Mathematics/Alder/DiffGeom.pdf