Hi, So I am an undergraduate math major who goes to a state university, and often time I get into conversations like the following: My friend: "What are you taking this summer?" Me: "I'll be taking (abstract) linear algebra." Case 1: My friend: "Hmm... 'linear' and 'algebra'? That doesn't sound really hard! How is that different from college algebra?" Me: "Well..." Case 2: My friend: "Well... what's that?" Me: "Um... you know... it has to do with vector spaces and... stuff like 10-dimensional space and..." My friend: "Wait, 10-dimensional what?" (Of course, there's the third case where s/he looks at you strangely and ask why I want to do that, but I omit that case here since it's not important in this discussion.) The thing with this is that I am having a hard time explaining the things I do in my math classes to someone who only knows math like college algebra or calculus (at best). This is sad because I want to describe what I'm actually doing so that they don't think math is a mysterious subject that only some people can appreciate it (well, this might be true, but still...). And I also think it is a good skill for me to explain what I'm actually studying to someone who is not a math major. I had a friend who was a physics major who was really talented at this--he could explain physics to someone who only knows a little bit about classical mechanics. I want to be able to do the same with math as well. So, if you have any suggestion/comment, please let me know. Thanks.