Well, I'm not going to pretend that my sister's life depends on learning maths - at the moment it's not that important to me, and I'm going to have to muddle through on lesser motivations. That is a very useful thread, though (I realise now that I have been thinking of maths as hierarchical - thinking that either you learned it in a particular order, each tier supported by the previous, or you didn't learn it at all) - thanks very much.
Math is sort of hierarchical in that advanced maths depend on earlier math learning. But the overall map is more a huge confused network, with topics splitting and feeding into each other all over the place. I am going to repeat my regular recommendation, if you want to study maths on your own, do some reading about math. My recommendation, as always, is Philip Davis and Reuben Hersh, The Mathematical Experience. It is the broadest discussion of maths that I know of. I haven't seen the new edition which has been enlarged and with exercises for studyin...
This will not be a long post; I have a simple question to ask: if you wanted to educate yourself to graduate level in mathematics, but didn't actually want to go to university, what would you do? I would ask for text-book recommendations, but I don't want to limit your responses (however, bear in mind that the wikipedia articles on, say, cardinality or well-ordering go over my head – they may skim my hairline, but over they go). Also bear in mind that while I personally have A-levels (British qualifications) in both Maths and Further Maths (which is to say, I know some calculus at least), there are probably plenty of people on lesswrong who don't and who desire the same information – so assume as much ignorance as you feel necessary (it's a shame, actually, that there isn't a sequence here on lesswrong for maths). What do you advise (if you think the query ill-defined, I would like to know that as well)?