Teaching abstract maths concepts to young children. I am interested in opinions and, if possible, references for published research, about the pros and cons of teaching abstract maths concepts to young children. My younger brother (five years old) understands negative numbers and square roots so I was thinking of trying to teach him about complex numbers and maybe some other concepts, but my elder brother (who is doing a maths/stats degree) said it was a crazy idea (without elaborating, but that's what he's like). 
Update: I quizzed my brother on why his thinks it is crazy and his response was "Don't you think there is a reason why 99% of maths teachers have a degree in maths ?"  I'm at high school by the way.
 A: Who cares what your elder brother says? Try it and see if your younger brother enjoys it. If he doesn't, try something else.  You're not going to do him any damage, so if he seems to be interested, why is there any question? 
A: There is no merit in just teaching abstract concepts...you must follow the "story" of maths. Things are always made for a reason and most people (including myself) will not appreciate abstract things unless I see the thought processes and intuitions behind them.
A: It really depends on what kind of a person your younger brother is.  If he has the mind of a mathematician (which sounds likely, given his advanced knowledge and your family history), then go for it.  It will intrigue him, and teach him to look at things in mathematically different ways.  On the other hand, if he is like 99% of the population, it would probably only serve to confuse and/or bore him. 
So I'd say go for it, and if he doesn't byte, then wait for him to get older.  It's not like you're making stuff up (there really are such things as imagainary numbers, and he probably will end up learning about them eventually).  
And remember that he is five, so unless he's a Gauss, he will need things explained slowly and in the most basic of terms.  He won't have taken mathematical concepts for granted yet that you probably have (for example, I remember when I was five or so, I did not yet have it ingrained in my mind that addition and multiplication must be commutative).
Finally, if he doesn't seem to be able to understand it, there are other "advanced" mathematical concepts that are more accessable to younger children because they are less abstract.  Prime numbers, for example.
A: I’d not set out to teach him anything; I’d make accessible mathematics available to him and let him choose what interests him. Enzensberger’s The Number Devil introduces a wide variety of interesting mathematical ideas in a very accessible way. If (or when) his reading is up to it, Martin Gardner’s collections of columns from Scientific American are good.
The main point is that it should be up to him. 
There’s all manner of accessible mathematics that might prove to be more interesting or more fun: Fibonacci numbers and their patterns come to mind immediately. Representation in other bases can be fun early on; I especially like binary (as the system that arises naturally when you want the most efficient set of counterweights for an equal-arm balance when the object being weighed and the weights must go in opposite pans) and balanced ternary (as the system that arises naturally when the weights may also be placed in the same pan as the object).
A: The problem here is that you need to understand a good order of concepts.  It may be pretty hard to just pick some topic you like and start teaching it.  So, I recommend
Kitchen Table Math by Chris Wright
There are actually 3 volumes which build on each other.  The way I see these is, someone who has never taught elementary math could get these and figure out a good order to teach things and also get ideas for how to teach these things.  This way you can always find interesting topics your brother still does not know without going into things that are inappropriate for his level.  You can check the table of contents to see which one to start with, maybe volume 2.  Notice these books get into fractions, probability, number theory, so there is plenty of interesting stuff.
A: Many good answers here already. In the same vein, if your little brother is interested, you should engage him as much as he likes it. I do recommend an old, but absolutely delicious, book Mathematics For The Millions by Lancelot Hogben. He presents how we evolved the concept of mathematics from scratch up to calculus, passing by astrolabs (and how to build one yourself!). The self declared goal of the book is to teach mathematics to the masses, and as and idea of how to present, and in which order, the basics of mathematics, this book is really great. Also with great humor (although I might not present complex number to your 5years old brother as he does, drunk numbers).
A: You should also look at Mathematics and the Imagination, by Kasner and Newman (do a web search for information). That was the book which introduced the term google. 
