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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.

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I think he's right. The intuition behind complex numbers is that they allow you to solve polynomial equations. Teaching someone who doesn't understand polynomials (or even real numbers!) about them would be breaking apart from the intuition. Mathematics should be taught in ways that relate somehow to intuition, while gradually expanding it. Otherwise, it's just meaningless symbol manipulation. That's what I think anyhow. –  tomasz Jun 22 '12 at 18:08
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@tomazs I think you could motivate complex numbers geometrically, understanding $i$ not as the root of $x^2+1$ but as the operation of rotation by a quarter-turn. –  MJD Jun 22 '12 at 18:22
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@tomasz : You've missed the point. "The intuition behind complex numbers is that they allow you to solve polynomial equations." I don't think so. Multiplying by a complex number consists of rotating and dilating. That's how I learned them as a child long before I saw heard that they contain solutions of polynomial equations with real coefficients. –  Michael Hardy Jun 22 '12 at 18:30
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@tomasz But if you don't understand them as numbers, then you don't see why they can be added or multiplied. But the fact that you can multiply $(1+i)(1-i)$ in the ordinary way and get 2 is exactly what is great about them! –  MJD Jun 22 '12 at 18:46
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Your older brother is badly misinformed if he thinks that 99% of high school (maths) teachers have a degree in maths. Maybe in Oz (as in the Wizard of). Or Finland. –  user12477 Jun 22 '12 at 21:31
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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).

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Binary is also awesome for counting to 1023 with your fingers. –  Jonas Jun 22 '12 at 19:37
    
@Jonas: I used to count laps that way when I ran on the indoor track at my university, though I don’t think that I ever had to go above $104$ or so, that being $10$ miles. –  Brian M. Scott Jun 22 '12 at 19:42
    
It's awesome for counting stair steps. And as a distraction when you're walking. It's interesting to notice how far you get in 1k steps (for an adult, it's about 700 meters). –  Jonas Jun 22 '12 at 19:43
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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?

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You MIGHT damage him (permanently, even) by making him dislike mathematics instead of letting him see the beauty of it. Just don't push too hard. –  tomasz Jun 22 '12 at 18:28
    
@tomasz : I would agree that one shouldn't try to force him to learn this, but if he wants to learn more math, I don't know why this shouldn't be included. –  Michael Hardy Jun 22 '12 at 18:31
    
@tomasz the thing is, he loves numbers. He's learned times tables that they don't normally learn in school for another 5 years and loves to practice them and simple sums in his head. He often just runs up to me and says things like "Joe ! Twelve minus twenty is minus eight" and then runs off again. –  Joe King Jun 22 '12 at 18:33
    
@Joe My kid is the same way. When she was three we heard her counting on her fingers in the back seat of the car: "Three and four is seven," she announced. Then later, in amazement: "Four and three is also seven!" I agree that you should be careful to help him go where he likes and not push him too hard. –  MJD Jun 22 '12 at 18:39
    
Mark, he worked out the commutative law for himself too :) It's a joy to behold. He also says things like "look, the square root of five is here", pointing to somewhere between 2 and 3 on a number line he's drawn. Then today he did the same thing with -5 and I had to tell him that it is a bit different with negative numbers. That's why I posted, thinking to introduce him to complex numbers, because I thought he was actually interested in it. I can just imagine that he will enjoy doing complex number arithmetic for the sake of it. But I'm sure he will be interested in other things as well. –  Joe King Jun 22 '12 at 19:22
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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.

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I don't mean to be argumentative, but isn't it a little different for young children ? Negative numbers are an abstract concept to five year olds, yet he just loves to do simple sums involving negative numbers - he seems to enjoy it just for the sake of it. As I mentioned in another comment, he also points on the number line to where he thinks the square root of a number is, if it's not a whole number. He has also done the same with negative numbers, and I've had to say that was wrong, and that's when I thought about introducing him to complex numbers. –  Joe King Jun 23 '12 at 12:07
    
I would say it is more so for younger children...I know more maths so I am more likely to see the point of something abstract immediately. Maths is not just about ability to do abstract things but the ability to use and understand their place. Anyone can learn how to add complex numbers given time, but fewer people can "understand" complex numbers and be able to use them to solve problems. Your brother sounds very gifted and you are right to encourage this but really you need a mathematician to decide how to build up his skills. –  fretty Jun 23 '12 at 13:57
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The best way is to work on problem solving using what he knows...rather than by just introducing more things to calculate with. There is a limit to which he will not understand the point of what he is doing. Maybe introduce him to algebra, solving problems using equations and negative numbers and get him to see WHY he cannot square root negative numbers on the real line (rather than simply being told it). Then he might want to know about complex numbers. –  fretty Jun 23 '12 at 14:13
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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.

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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.

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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).

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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.

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