I'm talking in the range of 10-12 years old, but this question isn't limited to only that range.
Do you have any advice on cool things to show kids that might spark their interest in spending more time with math? The difficulty for some to learn math can be pretty overwhelming. Do you have any teaching techniques that you find valuable?
Graph theory! It's essentially connecting the dots, but with theorems working wonders behind the scenes for when they're old enough. Simple exercises like asking how many colors you need to color the faces or vertices of a graph are often fun (so I hear). (Also, most people won't believe the 4-color theorem.)
I recently taught a once-a-week geometry class to grades 7-10 where we did some graph theory, some surfaces, some spherical geometry, a lot of complex numbers, and basic ideas of homeomorphism and homotopy. Most couldn't work with complex numbers at the beginning: showing their use in proofs of analytic geometry results had the benefit of not boring the kids who had already seen and worked with complex numbers, and giving the kids who hadn't a reason to learn the technicalities (getting under the hood of these pretty pictures).
Euler's formula and the relation of planar graphs to polyhedra showed them the basics of a connection between geometric and algebraic intuition, outside the coordinate geometry many were used to, and got them thinking about ways to define familiar things mathematically (space), and what we can learn about familiar things through mathematics (surprising things like orientability through the Möbius strip, or sphere eversion).
In general I've found geometry to be a very good place to start with people with a professed fear or disinterest in mathematics. For me, it's the quickest way to show the distance between what mathematicians play with when they're doing math and what was taught in high school, with the two column proof nonsense and the treatment of math as a branch of formal logic.
Of course, I suspect that I find this path the easiest because it is one of the ones I am most excited about in mathematics, and I further imagine that in many cases, the best thing to use to get people excited about math is something you yourself are really excited about, as long as you can think to translate it well.
Besides all of these mathematics topics, I would contend that it is equally as important to show them the more human side of the subject. Mathematics has a history and a culture that is accessible and exciting to some people. Of course there are the exciting lives of Pascal and Galois, but there is also the story of Bourbaki (and the individual stories of their various members).
Also, introducing children to actual mathematicians/engineers/scientists is a great way to motivate their interest in the subject. Let them see that mathematics has not been borne from the head of Zeus, that it is a living and breathing subject.
I have found most people liked math at some point, but something happened in their learning process that made them feel so stupid, they became disenfranchised with mathematics.
What tends to happen is students are presented with some mathematically result they are expected to memorize by route, which takes all the joy out of mathematics and prevents them from approaching mathematics intuitively.
So I would first try to zero in on what they don't like and what parts of mathematics they have had to take on faith. You might not have to excite them if you help them learn mathematics intuitively.
As per the suggestions, I would show them how mathematically equations make pretty shapes in Processing.
I think ideas like cardinality are suitable for getting kids interested in math -- the notion of a one-to-one correspondence is easy to grasp, and the proof of the uncountability of the reals is great fun.
Also, elementary number theory can be useful. This is the sort of thing that would be especially useful for, say, contest math, which is where a lot of kids get interested in math.
In general, popular math books can be a good source of ideas.
Personal experiences as an elementary school math resource teacher and a 1st grade teacher have led me to believe that mathematics learning should be engaging and fun at any level (including the graduate level), but clearly it is of the utmost importance to effectively engage students at an early age, so as to hopefully instill in them a lifelong appreciation for the subject. Most children are quite active in nature, so providing hands-on activities that keep their minds and bodies actively involved is an important task. It seems that geometry is like the fail-proof mathematical concept that can pique the interest of even the toughest audiences. Probably because geometry is naturally hands-on and therefore easily applicable to real-life situations. Children can see and touch geometric concepts. It may seem trivial to adults, but numbers and number systems can in reality be quite abstract concepts to wrap some little brains around.
Another way to engage children is to give them a sense of ownership in the mathematics learning process, providing them with a keener awareness of mathematical applications, and thereby providing them with a more personal encounter with learning mathematics. Children feel empowered when they are permitted to study things that are of interest to them. Mathematical connections can be made to just about anything that children find curious in our present day society. Be it the arts, dance, music, video games, computers, ipods, facebook, twitter, sports, or games, organic "aha" or "teachable" moments can easily be generated by harnessing children's intuitve curiosities.
I have found literature (specifically picture books) a useful tool for harnessing some of these curiosities in children of all ages. Additionally, actively listening to a narrative can engage the mind and promote purposeful thinking, learning, and even critical analysis.
Anyway, another reason that I enjoy using literature to teach math is that literature tends to provide a portal for asking and generating thought provoking and meaningful questions. One can kind of let the narrative guide the instruction. If the kids aren't interested in it, there is no point in belaboring the issue, so move on until you find a piece of literature that does interest them. Some books that I would recommend (based on student ratings) are The Very Greedy Triangle, The Number Devil, Anno's Mysterious Multiplying Jar, Amanda Bean's Amazing Dream, The Grapes of Math, How Much Is a Million, Sir Cumference and the Dragon of Pi, The Best of Times, One Grain of Rice, Spaghetti and Meatballs For All, and we can't forget how Lewis Carroll's Alice in Wonderland is densely saturated with mathematical concepts...There are oodles more! However, authors that I tend to come back to on a consistent basis are Greg Tang, Jon Scieszka, Marilyn Burns, Cindy Nueschwander, and David M. Schwartz.
At the end of the day, just try to find ways to make mathematics learning engaging and thought provoking but also meaningful and fun! And by all means, one doesn't always have to use a mathematics picture book to teach math concepts. Once I read, It's Pumpkin Time, a book about the life cycle of pumpkins to my students and ended up creating measurement activities that corresponded with the book rather than science actitivies. Just get creative! Children love that!
More resources if you're interested.
My biggest hobby throughout junior high was video game development. I taught myself how to program in C/C++ and first wrote pong. This introduced me to vectors and I saw first-hand how convenient they could be for geometric computations. I then gradually made more and more complex programs. By high school I was mainly interested in computer graphics, which led to my learning linear algebra.
Having lots of programming experience is in itself extremely beneficial. In today's computer age, you can never know enough about programming.
So this is obviously a quite indirect approach and it's not exactly a teaching technique, but I would highly recommend any young child at least look into game development. It's very intellectually stimulating, it will expose them to a variety of fields that they might otherwise have to wait years and years to be exposed to, it will condition them to think logically and rigorously from a young age, and above all it's lots of fun to see computer science and mathematics work together to produce a video game--which, by the way, you can then play!
These days there are simply incredible resources for teaching oneself the techniques of game development. Anyone interested should visit http://www.gamedev.net.
If they're fairly mathematically inclined anyway, then try to get them solving interesting problems with an obvious mathematical content, if they're less mathematically inclined try to find problems where the usage of maths isn't as explicit.
Problems with a very mathematical bent can be found at places like NRich, they update their problems monthly, the Stage 2 and 3 problems cover that age range. Other sources for problems could be video games (resource management based games require mathematical thinking), code breaking or programming a game (a simple driving game in Flash requires a lot of maths based problem solving).
If they've convinced themselves that they can't do Maths then you basically need to smuggle the maths into your chosen activity without them realising that they're doing maths to solve the problem.
When I was that age, I discovered Raymond Smullyan's classic logic puzzle books in the library (such as What is the name of this book?), and really got into it. I remember my amazement when I first understood how a complicated logic puzzle could become trivial, just symbolic manipulation really, with the right notation.
Some of my favorite ways of engaging students from jump street are:
Note that the curriculum need not be improved or reformed in order to drop one or two of these in at an odd moment or two during class.
Regards, - Mike Jones
Show them large buildings, tallest buildings, pyramids, visualize them about different structure.
Explain importance of maths
Brain remember things easily when comparison is done.
This really depend on how smart the kid is.
I lean toward discrete math, elementary number theory related topics when talking to non-math people about math. They requires little background knowledge. There are some fun problems in discrete math, especially combinatorics.
Simple probability is also nice. So are logic problem.
Both topics can be used to formulate some simple puzzles.
A simple number theory puzzle
How many zeros are there in 20!
I assume a bright 10 year old can solve it.
Speaking from my own experience with elementary mathematics, yours is a very hard question to answer because there is little in the elementary math curriculum worth getting excited about. Before students are going to get excited about math, the math curriculum has to be changed so that the process of doing mathematics is made to be engaging and thought provoking - not tedious. I think every teacher of elementary mathematics needs to watch this video about how math education can and should be reformed.
Seymour Papert's ideas about "maths you can build". Look at Scratch (much inspired by Papert), and read Mindstorms; note that quite a few of his ideas there are contentious, but there is a lot of maths you can make exciting this way.
Check out the book «Mathematical circles». It's written by some Russian mathematicians who used to run "math circles" for middle and high school students and has a lot of interesting material.
For people this age and older I know of no better way to introduce someone to mathematics than to have them look at Martin Gardner's books. A sample of his books are listed here:
http://www.york.cuny.edu/~malk/biblio/martin-gardner-biblio.html
Explaining counter-intuitive facts using mathematics can be useful to sparkle mathematical interest.
For example, in most cases, the height of a tumbler is less than its perimeter which is pretty counter-intuitive. This is a nice way to introduce about $\pi$ to children.