How to present math as something interesting? If you are helping someone with mathematics in high school, what do you need to do to win his attention so she/he can focus on curriculum?
 A: I've had success only two times that I can remember in getting students interested. The first time was with a younger student in fourth grade. We were talking about something unrelated to his interest in math when he asked me about bigger numbers. I explained that Google was named after a fairly large number (albeit mispelled) and that there is an even larger number, googolplex, which cannot be written out as a 1 followed by zeroes because there are fewer atoms in the universe than there are zeroes in googolplex. Somehow that changed his whole attitude about math. My theory is that he was able to go to school with that knowledge and impress his friends, which made him see math as something useful in his life. I think somehow exposing a child to the majesty of mathematics without having a homework problem or any "work" involved can really help. You don't have to tell someone a flower is beautiful or smells nice, and telling them to go to an arboretum won't be as effective as bringing a flower right to them. Then they will see for themselves.
The second time was with a high school boy who asked the age-old question of why we study math at all if we're never going to use it. I compared the study of math (and Shakespeare and making telegraphs with batteries and lightbulbs) to a muscle building workout. We might do bench presses to strengthen our muscles to help with playing sports or wrestling a bear, but when we are throwing a football or fighting a bear, the actual motions we make are nothing like a bench press. Likewise with math, we are doing brain exercises that strengthen certain parts of our brain, but when we actually use those parts, it might have nothing to do directly with mathematics. In this case, I think the student respected the fact that I didn't try to tell him that he would be calculating his gas mileage in the future, and not glorifying math above other subjects like english or science connected with his teenage cynicism. And maybe in his case he did not start finding math interesting as much as he respected math education in school enough to start doing his homework.
Both of those events were more luck than design. I do have some thoughts on the subject that are more direct:


*

*Be an interesting person who is honestly interested in math, and figure out why you find it interesting.

*Connect with your students and learn about them. Education starts with a relationship between a teacher and one or more students. Build that relationship and trust. How to Win Friends and Influence People by Dale Carnegie is a good, short book on connecting with people. Being interested in them helps them be interested in you and what you're interested in.

*Don't expect older students to ever learn to love math. Some might, most who don't already love it probably won't learn to love it. Claims that they will use it in some way usually ring false to these students who have been trained to be addicted to calculators and computers. Sometimes merely justifying the learning of it for grades needed to graduate and do whatever the student really wants to do are enough to get them to put in some work.

*Some students are not ready to be reached. I'm currently tutoring a high school girl whose father just passed away. If our lessons distract her from her grief for just a few minutes I consider that a win. Some students have much bigger problems than math, grades, graduation, or thinking about the future, and it might take more than we can offer to take their minds off of their other problems.

A: If you have not already, then I highly recommend you read A Mathematician's Lament. To quote directly from the paper:

"You don’t need to make math interesting— it’s already more
  interesting than we can handle! And the glory of it is its complete irrelevance to our lives. That’s why it’s so fun!"

It might not directly answer your question, but I'm sure it will give you something to think about.
A: I'd say the best way to stir interest in math is to start with applications of math to a subjects they're already interested in. If someone, say, shows an interest in politics, than the math behind elections, the effects of gerrymandering, and the like could be a start. Maybe introduce him or her to the theorem which, under pretty mild assumptions, shows that no satisfactory voting system exists.
If someone shows an interest in biology, maybe population dynamics would be a good start. Or some of the methods usually referred to as bioinformatics.
The most important part is to show that math can be a very usefull tool to gain insights into a lot of different problems in different fields. In my experience, the reason so many people dislike maths is that is is percieved as a subject made up of arbitrary rules, bearing no connection to the actual world. So overcoming that has to be the first and foremost goal.
A: Mathematics is INHERENTLY interesting if it is UNDERSTOOD.
"Understand" does not mean ability to remember and recite and mechanically perform and willingness to conform and follow directions. 
"Understand" means that you see what we are after, what we are about, what we want, what there IS to want, where we are going, what we are doing.
Mathematics as rules and arguments does reveal that or even hint at it. Neither does mathematics as rigor or reasoning or application or cognition or history or success or mathematicians or philosophy. Mathematics is about constructing and exploring. Promoting successes gives no clue to what it's about. It doesn't matter how many findings one remembers. Mathematics is ABOUT creating and searching, not pride in what was created and found. The relevance of mathematics is not in its application.
Mathematics is the physical science of quantity and the phenomenal developments that proceed from naming and symbolizing quantities, operations with quantities, relationships between and among quantities, and so on, including interest in the things to which quantity applies - mainly sets and fields and rings and algebras and whatever - and the astounding implications and properties of all of these and their descendants. The claim that mathematics is abstract surgically removes all possible interest in it except for exceptional jugglers or those who see beyond the claim. It is no more abstract than any other science. The fact that many of its findings can be proven is due to the nature of quantity and not the esotericness (esotericality, esotericosity, esotericicity) of the subject. 
A: I suppose a good way is to point to them concrete things that math has done to improve our lives. The fundamental operating basis for an MRI machine (and a variety of other sophisticated medical imaging equipment) is based on results straight out of mathematical physics. If you ever need to go to the airport, a variety of large-scale optimization algorithms are being employed to schedule your flight in the most optimal and robust fashion. Additionally, optimal control theory is being used to guide the aircraft's flight (under the pilot's management). Then there is a whole host of mathematics behind the operation of cellular telephone networks, which I am not qualified to write about in great depth (other than that it is heavily based on phase-locked-loops and the Viterbi algorithm). An excellent, interdisciplinary example was demonstrated a few months ago at a presentation given on optimization of wind-turbines. In that work, they parameterized the geometry of the turbine blades to a vector of parameters. Then, they plugged their turbine blade into a computational fluid dynamics simulation, which computed torque. They iterated on the surface parameters until convergence to a maximum-torque blade design was revealed. That example pulls together an immense amount of mathematical understanding - the equations of fluid dynamics, geometry, optimization algorithms, Newton's Laws, electrical power systems engineering, and the numerical solution thereof - in order to increase electrical power output.
