To a vast majority of people, the boring statement is turned around-pure math is dreadfully boring and a couple of minutes of such math talk would be more than enough to cure insomnia. Like most things, presentation has a great deal to do with the boredom factor. I would imagine from Mark Saul's postings that he could take a remedial class on trip through the mathematical celestial heavens without relating math to practical applications and leave the students with stars in their eyes. John Conway has enough knowledge of geometry and the other maths to spellbind professional math audiences as well as the unknowing crowds. If that was the state of mathematics education as a whole in this country, then pure mathematics classes would be Standing Room Only. Yet it is one of the most reviled subjects in school and out, because very few people ever gaze knowing upon its beauty.
The beauty of math is that it works for all things big and small. In my area, it works for the geometric design of the large sweeping landscape and architectural layout of a site and for the smallest screw as well. There is not a single part of the whole site that can be broken down or built up without the use of math. Mathematics is essential in every job, present or future. To skim over the important relationships of math to other subjects is to belittle the essence of a good education to anyone beside the pure mathematician.
Am I saying teach real world problems in math class? No, the fact of it is, math teachers are seldom qualified for such teaching and would probably bore the pants off Victoria herself in trying. They can make math real to students through their own experiences, no matter what they are. There is no limit to the experiences because there is nothing that cannot be related. There is no interest, hobby, avocation, or work that cannot be brought alive through a strong knowledge of mathematics if one can see their world through that knowledge.
Unlike John's experience, I was brought in to give a six hour seminar on the relationship of work to mathematics for a dozen high school math teachers and we wore the subject out for those six hours. The most frequent written comment from the teachers was, we need more time with this next time. What was funny was at the beginning of the class, the teachers themselves were wanting to know where this stuff they were teaching was used. It is difficult to portray the conviction of usefulness when you don't know yourself. Perhaps we have pushed this math for math sake so far that the followers have lost sight of the goal. While I am sure that is not the case for most of the people on this forum, it does seem be beyond this place.
Is the purpose of studying math to glorify mathematics itself or to provide our young with tools and skills to make a place for themselves in the world? I agree that it is not really up to the math educator to explain the uses of mathematics for the perennial questioner. The question should be directed to people who do use mathematics in their work. The problem here is very few people understand their work in a mathematically stated way, therefore, cannot make the needed relationship for the student. I think it will take a great effort to change that Catch 22 situation anytime soon.
Beside the rigid stand that math must be taught for math sake, is there some way that math can be taught so its applications will be apparent in the future?