A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas
Mathematicians are often asked by friends, family, colleagues in other fields, and strangers: "What do mathematicians do?" Here are some resources and facts that may help answer that question.
Mathematicians make it possible to send secure emails and buy things online. Mathematicians are essential to analyze data and design accurate models in fields as diverse as biology and finance. Mathematicians enabled researchers to complete The Human Genome Project quickly. And because of the prevalence of the computer at work and at play, mathematicians will continue to touch everyone in modern society.
There are over 35,800 individual members of the four leading professional mathematical sciences societies in the U.S.---the AMS, the Association for Women in Mathematics, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. Most would call themselves mathematicians; many received their doctoral degrees outside the U.S. There are at least 10,000 more members of the societies who are graduate students or in other categories, and there are also mathematicians who are not members of any of these societies.
Although they have advanced degrees in mathematics, many of those employed in academia might call themselves professors instead of mathematicians, and similarly, those in industry and government may not have "mathematician" in their job title. The job title doesn't tell the whole story, however: These people are doing mathematics and are indeed mathematicians. Furthermore, the number of mathematicians is increasing. The number of new Ph.D.'s in the U.S. has gone up every year since 2002.
Mathematicians are people of all ages and from all over the world who enjoy the challenge of a problem, who see the beauty in a pattern, a shape, a proof, a concept. Some of the best young mathematicians compete in math olympiads, state and national science fairs, or the fun Who Wants to Be a Mathematician game. Some high school mathematicians go to summer Math Camps to learn more and work with teams on projects; undergraduates participate in Summer Research Experiences. Many carry on their research and teach at colleges and universities, while others apply their skills in all kinds of professions. (See Sloan Career Cornerstone for profiles of mathematicians at all stages of their careers.) There's probably a bit of the "mathematician" in all of us and we don't even realize it. Keith Devlin poses this idea in his book, The Math Instinct: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs). In any case, those who are not "mathematicians" can appreciate the subject by reading reviews of books about mathematicians, how mathematicians think, breakthroughs in mathematics, and current applications at Math in the Media.