It’s a joyful but widely misunderstood subject. Many people, especially in the United States, grow up to believe that mathematics is a dry, esoteric subject that only ”math people” learn and excel at. This is usually because of how school mathematics is taught: by lectures and drills rather than exploration and problem-solving. People tend to think of mathematics as a body of content knowledge, but it is really a practice and an art. Teaching and learning this practice of mathematics both have their difficulties, but the problem-solving process and the discovery of unexpected patterns can bring a lot of joy. Personally, to help others discover this joy is my primary reason for teaching mathematics. Even if students do not learn mathematics in school, they receive many negative messages about it from society, so countering these messages and replacing them with positive messages in the maths classroom is paramount.
Mathematics has applications in many endeavors. Some students go on to directly use mathematical skills in their careers, whether as mathematicians, statisticians, experimental scientists, accountants, teachers of maths or science, etc. Their mathematics classes not only may be the foundation of their skill development, but also of their interest in such careers. Other students go into careers that are less obviously dependent upon mathematics but may still need mathematical knowledge along the way. When one of my music professors graduated from college, he first took a job at a bank before undergoing graduate study in music; his mathematics courses were key for that job, which allowed him to sustain himself while he practiced music in the evenings. Mathematics knowledge makes a person more versatile in an ever-changing job market. Thus teaching mathematics is particularly important in underprivileged communities, as mathematical mastery can give students from these communities a better chance of success and personal fulfillment. Students can also start applying their mathematical skills to real-world problems while in school, such as through research projects to make change in their own communities.
If done well, it trains you intellectually, emotionally, and socially. Even if students do not ever directly use mathematics after completing their education, their mathematical training can indirectly serve them well. Through mathematics, one learns deductive and inductive reasoning skills that are useful for all academic subjects (deduction being the logic of mathematics, induction being part of the problem-solving process as Pólya observed). Through the solving of difficult problems, you learn perseverance, creative thinking, how to deal with imposter syndrome, and generally how to grapple with intellectual and emotional challenges and not give up. Through collaboration on problems and communication of ideas and solutions, you become a better collaborator and communicator in general and also learn to appreciate other people’s ideas and unique capacities.