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This question is about what a mathematics instructor does.
Mathematicians work in a variety of settings, such as at universities, engineering firms, and banks. Depending on where they work, mathematicians will do a variety of things. They will also be required to focus on additional skills in some environments. A mathematician working at a bank will be required to have an understanding of finance as well as mathematics.
In the university setting, mathematicians will be working solely in mathematics. They will be working with both faculty and students. If you are interested in research and publishing work, working through a university will be great as this will be required along with your lecturing and teaching time.
If working in applied mathematics interests you more, finding a position at an engineering firm could be a great fit. Mathematical engineers often work either independently or with other engineers in office settings. They will focus on the practical application of mathematics and real-life implementation of mathematics.

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