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Choosing your audience
If you want to become a math teacher, the first step is obviously to decide what level of mathematics you want to teach, as different positions will have their own set of requirements. While someone planning to teach college math will obviously need a PhD in mathematics, and someone teaching high school math will need at least a bachelor's degree in the area, someone teaching elementary school may be better off with a degree in elementary education with a math minor.
While your classes are likely to be at a much higher level than anything you're going to be teaching (unless you teach graduate students), your goal is to have a thorough understanding of not just how to do the math, but the reasoning behind the problem-solving algorithms and how everything works together.
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The author's experience has been that many people who end up teaching elementary math don't particularly like the subject or have a great understanding of it; don't let that be you! If you're going to teach math at all, you should be comfortable with arithmetic and basic algebra.
One of the main reasons people struggle with math is that they never get the basics down, so when they have an algebra problem they're too busy trying to remember how to do multiplication and division to concentrate on the overall problem structure. If everything up to and including factoring isn't second nature to you, keep practicing until it is. Don't just take "Math for elementary school teachers"; sign up for a course in college algebra as well.
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If you're planning to teach high school math, you should have at least a minor in mathematics and preferably a major. You'll want to be familiar with algebra, geometry, and trigonometry, as well as possibly some offshoots like business math. You should also be comfortable using a graphing calculator.
High school (and college) students are often more interested in the practical side of things than in appreciating a subject for its own sake, so if you get a chance, it's not a bad idea to learn about how each subject is actually used in the real world. This also lets you make up more interesting problems as you can relate the math to careers that your students may be interested in.
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If you plan to teach college math, you'll need at least a master's degree (to teach community college) up to a PhD (to become a professor at a university). What subfield you specialize in is pretty much up to you; if you're going to teach community college, you no doubt took more advanced classes getting your bachelor's degree than the community college offers. Similarly, teaching at a university you won't get into the complex math you became accustomed to during your graduate studies unless you start teaching graduate students, in which case you'll be assigned classes in your area of specialty. Thus, while it's always helpful if your interests and background happen to line up with whatever the schools are looking for when you apply for a job, you're pretty much free to study any area you like. Regardless of the area though, you should endeavor to become an expert so that you're the obvious choice when a position teaching that particular part of mathematics opens up.
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About the Author
William spend several years teaching middle school math before returning to school, where he taught undergraduate classes for four years while working on his PhD in computer science.