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## Taking Lots of Math in High School: A Reflection

How should a world-class secondary education in mathematics be structured? Of course, that depends on who you ask. Many parents and students likely desire an applicable education, one that facilitates, among other outcomes, enrollment in a competitive 4-year college with the possibility of an internship, or an education that allows for the study of multiple content areas (such as physics or chemistry). through a mathematical lens. Some parents, perhaps, just want their children to pass math and "get on with it," important in something applicable, treat the P, NP conjecture with healthy independent skepticism, and move on to a plush automated trading job. This article is in no way intended to offend the final camp. After all, who in the world would want their 22 year old making 6 figures with a course or two in Numerical Analysis or Stochastics or Healthy Modeling and going about life’s tasks?

My personal opinion is that high school math can be fun and rigorous at the same time. When the mind is young and voracious, it is especially flexible: many excellent mathematics can be learned, and great results can be demonstrated with effort and diligence. Aside from national and international standards (which a world-class math education will pass) and standardized test questions (which won’t apply to someone who completes a really rigorous math education—they’ll pass those tests with little effort), a math really excellent Education must include very well: mathematics. Maybe 10-15 courses in high school would be possible.

No, that’s not a typo. For the most rigorous and modern high school mathematics education for students who desire future studies in mathematics, engineering, theoretical physics, or computer science, it is certainly possible to accumulate as many courses (or more) in some (but not all) cases, and it may not actually be that hard to do, especially if your child is homeschooled or attends certain "classic schools" (see below). Please note that I am not claiming that a 4 or 5 course sequence is insufficient to attend a great university – many people have had great success with this preparation. In my own education, it was better to start with 4 semesters of readings in analysis (advanced calculus) from scratch, working out all the results of sequences, series and approximations, and then progress to theoretical physics via vector calculus, the theory of flows, differential geometry, and partial differential equations (all fields of advanced mathematics that seem terrifying but are more than amenable to study by a highly motivated high school student). With this foundation in place, it was possible to study advanced topics such as manifold theory, algebraic topology, set theory, category theory, logic, module theory, measure theory, and group structure theory (all these are mathematical concepts typically taught in undergraduate and graduate studies). What is the point of all these studies? If this question doesn’t answer itself, don’t try this at home. The point is to learn some good math at a young age so that one can (hopefully) contribute to the "conversation" of the mathematical discourse discovering something beautiful. Of course, a person with the courage and interest to pursue this curriculum could get into IB, law, medicine, etc. later in his training, having been much better at learning a lot at a young age. It certainly couldn’t hurt a student to get an education like that.

But a parent may object that he could never find a syllabus for these courses. And they would be right. You won’t find any canned curriculum (in my experience) that teaches pretty math. Sorry. But this does not mean that this learning is impossible. If you’re lucky, you might have one (or more) of the following available to you, depending on where you live. For each option, here are some suggestions if your family lives near that option:

If your child already attends or has been accepted to an elite school or boarding school whose graduates routinely change the world and have been driving global trade and politics for several centuries. In this case, the school likely has some excellent options at home and probably has some very competent teachers to administer them. You may still need to negotiate some independent study courses with professors. If you live near a classical Christian school. Honestly, these are a goldmine. These schools give students the opportunity to read great books in a tightly targeted context, and often produce incredibly bright graduates who attend strong universities and do well. In full disclosure, I teach at one of these schools right now, but I’m not getting paid to say any of this. It’s my honest opinion. The professors at these schools typically have a strong mathematical background, and some of these schools may have a faculty member with a graduate degree in pure mathematics (a comparative rarity at other types of schools) who will be happy to supervise your child’s high school math education. . Even if you don’t have your children in school, you may be able to pay a fee directly to the teacher, who could develop learning materials for your student and assess the relevant courses. If you live near a university and can find a teacher who is interested in helping your child develop this curriculum. I was lucky to have many instructors like him. Don’t be shy: Email a pure math teacher and ask them directly if they’d be interested in mentoring and helping build courses for your child. Offer to pay for the help. Tell them about your child’s needs and ask for advice. Tell them you want your child to have a top math education. Don’t be surprised if the teachers help you. They will probably be so amazed at your child’s eagerness to learn that they can really help you. In future articles, I (or my colleagues) will clarify grade-by-grade the type of education we propose for highly motivated high school students, as well as answer current questions about rigorous mathematics in high school. In addition, I will answer questions about state standards, testing, etc.

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