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Taking Lots of Math in High School: A Reflection
How to structure a world-class secondary mathematics education? Of course, it depends on who you ask. Many parents and students probably want an applicable education – an education that facilitates, among other outcomes, enrollment in a competitive 4-year college with the possibility of internships, or an education that allows for the study of various content areas ( such as physics or chemistry) through a mathematical lens. Some parents, perhaps, just want their children to do well in math and "Go for it," major in something applicable, treat the P, NP conjecture with detached and healthy skepticism, and move on to some plush automated trading work. This article is in no way intended to offend the final camp. After all, who in the world would want their 22-year-old to make 6 figures with a course or two in numerical analysis, stochastics, or healthy modeling and get on with the chores of life?
My personal opinion is that high school math can be both fun and rigorous. When the mind is young and voracious, it is particularly flexible – much great mathematics can be learned and great results can be proved with effort and diligence. Aside from national and international standards (which a world-class math education will exceed) and standardized test questions (which won’t be applicable to someone completing a truly rigorous math education – they’ll pass these tests with little effort), a really superb math education should include a lot of good – math. Maybe 10 to 15 classes in high school would be possible.
No, it’s not a typo. For modern and rigorous secondary mathematics education for students wishing to pursue studies in mathematics, engineering, theoretical physics or computer science, it is certainly possible to accumulate as many courses (or more) in some cases (but not all), and it might not be that hard to do, especially if your child is home-schooled or attends some "classical schools" (see below). Please note that I am not claiming that a sequence of 4 or 5 courses is insufficient to attend a major university – many people have achieved such preparation brilliantly. In my own education, it was best to start with 4 semesters of readings in analysis (advanced calculus) from scratch, working all the results of sequences, series and approximations, then moving on to theoretical physics via the vector calculus, flux theory, differential geometry, and partial differential equations (all areas of advanced math that sound scary but are more than likely to be studied by a highly motivated high school student). With this foundation in place, it was possible to study advanced topics like manifold theory, algebraic topology, set theory, category theory, logic, module theory, measure theory, and theory. group structure (all of these mathematical concepts are usually taught in university and graduate courses). What is the point of all these studies? If this question doesn’t answer for you, don’t try this at home. The goal is to learn great math at a young age so that one can (hopefully) contribute to the "conversation" of mathematical discourse by discovering something beautiful. Of course, someone with the courage and interest to pursue such a program could enroll in IB, law, medicine, etc. later in his education, having been all the more apt to learn much at a young age. It certainly couldn’t hurt a student to get an education like this.
But a parent may object that he could never find a program for such courses. And they would be right. You won’t find any canned curriculum (in my experience) that teaches great math. Sorry. But that does not mean that such learning is impossible. If you’re lucky, you may have one (or more) of the following available to you, depending on where you live. For each option, I present some suggestions if your family lives near this option:
If your child is already attending or has been accepted by an elite day or boarding school whose graduates have been steadily changing the world and have been driving global commerce and politics for centuries. In this case, the school will likely have great in-house options and will likely have very capable teachers to administer them. You will likely still need to negotiate independent study courses with teachers. If you live near a mainstream Christian school. Honestly, these are a gold mine. These schools give students the opportunity to read excellent books in a tightly supervised setting and often produce incredibly bright graduates who attend solid universities and do well there. For full disclosure, I teach at one of these schools right now, but I’m not paid to say anything about any of this. This is my honest opinion. Typically, the teachers at these schools have excellent math training themselves, and some of these schools may have a faculty member with a graduate degree in pure math (a relative rarity in other types of schools) who would be happy to oversee your child’s secondary math education. . Even if you don’t have your kids in school, you might be able to pay a fee directly to the teacher, who could produce learning materials for your student and assess relevant lessons. If you live near a university and can find a teacher who would like to help your child develop such a program. I have been especially lucky to have many instructors like this. Don’t be shy: email a pure math faculty member and ask them directly if they’d be interested in mentoring and helping build lessons for your child. Offer payment for assistance. Talk to them about your child’s needs and ask for advice. Tell them you want your child to have a great math education. Don’t be surprised if the teachers help you. They will probably be so amazed at your child’s appetite for learning that they could very well help you. In future articles, I (or my colleagues) will clarify on a lesson-by-lesson basis what kind of education we offer highly motivated high school students, as well as answer current questions about rigorous high school math. Additionally, I will answer questions regarding state standards, testing, etc.
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