How should a world-class secondary education in mathematics be structured? Of course, that depends on who you ask. Many parents and students are likely to want an applicable education, one 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 multiple content areas (such as physics or chemistry) through a mathematical lens. Some parents may just want their children to pass math and "let’s do it," specializing in something applicable, treating the P, NP guess with healthy distant skepticism, and moving on to some fancy automated trading job. This article is not intended to offend the final camp. After all, who in the world would want their 22 year old to make 6 figures with a healthy Numerical Analysis or Stochastic or Modeling course or two and go about the business of life?
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: much excellent mathematics can be learned and great results can be demonstrated with effort and diligence. Aside from national and international standards (which a top-tier math education will pass) and standardized test questions (which will not apply to someone who completes a truly rigorous math education; they will pass these tests with little effort), a truly excellent education has to include a lot of good math. Perhaps 10-15 courses in high school would be possible.
No, that is not a mistake. For the most rigorous and modern secondary mathematics education for students who wish to study in the future in mathematics, engineering, theoretical physics or computer science, it is certainly possible to accumulate this number of 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 "classical schools" (see below). Please note that I am not claiming that a sequence of 4 or 5 courses is insufficient to attend a great university; many people have done brilliantly with such preparation. In my own education, it was better to start with 4 semesters of readings in analysis (advanced calculus) from scratch, calculating all the results of sequences, series and approximations, and then progressing to theoretical physics via vector calculus, theory of flows , differential geometry, and partial differential equations (all advanced math fields that sound scary but are more than likely to be studied by a highly motivated high school student). With this foundation established, it was possible to study advanced topics such as manifold theory, algebraic topology, set theory, category theory, logic, modulus theory, measure theory, and quantity theory. group structure (these are all mathematical concepts typically taught in undergraduate and graduate courses). 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 beautiful mathematics at a young age so that one can (hopefully) contribute to the "conversation" of mathematical discourse when discovering something beautiful. Of course, a person with the courage and interest to pursue such a curriculum could go into IB, law, medicine, etc. later in his education, having been much better off having learned so 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 or she could never find a curriculum for such courses. And they would be right. He won’t find any canned curriculum (in my experience) that teaches beautiful mathematics. I am sorry. But this does not mean that such learning is impossible. If you’re lucky, you may have one (or more) of the following at your disposal, depending on where you live. For each option, I’ve outlined some suggestions for you if your family lives near that option:
If your child already attends or has been accepted into an elite day 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 probably has excellent internal options and probably has very competent teachers to manage them. You may still have to negotiate some independent study courses with professors. If you live near a classical Christian school. Honestly, these are a gold mine. These schools give students the chance to read great books in a context of strict tutoring and often produce amazingly bright graduates who go on to solid universities and do well there. For full disclosure, I teach at one of these schools right now, but I don’t get paid to say any of this. It is my honest opinion. Teachers in such schools are generally highly educated in mathematics, and some of these schools may have a faculty member with a postgraduate degree in pure mathematics (a comparative rarity in other types of schools) who would be happy to oversee the teaching. your child’s secondary math education. . Even if your children are not in school, you may be able to pay a fee directly to the teacher, who can produce learning materials for your student and assess relevant courses. If you live near a university and can find a teacher who is interested in helping your child develop such a curriculum. I was particularly lucky to have many instructors like this. Don’t be shy: email a member of the pure mathematics faculty and ask them directly if they would be interested in advising and helping build courses for your child. Offer payment for help. Tell them about your child’s need and ask for their advice. Tell them you want your child to have a top-notch math education. Don’t be surprised if teachers help you. They are likely to be so amazed by your child’s appetite for learning that they may very well help. In future articles, I (or my colleagues) will clarify course by course the type of education we propose for highly motivated high school students, as well as answer topical questions about rigorous mathematics in high school. Also, I will answer questions about state standards, tests, etc.
