How should a world-class secondary-school education in mathematics be structured? Of course, that depends on who you ask. Many parents and students likely desire an applicable education – one which facilitates, among other outcomes, matriculation at a competitive 4-year college with the possibility for internships, or an education that permits the study of diverse content areas (such as physics or chemistry) through a mathematical lens. Some parents, perhaps, just desire their children to pass mathematics and “get on with it,” major in something applicable, treat the P, NP conjecture with a detached and healthy 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 to make 6 figures with a healthy Numerical Analysis or Stochastics or Modeling course or two and get about the tasks of life?
My personal opinion is that high-school mathematics can be fun and rigorous at the same time. When the mind is young and voracious, it is especially supple – lots of 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 testing questions (which will not be applicable to someone completing a truly rigorous math education – they will ace these tests with little effort), a truly superb math education has to include a lot of well – mathematics. Perhaps 10-15 courses in high school would be possible.
No, that is not a typo. For the most rigorous, modern secondary math education for students desiring future study in math, engineering, theoretical physics, or computer science, it is certainly possible to accumulate this many courses (or more) in some (but by no means all) cases, and it’s actually perhaps not that difficult to do so, especially if your child is home-schooled or attends certain “classical 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 fared brilliantly with such preparation. In my own education, it was best 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 advancing to theoretical physics via vector calculus, the theory of flows, differential geometry, and partial differential equations (all fields of advanced math that sound scary but are more than susceptible to study by a high-school student of high motivation). With this foundation in place, it was possible to study advanced topics like manifold theory, algebraic topology, set theory, category theory, logic, moduli theory, measure theory, and the structure theory of groups (all of these are math concepts typically taught in university and graduate courses). What is the point of all of these studies? If this question does not answer itself for you, 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 by discovering something beautiful. Of course, a person with the pluck and interest to pursue such a curriculum could go into IB, law, medicine, etc. later on in his or her education, having been all the better for learning a lot at a young age. It certainly could not hurt a student to get an education like this.
But a parent may object that he or she could never find curriculum for such courses. And they’d be right. You won’t find any canned curriculum (in my experience) that teaches beautiful mathematics. 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 upon where you live. For each option, I outline some suggestions for you if your family lives near that option:
If your child already attends, or has been accepted by, an elite day- or boarding-school whose graduates routinely change the world and have been driving global commerce and politics for several centuries. In this case, the school will likely have some superb options in-house, and will likely have some very competent teachers to administer them. You’ll still likely have to negotiate some independent study courses with the teachers. If you live near a classical Christian school. Honestly, these are a goldmine. These schools give students the opportunities to read great books in a tightly mentored context, and often produce astoundingly 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 any of this. It is my honest opinion. Typically, teachers at such schools have great mathematics educations themselves and some of such schools might possess a faculty-member with a graduate degree in pure mathematics (a comparative rarity in other sorts of schools) who would be happy to supervise your child’s secondary mathematics education. Even if you don’t have your children attend the school, you could perhaps pay a fee directly to the teacher, who could produce learning materials for your student and assess the relevant courses. If you live near a university and can find a professor who is interested in helping your child craft such a curriculum. I was particularly lucky to have many instructors like this. Don’t be shy: email a faculty member in pure mathematics and ask them directly if they’d be interested in mentoring and helping to construct courses for your child. Offer payment for the help. Tell them about your child’s need and ask for advice. Tell them you want your child to have a top-notch mathematics education. Don’t be surprised if the professors help you. They will likely be so astounded at your child’s appetite for learning that they very well may help you. In future articles, I (or my colleagues) will clarify course-by-course the sort of education we propose for very driven secondary-school students, as well as answer topical questions regarding rigorous mathematics in high school. Also, I will answer questions regarding state standards, testing, etc.