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 university with the possibility of internships, or an education that allows for the study of a variety of 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," specialize in something applicable, treat the P,NP conjecture with aloof and healthy skepticism, and move on to a fancy job of automated trading. This article is not intended to offend the end camp. After all, who in the world would want their 22-year-old son to make 6 figures with a healthy course in Numerical Analysis or Stochastics or Modeling or two and get on with 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: much excellent mathematics can be learned and great results can be demonstrated with effort and diligence. In addition to national and international standards (which a first-rate mathematics education will pass) and standardized test questions (which will not be applied to someone who completes a truly rigorous mathematics education; they will pass these tests with little effort), a truly excellent education has to include a lot of well-mathematics. Maybe 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 many courses (or more) in some (but not all) cases, and it may actually not be that hard to do, especially if your child is homeschooled or attends certain "classical schools" (look down). Please note that I am not claiming that a 4 or 5 course sequence is insufficient to attend a great university; many people have done brilliantly with such a preparation. In my own education, it was best to start with 4 semesters of reading in analysis (advanced calculus) from scratch, calculating all the results of sequences, series, and approximations, and then moving on to theoretical physics via vector calculus, flow theory , differential geometry, and partial differential equations (all advanced math fields that sound scary but are more than amenable to study 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, module theory, measure theory, and the theory of group structure (these are all mathematical concepts that are 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 an early 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 get into IB, law, medicine, etc. later in his education, having been much better for 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 math. Forgiveness. But this does not mean that such learning is impossible. If you’re lucky, he may have one (or more) of the following at his disposal, depending on 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 to an elite day school 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 likely has excellent internal options and probably very competent teachers to manage them. You may still have to negotiate some independent study courses with your professors. If you live near a classical Christian school. Honestly these are a gold mine. These schools give students the opportunity to read great books in a context of strict tutoring, and often produce amazingly bright graduates who attend and do well at strong universities. 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 generally have an excellent mathematical education 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 high school math education. . Even if your children aren’t in school, you might be able to pay a fee directly to the teacher, who could produce learning materials for your student and test 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 Pure Mathematics faculty member and ask them directly if they would be interested in mentoring and helping build courses for your child. Offer to pay for help. Tell them about your child’s need and ask for their advice. Tell them that you want your child to have a top-notch math education. Don’t be surprised if teachers help you. They are likely so amazed by your child’s appetite for learning that they may 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 current questions about rigorous high school mathematics. Also, I will answer questions about state standards, tests, etc.