Advice for an independently-learning pre-college student.

Back again, I am....

Here was a question I received recently:

Hello, ... I recently finished my sophomore [year in high school].... I have, for the past year, learned mathematics independently, taking trig, pre-calculus, calculus BC, and multi-variable calculus on my own. For the first two courses, I used an online provider. The third, I took an AP test to demonstrate that I have sufficiently learned the material so that I might receive credit for it when I go to college.

For the last course, multi-variable calculus, I would like to find a way to gain either recognition (such that I would not have to take the class in college) or credit for the course before I enter college. This is not the only class that I will have taken independently for which I cannot take an AP exam to be granted recognition. Also, I plan on starting other math courses independently (Linear algebra, differential calculus, etc.), so there will be multiple classes which I will have learned, but nothing to show for them.

Is there a way, through Johns Hopkins, I could acquire either credit for having learned college-level courses independently? If not through Johns Hopkins, do you know of a way to do this using different means?

A good question, as there are many pre-college out there struggling to quench their thirst for mathematical knowledge amidst a dry, arid environment void of opportunity. My reply:

While I like your initiative, and value your capabilities, I am wondering why you are trying to burn through all of this material at such a high speed. The AP exams, while a nice system for providing advanced training in mathematics to pre-university students, do not really measure proficiency in calculus. Rather, they measure your ability to apply proper techniques to appropriate problem types. While this is helpful, it is not really what mathematics is all about.

In your case, looking for opportunities outside high school for advanced training (as you are doing through self-study) is a good idea. But simply relying on an online course or a book and a standard exam may wind up giving you a false indication of your true knowledge base in these subjects. And if you foundation is not strong in basic subjects, you may find yourself faltering later on at the higher levels.

Some questions: (1) Do you have a mentor at your high school, or nearby, a math instructor, or mathematician to help guide you through your self studies? Someone who can see your "path" from above while you walk it is very important to your training. (2) Is there a goal in your life, which provides the reason for going from trigonometry to vector-calculus and beyond in a single year? These are beautiful subjects full of amazing insight and deep conceptual meaning. Burning through them at top speed is really selling the individual topics short. This is like driving through a safari park at 80 miles an hour. You have done the park, but have you really spent time learning about the animals. (3) Have you looked at simply taking courses on these topics at your local university, one at a time, and with live instruction? Even at the community college level, there are very good instructors whose lectures in class and conversations outside of class can be extremely helpful in seeing more then the techniques.

Yes, we here at Hopkins have many ways of evaluating the proper level for students to start at their first semester here. And we are committed to ensuring that students are not taking courses they are clearly too advanced to take. Acknowledging a students proficiency in a mathematics course may not always involves credits for the course (maybe just a waiver), but most of our evaluation involves some sort of comprehensive documentation of prior training, and not just an exam. Exams are not usually very good indicators of real understanding.

I hope this helps. Good luck in your training.

Spirit, initiative and resourcefulness are primary qualities of the budding scholar. Having and/or finding a mentor or guide is absolutely fundamental (even Harry Potter wouldn't have made it on his own!) And taking your time to digest what you are learning always leads to "better nutrition", no?