Skip the Basics
My younger brother recently left for his sophomore year at MIT. While he was still here, we got together several times and had several lengthy discussions about college life had been for him over the past year. One interesting point he mentioned is how placement works at MIT:
When my brother matriculated at MIT, he had already taken honors multivariable calculus at Stanford (receiving an A). However, MIT refused to accept the Stanford course as credit for it’s own (non-honors) multi-variable calculus course . Instead, he would have to take a placement exam along with all of the other freshmen hoping to place out of multi-variable calculus. No problem, I figured. As I saw it, his preparation should have been more than enough.
Well, here’s my paraphrase of my brother’s story:
Ten minutes before his multivariable calculus exam, I walked to the exam room and noticed a girl sitting outside immersed in her crib sheet. She had meticulously copied all of the formulas that she had learned down on it and was last-minute cramming before the exam. I asked her if I could take a look at her preparation sheet. Upon examining her preparation sheet, I quickly realized that I did not know a single formula on it. Many of the formulas were for double integrals.
Minutes before the exam, I did not actually know how to actually take a double integral.
In the quarter I had spent doing honors multi-variable calculus at Stanford, I had done literally nothing but proofs. While the class had been exceedingly difficult and had required us to prove many non-trivial properties about double integrals, we had never actually taken one. The professor deemed this material as unsuitable for the theoretician to waste his working memory on. He consciously never taught it.
In the next five minutes preceding the exam, my friend Nathan (who had also taken 51H at Stanford) and I learned how to take a double integral. Following that, we both prayed to the god of partial credit and headed into the exam room.
The actual exam contained quite a few double integrals that I was unable to solve. So what I did instead was exactly what I had learned to do in 51H: I proved the theorem that they wanted us to apply. Then (as I’d never learned any of the formulas) I’d just leave the problem there: with a proof in English and no numeric answer.
Given that he never wrote an answer for a good part of the test, you might expect that my brother bombed his exam and had to retake the class. However, he instead passed his math exam with flying colors, receiving one of the highest scores out of his placement group.
In my brother’s case, his high-level theoretical understanding helped him to quickly comprehend the low-level applications that he had to solve . There would simply have been no point for him to spend extra time learning how to plug-and-chug numbers into what already knew. Furthermore, I assert that the teaching philosophy of his Math professor is spot on. I’ve believed for a long time that (if you can do it), it’s always better to skip the basics and go headfirst into the advanced stuff. If something is beneath you, don’t learn it. If it isn’t worth your time, then don’t spend your time on it. My philosophy is that you should learn how to tackle the hardest problems that you can. And the contrapositive to that is that you should triage everything else, as much as you can.
I used to live with an entrepreneur who literally never cleaned his room or cooked his own food. What he did do was use TaskRabbit to hire a helper, who kept his room immaculate and would cook him plain chicken and brown rice. Now I’m not a chef, but chicken and brown rice is not the toughest meal to make. I’m sure he does this because he realizes that there is no point in wasting his time doing tedious life maintenance when it would be better spent making business deals, meeting new contacts, or spending time with his girlfriend. And although he is still young, his track record for focusing on these aspects is already quite good. Already, his decision to skip the basics has helped to make him one of the most seriously professional and well-networked people who I know.
Modern education has an obsession with fundamentals. We persist in preaching a canon of knowledge that includes that is almost entirely basic. The general consensus seems to be that a student should first learn the basics, slowly increasing difficulty with higher levels until those too become basic. While this may be true in sports (Lebron James still practices his dribble), it doesn’t seem likely to me to be true for most of knowledge .
 This would have been reasonable to me, had my brother taken the normal multi-variable course as opposed to the honors one. Objectively speaking, I do think that MIT is a more theoretically rigorous school than Stanford. There is a reason why many more Math and Science Olympiad kids go to MIT, and I think this is in part because the undergraduate education is more intellectually invigorating. The students are more nerdy, and the classes are harder (of course, people go to Stanford for other reasons).
 I am often skeptical about much of what I hear. However, my own experience with Math leads me to believe that this is not actually as difficult as it sounds. Math is much more so a system of logic than it is a system of numbers. In fact, real mathematicians never use numbers.
 The former understanding of knowledge is something like a line. You progress a little and move a bit forward on the line. We like thinking about it this way because one-dimension is easy for us to intuitively understand. However, we already know that our brain - composed of highly-dimensional neural networks - does not fit this pattern at all. If we tried to understand knowledge this way, it’d be something more like evolution: highly punctuated and irregular, with random connections from all over the place, full of equally surprising successes and failures. In this case, everyone’s learning pattern would be different, as it should be.