Today, YouTube recommended a video titled Academician Lin Qun: Textbooks Make It Too Complicated – You Only Need One Example to Learn Calculus. Curious, I clicked in.

A story about learning calculus

Why was I curious? Because when I was studying calculus as an undergraduate, I noticed something: the textbooks and lecturers would launch straight into proof after proof from the very beginning, spending an entire class session deriving formulas on the board. I thought to myself, these formulas are all in the book already – what’s the point of re-deriving them in class? More importantly, all that energy and attention got swallowed up by lengthy proofs, while the motivation and reasoning behind the theorems were glossed over in a sentence or two. After a full semester, all I remembered were formulas – forget about knowing how to apply any of it flexibly in real life or at work.

I didn’t truly start to understand calculus until I took several elective courses like “The Beauty of Mathematics.” In those courses, I learned the history of mathematics. I learned how masters like Newton and Leibniz started from specific, practical problems and gradually achieved intellectual breakthroughs that eventually developed into a powerful mathematical tool.

Only then did I truly realize that the most fundamental ideas of calculus – concepts like “infinitesimal” and “infinity” – are not things people can intuitively grasp from everyday life. Those terms that textbooks and lectures breeze past in a line or two are simply impossible for a mind encountering them for the first time to immediately understand.

If you could truly, deeply understand these concepts the very first time you encountered them, with not a single doubt in your mind, then I can only say you’re more brilliant than Newton and Leibniz combined. I’m just an ordinary person, and I’d be in awe. Though I suspect there’s another possibility: you simply didn’t realize at the time that you hadn’t truly understood the concept.

Only then did I realize that feeling like some things hadn’t quite clicked was perfectly normal. What’s more, it actually showed that I was relatively perceptive – that I could tell the difference between knowledge I truly understood and knowledge I hadn’t fully grasped.

Only then did I discover that those classmates who were noisily solving extremely difficult calculus problems in class during freshman year must have studied ahead and practiced extensively beforehand. Because some of those problem-solving principles took the great mathematicians in history considerable effort to figure out – solving a genuinely new problem is not something you come up with on the spot in half a class period.

Only then did I realize that I wasn’t actually that stupid, and my classmates weren’t actually that smart. But I was stupid enough in one regard: it wasn’t until college that I figured out how to “study for exams” – that there’s a significant gap between truly learning and understanding knowledge and preparing for a test, and how to systematically prepare for exams.

A story about exams

Of course, once I figured it out, I improved pretty fast. At one point I’d put together a whole set of test-taking strategies and theories that carried me through exam after exam. This system of mine worked pretty well (I should write it down next time I’m in the mood, before I forget everything – save it for my daughter), so let me brag a bit:

• Chinese Actuarial Exam: 9 associate-level subjects, 1 fellowship-level subject
• SOA (North American) Actuarial Exam: 5 preliminary subjects, 1 advanced subject (Stochastic Processes)
• Graduate school entrance exam
• CFA: all three levels

Sure, these are just ordinary exams. But why do I think my system is decent? First, I sat for a lot of exams. Second, I always prepared in very little time – thanks to severe procrastination. Third, the results were consistently good. See? Theory guided practice, and practice delivered results. Can you blame me for bragging? Ha.

Yes, I really am this scatterbrained. I was pretty lost during college, not sure what I wanted to do. I started by casually taking the Chinese actuarial exams, and after finishing the associate level I figured I could do actuarial work, so I went ahead and finished the preliminary SOA exams too. Then, because I’d done poorly in my freshman Marxism-Leninism courses, even though my grades were excellent in later years, I still couldn’t get a direct graduate admission to my own university – so I took the entrance exam. After that I started interning. The internship made me feel actuarial work was too boring, so I moved to the investment department of an insurance company. They happened to be building a portfolio management system in VB.Net, and that completely unleashed my passion for programming. I morphed into a “coder + finance guy” hybrid, and naturally I went ahead and took the CFA too.

Off topic – this is just how I amuse myself

So why did I want to write something? Because this video raised many criticisms of textbooks that resonated with me deeply¹:

Our textbooks teach in an overly tedious way. What should have been explained through a single clear example gets no example at all – instead they teach you how to prove things, how to state theorems. It’s all hollow. In truth, proofs come from invention, and theorems come from sense. So we should first teach students the sense, teach the invention – not the theorems, not the proofs. When we’ve made the sense clear, the proof, in most cases, becomes clear on its own. … But today’s textbooks don’t teach sense – they only teach proofs. Look at current textbooks: proving a theorem like the one just shown takes an entire book, because they refuse to teach the sense and insist on teaching theorems, refuse to teach invention and insist on teaching proofs. So, in our learning, we must deeply avoid this: teaching only proofs without invention, teaching only theorems without sense.

I’ve never been a particularly confident person – I’m someone who frequently doubts himself. So for the first twenty years of my learning journey, I always suspected that the reason I couldn’t follow along or couldn’t figure things out was that I simply wasn’t as good as everyone else. But now, with years of work experience under my belt, I’ve gradually built up a little confidence. I now know that I’m someone who can learn flexibly and apply knowledge creatively, connecting ideas and concepts from all sorts of sources.

So now I can say with confidence: the knowledge that ultimately stays with you, the knowledge that can actually be applied in life, can only be the knowledge you truly understood, the knowledge that genuinely resonated with you – the kind that “seems to just open a window in your mind, springing forth from within your own soul, as though you already knew it all along.”

To close – a concept that has puzzled me since high school: the vector decomposition of forces

In my first year of high school, the physics teacher told us that forces can be decomposed using the parallelogram rule. Yes, I could solve the problems, and my physics grades were fine. But to this day, I don’t understand why the mathematical decomposition of vectors can be directly applied to the physical world. That most fundamental point – I still don’t get it.

I guess I really am too slow!