Now I place the concrete over everything else; theory is nice when it can illuminate why the practice works. Otherwise, it’s just words.

The most frustrating is when I have friends who have derived their entire understanding of a subject I know as a practitioner (typically something tech/programming related) from watching YouTube videos/listening to podcasts.

Because they’ve heard hours and hours from experts, they have a feeling of deep understanding. But talking to them about this topic is extremely frustrating because their knowledge clearly has never had to be applied to the real world, and is grounded in nothingness, so they misunderstand lots, but they feel like they know what they’re talking about as much as you do.

> Now I place the concrete over everything else; theory is nice when it can illuminate why the practice works. Otherwise, it’s just words.

That mirrors the trajectory of all humankind, doesn’t it? From lofty Platonic ideals to nitty gritty empiricism and experimentation.

You reference Plato, but you could easily phrase it in the other direction. "From Roman engineering to Wittgenstein blithering on about meaning."

> Because they’ve heard hours and hours from experts, they have a feeling of deep understanding. But talking to them about this topic is extremely frustrating because their knowledge clearly has never had to be applied to the real world, and is grounded in nothingness, so they misunderstand lots, but they feel like they know what they’re talking about as much as you do.

You just gave the most accurate description of HN userbase in two paragraphs.

I would also deemphasize the more mathy parts of calculus - do you really need a deep dive into continuity or the fundamental theorem of calculus? Eventually, yes. But it's just like programming: you're not going to need to understand language theory or ADTs or category theory or lambda calculus to write your first program. Or your second. And, IMHO, you should only reach for this understanding when you realize you need it. Otherwise, it won't integrate well into your toolkit.

  > If you present the hard problem first, the student may flail around and realize: I need something to help with this! 

As I've grabbed my horse and lance and rushed at this windmill from assorted directions, I quickly run into my limitations that prevent the problem being solved. I run into vocabulary problems with the math, the fact that I simply don't have the math to approach the problem (which appears to be some vector calculus -- I think. "No, you idiot, it's XYZ instead", but I don't know enough to know that it's not vector calculus, if, indeed, it isn't). I try to apply basic kinematics to the problem, but I don't know if that's enough. And, finally, it could be all of those things plus, oh, some optimization issues and, also, would you like to be introduced to the several different techniques for computing numeric integration and the differential equation solvers?

"Eeep!"

To quote the film "Addams Family Values":

  Wednesday: Pugsley, the baby weighs 10 pounds, the cannonball weighs 20 pounds. Which will hit the stone walkway first?
  Pugsley: I'm still on fractions.

The videos that it was essentially transcribed from are great tho.

If it exists, you could probably replace most of the first book with just a really good explanation of the Lagrangian, with lots of examples, I think.

Yes.

> If you present the hard problem first, the student may flail around and realize: I need something to help with this! Now that they know they need it, and why, you can give them the tool that fits the bill.

No.

If an instructor deliberately gives a student a problem that they know the student _cannot_ solve, then it rightfully destroys trust.

I never taught at the university level, but with middle and high school math students I taught them to how (re)discover the solutions, rather than teaching them the solutions directly.

As a practical matter, many of my college classes went too quickly to do anything _but_ teach the solution -- or tell us to learn it between classes and bring questions back.

This does not destroy trust, but gives the student an important lesson: we only have the techniques to solve, say, 0.0000001% of the problems. So you have to learn brutally hard for the next many years (or rather decades) to have the minimal qualifications to be able to invent whole new techniques that no person has ever come up in history before to increase this ratio from, say,to 0.000000100000000001% (even this would trigger a whole new aera in the history of science).

  > As a practical matter, many of my college classes went too quickly to do anything _but_ teach the solution

I dropped that class. I always felt it was a mistake not taking calculus in High School. I had a very good relationship with the math teacher there, and we could have done it, to some level, casually between classes. I just didn't take him up on it.

I've never learned calculus.

Issue is that often times people don't know that a certain tool exists, so they re-invent a 100x worse version and just hack something.

There is no substitute for solving problems.

Some anecdotal evidence for this:

https://www.dwarkeshpatel.com/p/dario-amodei#details

> Dario Amodei: We have generally found that if we hire someone who is a Physics PhD or something, that they can learn ML and contribute just very quickly in most cases.

In math/physics, it often won't. Solving lots of problems serves two particular purposes: To really solidify the concepts in your mind so you won't forget, and ensuring you learn the techniques and not just the knowledge.

For the former, you may find yourself in the position where you find yourself way over your head, and won't know where to start. You usually will not have a single gap, but many. You'll find yourself realizing you'll need to look up material from several textbooks to regain the knowledge you've lost. Once you begin that process, you'll pick up one of your old textbooks and while the physics knowledge may be absorbed, you'll realize you've forgotten much of the math needed to solve such problems. In the unlikely event you'll retain enough to follow the textbook, it is very unlikely you'll know the techniques well enough to solve the real world problem.

And your colleagues will. You'll be alone, and you'll drop out of that group. With physics/math, there often are hard boundaries in these groups. Those who meet the bar are in. Those who don't drop out, because it really sucks being the only person in the group who is struggling with what everyone else considers as basic.

SW engineering has a much more gradual change in skills amongst people, and usually the problems most people work on are fairly learnable in a short amount of time.

Math and its applications are a contact sport. You don’t truly appreciate it until you try to use it yourself.

I'm going to ruminate on this.

But you will spend the rest of your life arguing with people who insist that there must be a ("quick and dirty") substitute and you're responsible for finding it.

Be it microservices, coronavirus or taxation.

I emphasize this story to my kids because knowing isn’t important because everyone eventually figure it out. It’s the ones who can do the problems and get good marks that succeed in the end.

Then again, in the process of teaching I always found myself teaching people to work problems, which required me to be able to work the problems myself. In a way, it's kind of impressive you managed to avoid doing that.

The gap between knowers and doers will only get larger as math explainers improve their content.

Nowadays it's a lot easier when there are so many free materials from top school online. And stack exchange and reddit is available almost 24-7 if one ever has a question.

Many textbooks still employ the deplorable practice of not presenting the answer to all exercises at the end, unfortunately.

The reason is that you think you understand what you read, but as Richard Feynman said:

> The first principle is that you must not fool yourself, and you are the easiest person to fool.

You think you understand 90% of what you read, but in reality it's probably only 20-30%. By doing the exercises, at the very least you'll know that you don't know that much. And if you then reread the materials a few pages before, you'll realize that you have skimmed (or worse, skipped) some parts because you mistakenly thought you already understood it.

Another tips from my personal experience: When you're reading a textbook, keep asking in your mind questions with the types of "what if" and "how about," which are sometimes not yet explained in the section you're reading. Also, keep associating what you've recently learned with what you've already known (days ago, years ago).

Be curious and validate that you really understand what you think you understand.