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Comment by prof-dr-ir | Hacker News Reader

I have no specific comment to make on the (first two chapters of the) book recommended by this blog. However I have become wary of people recommending a textbook because it finally helped them understand something they had tried to wrap their head around for years.

I heard a conjecture once that the best textbook you'll find on any given topic is your third. The point, of course, is that it simply takes about three serious attempts to make it click - but it is a fallacy to give all the credit to the third book.

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A reinforcing point to this is that I still have a lot of my textbooks from undergrad/grad school. And while I hadn't studied any math for 20 years or so, it was interesting when I went through some of these books -- how much easier it is to understand some of these concepts now. I don't know what it is that helped me understand these concepts, twenty years after graduating, better than I did when I was studying it everyday. But I would NOT attribute it to the textbook, given it is literally the same one.

> I don't know what it is that helped me understand these concepts, twenty years after graduating, better than I did when I was studying it everyday. But I would NOT attribute it to the textbook, given it is literally the same one.

Probably additional skills and experience you picked up all these years by solving/understanding hard problems

And more time for it to be mulled over. It was frustrating, but in college I found that the 16 week semester was rarely enough to actually learn something, only enough time to commit it to memory. It was often a year or more later that I'd have a sudden eureka moment while studying a separate topic. Perhaps something that was applying the unlearned material (like Physics I and Calculus, which were a semester apart for me since I took Chem I my first semester in college) or something related but not exactly the same (like Concrete Mathematics by Knuth et al. leading me to a realization about some aspects of calculus).

I had the experience of trying to relearn some math just a few years after first learning it in college and it was similarly much easier to learn it all a second time. I don't think I had a whole bunch of new experience. I think rather there must be some kind of residual understanding, even if it can't all be articulated cold.

Could this be a cognitive advantage of getting older?

I had the same experience. I think it's because school dumps on us solutions to problems we haven't had yet. It's difficult to correlate abstract concepts to tangible problems. But 20 years later those formulas are actually painting a picture of things we've experienced. It's like reading someone else putting perfectly into words thoughts you had. It clicks only if you had those thoughts; to other people it is devoid of meaning.

This is IMO what makes teaching well incredibly hard: you not only need to understand the topics at hand better than people who just apply it, you also need to remember the time before you understood what tou are teaching.

This is something my mentor had to come to grips with when I started, and something I've been working through the last few years as my team grows.

It is very difficult to take that step back and divorce myself from years of experience and tough lessons, and to present the subject matter in a way that can be grasped without an innate understanding that took me years to reach.

Maybe the ideas you learnt were "settling in" over the past 20 years.

Same here as well. From what I've found out is that, I know the problem space much better now, meaning, I know how some of it applies in real life and I can better visualize it now than when I was just trying to finding the right answer for grades. Maybe that's just for me but that's my reasoning anyway.

Spaced repetition, just with a very long space.

Very true. After spending lot of time looking for best books to learn algorithms and data structures, and buying more than 10 books I realised what I lacked was not the resources but rigor and discipline to pursue one of the tons of best resources. I am not telling that there aren't bad books, but most likely the limiting factor to acquire the skills isn't lack of resources, but the rigor to sit and plod through one ( or couple) of the best resources that you have zeroed on.

I've been collecting data structures and algorithms books lately. What titles have you read or own?

http://algorithms.wtf

It is free online. It starts with recursion and dynamic programming. I felt like dynamic programming really clicked for me after reading the first few chapters but this was probably the 3rd or 4th book I read on the subject.

Thanks!

" I am not telling that there aren't bad books, but most likely the limiting factor to acquire the skills isn't lack of resources, but the rigor to sit and plod through one ( or couple) of the best resources that you have zero"

Ah, but there are books, just making you want to go tp sleep by just looking at them and some are able to spark passion (in me).

My point being, it is definitely about motivation and discipline, but a good didactic book, helps with that.

And since we are all different (types of lerners), there definitely isn't one book to rule them all. And its been a while since I studied from a book, but I could usually tell from skimmimg over a few pages, of whether this book can help me, or not.

I’ve had a similar experience in my learning journey. This was really clear in High School: the physics taught by my poorly trained teachers or the “recommended” books were all targeted towards rote memorization and I really disliked the subject because of it. When I looked for other books that explained these concepts in a more accessible way, It became a joy to learn the subject.

Yes, you have a point. For example, if we consider algorithms, the grokkkng algorithms book is definitely easier to digest but maybe doesn't go as deep.

So it gets you over a hump which might make the more advanced / detailed books more accessible.

This reminds me of Hannah Fry’s TED talk on the mathematics of love, specifically point #2 on how to pick the perfect partner. [1] It was basically about not committing to the very first person you meet, but also not searching for the perfect person in perpetuity. To paraphrase, you should pick reject the first two people that come along, and then commit to the first person that is better than everyone you previously dated. It’s not a perfect similarity, but I think the point of making a good faith effort a couple times and then really going for it once you understand a little bit makes sense.

[1] https://youtu.be/yFVXsjVdvmY?t=431

Seconding this a little. The best way to learn math is to spend a long time with it and see it in as many ways as you can. You have to build an intuition for it so you can move past definitions and theorems and just "get it".

Although I also dont want to discourage anyone from trying this either. Anything that'll get people to learn is better than apathy. :)

While I agree this is definitely a thing, it would be a mistake to swing too far in the other direction and view any texts sharing a subject as roughly equivalent.

I've experienced the '3rd text' phenomenon, but I could also point to specific features affecting the wide variance in math text effectiveness.

And this book is a pretty good example of exactly that: it has specific unique features allowing it to fulfill its promise of being an effective 'translation' guide for a programmers to a bunch of otherwise typically implicit ideas relating to methods or foundational concepts in mathematics that can be extremely difficult stumbling blocks for the self-taught.

IMO a good strategy: take people's glowing praise about particular texts with a grain of salt—but, if specific beneficial features can be pointed out, which would be advantageous to you as a learner, know that can mean striking gold sometimes (in terms of not wasting time).

This is how I learned C++11. I didn't fully understand move semantics and smart pointers until I read the relevant chapter from 3 or 4 books. Whenever an inter/co-op student joined I'd have them read the specific chapters from multiple books plus a good article on smart pointers before having them touch any code. All during working hours of course.

I'm learning Rust now and feel like I need to do the same thing. Read "the rust book" plus Rust by Example plus one or two Rust books on the O'Reilly website. It takes a lot longer to read multiple books in parallel but I find I remember better this way.

This conjecture sounds like the Monad Burrito Fallacy on a broader scale.

https://www.google.com/amp/s/byorgey.wordpress.com/2009/01/1...

> I heard a conjecture once that the best textbook you'll find on any given topic is your third

Walter Rudin’s Principles of Mathematical Analysis (chapters 1 through 7). A mental torture on the first exposure, but like a fine wine when the palate is mature.

Don't they say the same about partners? It's optimal to marry the third one.

Does it mean that your third partner should have you as the their third too?

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