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How to Solve It (1945)

math.utah.edu

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Re: How to Solve It (1945)

#61

Post by ants_everywhere »

OldGuyInTheClub wrote: post #28
boppo1 wrote: post #25

This summary really undersells the book IMO. It's one of the more interesting books I've read in that it is not structured linearly. He introduces a few ideas which have very particular language he defines in the remainder of the book which is essentially a dictionary. If you don't come from a math-y background and you are trying to get into serious mathematics, this definitely helps to 'lift the veil' on how you mig…

"How To Solve It" does not account for actual ability. It was recommended in high school to those of us interested in math contests. My big lesson was that if you need this book, you're going to be roadkill in math contests. If you have the chops to be a successful mathlete, you don't need this book. And so it came to pass. There are some people who have the ability to see a problem well enough to analyze it and make…

> if you need this book, you're going to be roadkill in math contests

> I have not seen anyone including Polya successfully /teach/ this ability.

This is a pretty wild take. For alternate takes, see John Horton Conway's foreword to the most recent edition of the book. There he talks about how amazing the book is for both students and teachers, including the things he learned from it as both a student and teacher. Or see Terry Tao's blog post on solving mathematical problems [0], in which he says he learned from the Polya book when preparing for Mathematics Olympiads. Conway and Tao are widely considered outside the "roadkill" category of mathematicians.

> If you want to be good at math, piano, chess, sculpture, or whatever you need the talent and then it can be nurtured.

The "either you got it or you don't" theory used to be common wisdom, but it's not supported by empirical evidence and these days it's mostly relegated to grouchy coach stock character stereotypes. You need a sustained level of interest, you need practice, and you need some amount of courage, but there's no real evidence for an innate ability that some people got and some people don't.

[0] https://terrytao.wordpress.com/career-advice/solving-mathema...

Re: How to Solve It (1945)

#62

Post by jodrellblank »

js2 wrote: post #6

Ah yes, just four steps: 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Look back. --- Compare with the Fenyman Algorithm: 1. Write down the problem. 2. Think real hard. 3. Write down the solution. https://wiki.c2.com/?FeynmanAlgorithm (The discussion on FeynmanAlgorithm links back to Polya's book since not everyone is Feynman.)

You say 'ah yes' as if there's no value in these steps. Compare with more common steps:

1. Don't try to understand the problem, just stare at the given information blankly.

2. Instead of devising a plan or thinking real hard, try to "guess the teacher's password" and parrot any relevant sounding words for this ... or any related thing - https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t... and the linked https://www.lesswrong.com/posts/48WeP7oTec3kBEada/two-more-t... - as if the universe will accept the words (map) as the solution (territory).

3. Carry out incantations handed down from above, regardless of whether they could be applicable to this problem; they are not understood which leaves room for them being potentially powerful and relevant.

4. Give up. If the incantations of power didn't help, what else is there to do? Don't look back.

Are these steps from How To Solve It in 1945 so different from Eric Raymond's "how to ask smart questions" from the 1990s(?) or from Eric Lippert's "How to debug small programs" from 2014 ( https://ericlippert.com/2014/03/05/how-to-debug-small-progra... )? His example "I wrote this program for my assignment and it doesn’t work." is still a common approach to "problem solving" nearly a decade later, not just for student coders either.

People seem to be missing:

- the belief that problems are solvable. (Instead apparently believing that some people know the answer or that the problem cannot be solved; not understanding that someone can start unknowing, work to solve, and then know).

- that solving something involves a mental model of how a thing works, in some context. (Instead apparently believing that solving is random or guesswork or luck or flash of inspiration, but not solved by any kind of knowable process).

- that solving something involves a feedback loop of gathering information, testing hypotheses, learning from the result, looking back (Instead trying one thing and stopping, not seeing if anything could be learned from the result, not trying to expand a mental model from what was known).

Even that this is a pet frustration of mine with some of my coworkers and internet programming askers, I still resist writing down the problem or thinking real hard (as far as I can do such a thing).

Re: How to Solve It (1945)

#63

Post by OldGuyInTheClub »

ants_everywhere wrote: post #61
OldGuyInTheClub wrote: post #28

Earlier quoted context omitted.

"How To Solve It" does not account for actual ability. It was recommended in high school to those of us interested in math contests. My big lesson was that if you need this book, you're going to be roadkill in math contests. If you have the chops to be a successful mathlete, you don't need this book. And so it came to pass. There are some people who have the ability to see a problem well enough to analyze it and make…

> if you need this book, you're going to be roadkill in math contests > I have not seen anyone including Polya successfully /teach/ this ability. This is a pretty wild take. For alternate takes, see John Horton Conway's foreword to the most recent edition of the book. There he talks about how amazing the book is for both students and teachers, including the things he learned from it as both a student and teacher. Or…

> Conway and Tao are widely considered outside the "roadkill" category of mathematicians.

Ah, but they didn't really /need/ Polya's book.

Re: How to Solve It (1945)

#65

Post by scns »

ants_everywhere wrote: post #61
OldGuyInTheClub wrote: post #28

Earlier quoted context omitted.

"How To Solve It" does not account for actual ability. It was recommended in high school to those of us interested in math contests. My big lesson was that if you need this book, you're going to be roadkill in math contests. If you have the chops to be a successful mathlete, you don't need this book. And so it came to pass. There are some people who have the ability to see a problem well enough to analyze it and make…

> if you need this book, you're going to be roadkill in math contests > I have not seen anyone including Polya successfully /teach/ this ability. This is a pretty wild take. For alternate takes, see John Horton Conway's foreword to the most recent edition of the book. There he talks about how amazing the book is for both students and teachers, including the things he learned from it as both a student and teacher. Or…

"Genius is 1 percent inspiration and 99 percent perspiration."

https://quoteinvestigator.com/2012/12/14/genius-ratio/

Re: How to Solve It (1945)

#66

Post by raincole »

hgsgm wrote: post #40
steno132 wrote: post #35

The best people I work with don't use this stuff. They solve problems based on pattern matching. Having read hundreds of books and papers over the years, no problem seems new to them. They can rapidly find a solution, especially compared to someone trying to derive a answer from first principles like Polya suggests.

If what you say is true, why did it take so long to prove Fermat's Last Theorem? Polya's book is distilling hard fought PhD wisdom into a book children can understand. It's for solving novel hard problems, not trivial ones.

The best people the GP worked with are probably not capable to solve Fermat's Last Theorem at the first place...

Re: How to Solve It (1945)

#68

Post by travisjungroth »

OldGuyInTheClub wrote: post #63
ants_everywhere wrote: post #61

Earlier quoted context omitted.

> if you need this book, you're going to be roadkill in math contests > I have not seen anyone including Polya successfully /teach/ this ability. This is a pretty wild take. For alternate takes, see John Horton Conway's foreword to the most recent edition of the book. There he talks about how amazing the book is for both students and teachers, including the things he learned from it as both a student and teacher. Or…

> Conway and Tao are widely considered outside the "roadkill" category of mathematicians. Ah, but they didn't really /need/ Polya's book.

This “need” standard doesn’t seem useful. You could say they don’t need anything besides air, water, food and shelter.

Re: How to Solve It (1945)

#69

Post by bigbluesax »

OldGuyInTheClub wrote: post #28
boppo1 wrote: post #25

This summary really undersells the book IMO. It's one of the more interesting books I've read in that it is not structured linearly. He introduces a few ideas which have very particular language he defines in the remainder of the book which is essentially a dictionary. If you don't come from a math-y background and you are trying to get into serious mathematics, this definitely helps to 'lift the veil' on how you mig…

"How To Solve It" does not account for actual ability. It was recommended in high school to those of us interested in math contests. My big lesson was that if you need this book, you're going to be roadkill in math contests. If you have the chops to be a successful mathlete, you don't need this book. And so it came to pass. There are some people who have the ability to see a problem well enough to analyze it and make…

>If you want to be good at math, piano, chess, sculpture, or whatever you need the talent and then it can be nurtured.

No offense, but you sound like you don't really know anything about these disciplines, or about the science of learning (Peak is a good book to read). Getting a good piano teacher will absolutely accelerate your progress and being talented at piano as a kid is a pretty bad predictor of how good you'll be as an adult. Every outstanding Piano "talent" you see on stage today has been nurtured and nurtured themselves through practice, to hell and back.

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