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How to Solve It (1945)

math.utah.edu

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Re: How to Solve It (1945)

#21

Post by jiggawatts »

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.)

Reminds me of the official Microsoft guidance for big projects like migrating Exchange to the cloud, merging an Active Directory, or whatever.

They’re all verbatim the same, except for the product name.

1. Gather requirements.

2. Devise plan.

3. Run a proof of concept

4. Run production migration

…etc.

It’s about as helpful as someone telling you to “do your job”.

Re: How to Solve It (1945)

#22

Post by furyofantares »

I remember this book as being written for teachers and was about about prompting students as you help them solve problems.

I read it as a student, and felt it was somewhat formative even though I don't think I ever explicitly applied anything it said.

Re: How to Solve It (1945)

#23

Post by vl »

ergocoder wrote: post #8

I read it and it doesn't help much. What helps with solving problems like math and algorithmic problems is to go through a lot of problems to see different patterns and strategies of solving problems. I'm talking about going through thousands of problems. That is very effective.

I read it as a kid and it was transformative.

This is one of the rare "meta" books which really help when you are not exposed to meta concepts that much yet.

Re: How to Solve It (1945)

#24

Post by CBLT »

mathisfun123 wrote: post #20
ergocoder wrote: post #8

I read it and it doesn't help much. What helps with solving problems like math and algorithmic problems is to go through a lot of problems to see different patterns and strategies of solving problems. I'm talking about going through thousands of problems. That is very effective.

> I'm talking about going through thousands of problems. you don't need thousands of problems. you don't even need hundreds, unless, no offense, your medium-term memory is very poor. personal anecdote 1: in between undergrad and grad school i decided i was gonna try this "solve all of the problems" approach, as opposed to my usual "sit there and ponder approach", in order to prepare for eventual quals in grad school.…

100% agree. I went through a similar phase of being infatuated with work ethic, and all I got were the following three lessons learned:

1. Trying to "put more effort" into artifacts (e.g. note taking) while learning something the first time is counterproductive. Write something lightweight and put your focus on paying attention.

2. Do one intensive review session while you still remember the content. Either teach someone else or write a document that could teach someone else. It's an ineffective use of time to do that more than once, for later review of the material just skim what you wrote on your first review.

3. Everyone who offers advice is just telling you their life story, dressed up as a simple directive. "Work harder" often means they wished they worked harder in some part of their life. Treat it like any other life story: learn the lessons they did, but don't let it become baggage that you irrationally follow.

Back on topic of Polya's How to Solve It, I took a no-credit uni course where the professor taught in Socratic style that expected us to (loosely) follow Polya's process. It was a great experience, and I've always been fond of Polya's steps because of it.

Re: How to Solve It (1945)

#25

Post by boppo1 »

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 might think about math. Most of the criticism I've heard about the book comes from

Paraphrased from memory: "I'm not claiming to have discovered anything special here. What I think I'm doing is describing the mental operations that lots of people perform, consciously or unconsciously, to solve different sorts of problems. You probably already do some or all of what I'm describing."

I found that he was describing some thought patterns that I am extremely familiar with because I use them so often. Making them explicit was pretty cool. I haven't gotten around to his follow-up works yet, but I look forward to them.

Re: How to Solve It (1945)

#26

Post by mrelectric »

vl wrote: post #23
ergocoder wrote: post #8

I read it and it doesn't help much. What helps with solving problems like math and algorithmic problems is to go through a lot of problems to see different patterns and strategies of solving problems. I'm talking about going through thousands of problems. That is very effective.

I read it as a kid and it was transformative. This is one of the rare "meta" books which really help when you are not exposed to meta concepts that much yet.

Can you perhaps think of any other meta books in other (or same) domain?

Re: How to Solve It (1945)

#27

Post by sfpotter »

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…

If you don't come from a math-y background and are trying to "get into serious mathematics", the only way to do this is to get a PhD.

I realize this might come across as gatekeeping, but the reality is that each subfield of math (or any scientific discipline) has its own set of tactics for approaching problems, which have been developed over the years by the people actually in the trenches. These problem solving strategies and habits of mind aren't written up anywhere, and even if they were, they wouldn't be useful to read. The way you learn them is by getting the training that comes with a PhD: they are passed down as part of the mentoring process. It's not clear to me that it could happen any other way.

I think the right way to look at this is that someone getting a PhD is basically a "journeyman researcher", much like a journeyman in one of the trades. Unfortunately, this often goes sideways (particularly if an advisor is bad or if there aren't enough support systems). But much like the only way to become an electrician is to apprentice yourself, the same goes for becoming a researcher. I think this is for good reason.

Re: How to Solve It (1945)

#28

Post by OldGuyInTheClub »

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 progress. Some people can do this in math, some in other technical disciplines, others in the arts, and so forth. Umpteen years later, I have not seen anyone including Polya successfully /teach/ this ability. If you want to be good at math, piano, chess, sculpture, or whatever you need the talent and then it can be nurtured.

In college I took organic chemistry[0]. The exam problems would often have redundant and/or contradictory information. The student had to figure out what to NOT believe to get anywhere.

As others have mentioned, none of Polya matters for selecting and attacking research problems.

[0] It was taught well and none of the cliches of "It's all memorization" applied. I struggled but look back fondly on it because I had to think along many axes at the same time.

Re: How to Solve It (1945)

#30

Post by mathisfun123 »

CBLT wrote: post #24
mathisfun123 wrote: post #20

Earlier quoted context omitted.

> I'm talking about going through thousands of problems. you don't need thousands of problems. you don't even need hundreds, unless, no offense, your medium-term memory is very poor. personal anecdote 1: in between undergrad and grad school i decided i was gonna try this "solve all of the problems" approach, as opposed to my usual "sit there and ponder approach", in order to prepare for eventual quals in grad school.…

100% agree. I went through a similar phase of being infatuated with work ethic, and all I got were the following three lessons learned: 1. Trying to "put more effort" into artifacts (e.g. note taking) while learning something the first time is counterproductive. Write something lightweight and put your focus on paying attention. 2. Do one intensive review session while you still remember the content. Either teach som…

> Everyone who offers advice is just telling you their life story, dressed up as a simple directive. "Work harder" often means they wished they worked harder in some part of their life. Treat it like any other life story: learn the lessons they did, but don't let it become baggage that you irrationally follow.

I don't know how old you are but bravo - it took me a very very very long time to realize this. The converse to this is to always be very transparent and very careful when giving novices advice - when junior PhD students ask me about some kind success I've had I always make sure to emphasize the ways in which I got lucky first and only after talk about how I applied myself.

Edit: and in full transparency, I believe i had this epiphany after at some point listening to the funny poem/song/thing (from the 90s):

Everybody’s Free (to Wear SUNSCREEN): THE SUNSCREEN SONG (Class of ’99)

https://youtu.be/sTJ7AzBIJoI

which has a lyric

>Be careful whose advice you buy but be patient with those who supply it

>Advice is a form of nostalgia, dispensing it is a way of fishing the past

>From the disposal, wiping it off, painting over the ugly parts

>And recycling it for more than it's worth

I'm not one for sentimentality but that nicely captures it.

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