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So you want to study mathematics

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Re: So you want to study mathematics

#211

Post by jackthetab »

brimble wrote: post #31

Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake.…

YES! This is exactly my complaint about math. I find the abstract proof-oriented math kinda interesting (although How to Prove It is an exception that I discuss below) but I _really_ want practical real-word applications of these maths.

Even though I've learned linear algebra decades ago, Andrew Ng's example of using a matrix to encode 5,000 images then doing linear algebra on it blew my mind. I've since used that perspective in many other fields. Not once have I applied a proof to solve a programming problem.

I've thought of publishing, i.e.blogging, examples that I've come across but that would just be a mish-mash of stuff I've read elsewhere with no overarching theme/framework. Besides, someone else must have done this, no?

EDITED: Used the correct book title.

Re: So you want to study mathematics

#212

Post by Tomis02 »

jackallis wrote: post #37
brimble wrote: post #31

Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake.…

i second this. there should clear distinction between academic math and "real world usage" math.

Mathematics is a social activity, in the sense that mathematicians create proofs to convince others that their findings are valid. Some proofs are easier to verify than others. Some proofs are "believed to be true" because no one has yet managed to prove them wrong yet.

So if you think about "real world usage", you can either use the math results and implicitly trust that they "just work", or you can dive a bit deeper, see if you agree with those results, or at least gain some insights from the proofs that have been presented.

And just to be clear, there is little to no "real world usage" math without academic math.

Re: So you want to study mathematics

#213

Post by brimble »

jackthetab wrote: post #211
brimble wrote: post #31

Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake.…

YES! This is exactly my complaint about math. I find the abstract proof-oriented math kinda interesting (although How to Prove It is an exception that I discuss below) but I _really_ want practical real-word applications of these maths. Even though I've learned linear algebra decades ago, Andrew Ng's example of using a matrix to encode 5,000 images then doing linear algebra on it blew my mind. I've since used that pe…

> Not once have I applied a proof to solve a programming problem.

There are people in this very thread insisting that proofs are extremely useful in programming. I dunno if I just picked up the same skills elsewhere (Logic? Philosophy? Just... IDK, thinking and developing an absolute shitload of heuristics through years of experience?) or am entirely missing out and in fact don't have a clue how to program, but I don't see it (outside some rare niches where it probably is useful—coq exists, after all).

Sure, the word "exhaustive" can apply both to accounting for all (reasonably) possible problems in a block of code, and also to proofs, but the former doesn't feel at all like working on proofs, to me, to pick just one example (and some posts have seemed to imply that accounting for e.g. edge cases is exactly one case in which experience with proofs come in handy, but man, they feel like very different and barely-related activities to me).

Re: So you want to study mathematics

#214

Post by jackthetab »

jeffreyrogers wrote: post #79
tzs wrote: post #26

Overall a pretty decent list, although I would suggest considering some tweaks. For real analysis it recommends as essential Abbott's "Understanding Analysis" and Rudin's "Principles of Mathematical Analysis". If you "haven't gotten your fill of real analysis" from those it recommends Spivak's "Calculus". I'd consider promoting Spivak to essential, but using it for calculus rather than real analysis, replacing their…

I'll second this. "How to Prove It" gets recommended a lot, but I couldn't get through it. I found it terribly boring and unmotivated. Some people can power through dry material but I'm not one of them. I found it much easier to learn to write proofs when they were related to topics I was interested in.

I loved How to Prove It. Not for the proofs - which are interesting in a gazing-at-your-navel kinda way - but rather all the little _practical_ tidbits. "So THAT'S what a partially ordered set looks life in real life!"

And the last(?) chapter where he uses induction to determine how to place an L-shaped figure on a grid...I never knew how to even approach that kinda' problem.

So yeah, I want actual practical applications ("exercises" != "applications") for math.

Re: So you want to study mathematics

#215

Post by tmabraham »

I love that the author highlighted Prof. Robert Ghrist's great material. I took his calculus classes through Coursera maybe about 8-9 years ago. He just makes everything exciting and his visuals are just beautiful. For example, within the first few lectures, he made it feel like Taylor series was like the coolest thing ever. Highly recommend checking his lectures out. Check out his website: https://www2.math.upenn.edu/~ghrist/

Re: So you want to study mathematics

#216

Post by rotifer »

To the section "Popular Math Books" I would add almost anything by Julian Havil. John D. Cook referred to him as a writer of "serious recreational mathematics" [1]. I would probably put his "Nonplussed!" and "Impossible?" books in the "Level: Easy" group, with the others at least in "Level: Medium". "Gamma" is one of my favorite serious recreational math books.

If you like getting into the nitty-gritty of problem solving then check out the books of Paul Nahin. They vary between "Level: Medium" and "Level: Difficult", with many of them reveling in the solution of equations, and integrals in particular. Although he recognizes the need for proofs, he makes a point of avoiding them in his books.

[1] https://www.johndcook.com/blog/2019/09/29/a-sort-of-mathemat...

Re: So you want to study mathematics

#217

Post by Jhsto »

brimble wrote: post #213
jackthetab wrote: post #211

Earlier quoted context omitted.

YES! This is exactly my complaint about math. I find the abstract proof-oriented math kinda interesting (although How to Prove It is an exception that I discuss below) but I _really_ want practical real-word applications of these maths. Even though I've learned linear algebra decades ago, Andrew Ng's example of using a matrix to encode 5,000 images then doing linear algebra on it blew my mind. I've since used that pe…

> Not once have I applied a proof to solve a programming problem. There are people in this very thread insisting that proofs are extremely useful in programming. I dunno if I just picked up the same skills elsewhere (Logic? Philosophy? Just... IDK, thinking and developing an absolute shitload of heuristics through years of experience?) or am entirely missing out and in fact don't have a clue how to program, but I don…

In a way, writing types is proof-esque. It establishes some basic correctness guarantees regardless of the programming language. Some concepts like refinement types make it seem more "mathy", by requiring quantifiers within programs: https://ucsd-progsys.github.io/liquidhaskell/blogposts/2019-...

Someone without any background in the subject would probably find Dafny interesting.

Re: So you want to study mathematics

#218

Post by jackthetab »

RoddaWallPro wrote: post #86

I read a Murakami novel in high school, 1Q84. The protagonist is a math teacher who talked about math in a way that I had never seen before. I'd been told I was "good at math" beforehand(for whatever that means, I'm not a fields medalist or anything), but for ~6 months after reading that book, I was _really good_. Like, suddenly I did not have to do any homework in my sr. year calculus class. I loved sitting in class…

C'mon! With a build-up like that and you don't mention the name of the book?!

Don't leave us hanging, dude!

Re: So you want to study mathematics

#219

Post by [deleted] »

bee_rider wrote: post #195
SamReidHughes wrote: post #155

Earlier quoted context omitted.

Well, AP statistics covered the area under a probability distribution function and people seemed to understand that -- you look up the answer in a table (or use the TI-83 function). Presumably they'd do the same for a cumulative distribution function.

Highschool is pretty early to teach people that math is too complicated for them to actually evaluate their expressions.

[deleted]

Re: So you want to study mathematics

#220

Post by smugma »

joatmon-snoo wrote: post #49
brimble wrote: post #31

Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake.…

My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs.…

A few years ago USC’s MBA program replaced statistics with data science, which is further down the pragmatic chain.

Stats is not a generally useful skill (the concepts are, but can be taught in a data science course) but understanding how to work with data is.

Discrete math seemed to be the most applicable math class to my CS curriculum. I never took graphics so didn’t use linear algebra, and definitely never touched anything related to DiffEq.

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