So you want to study mathematics • Hacker News Board

Re: So you want to study mathematics

#181

Post by antishatter » Mon, Mar 07, 2022, 11:29 PM UTC

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.…

Statistics converges to calculus at its core. Source: am statistician

Re: So you want to study mathematics

#182

Post by antishatter » Mon, Mar 07, 2022, 11:31 PM UTC

zozbot234 wrote: post #150

Melatonic wrote: post #95

Earlier quoted context omitted.

Not sure I agree with that - you also have to take into account that calculus is a requirement for many other sciences. I suppose you could just of course make it a pre-req for things like that but I found that in highschool rarely did they go that deep. Physics becomes a hell of a lot easier with basic calculus for example. If anything should be dropped from a highschool level its all of the insane memorization you…

Calculus is also a requirement for statistics and probability. You simply cannot meaningfully talk about the latter without referencing the former.

I am a statistician and this guy gets it. Stats converges to calc at its core. The real issue is stats education is bad.

Re: So you want to study mathematics

#183

Post by antishatter » Mon, Mar 07, 2022, 11:32 PM UTC

ghaff wrote: post #161

jll29 wrote: post #62

Earlier quoted context omitted.

Calculus and linear algebra seem to _totally_ dominate the curriculum in most (all?) countrie. What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO. And I fully agree, in-depth knowledge of probability theory as well as descr…

Interestingly the ascent of linear algebra is a relatively recent thing. I have an engineering degree but while I was certainly exposed to basic matrix stuff, never took a linear algebra class. When I was an undergrad, you took differential equations in addition to basic calculus for engineering. (You needed for system dynamics among other things but it was very cookbook.) Linear algebra became a lot more interesting…

Computers made it insanely useful where before it was an interesting view of systematic math.

Re: So you want to study mathematics

#184

Post by susanrigetti » Mon, Mar 07, 2022, 11:33 PM UTC

mrjangles wrote: post #178

Py-o7 wrote: post #17

This felt like it was written by a physicist or engineer. Too much emphasis on differential equations and not enough on things like topology, functional analysis and/or non-introductory parts of algebra like say representation theory.

Yep, I came here to say this. I think it is important that anyone who wants to study math understand that real math is not at all like what you learn in a physics or engineering department. In these departments you will always hear people say things like >"proofs are not useful, all you have to do is memorize the 'trick' they use. Once you know which trick to use, it is easy" or you will hear them say. >"Math isn't a…

For what it’s worth, the curriculum in this guide is modeled after the math major maps of many universities, including the one I attended (Penn). I would be curious to know what part of an undergraduate math curriculum will lead people very far astray…

Re: So you want to study mathematics

#185

Post by mjfl » Mon, Mar 07, 2022, 11:40 PM UTC

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.…

The list presented here, calculus, odes + pdes, linear algebra, is essentially that, mathematics for people who just want to use it. It's all undergrad level. There's several layers on top of this - set theory, rigorous probability theory, algebraic geomeetry, topology, that are less useful but interesting to mathematicians.

Re: So you want to study mathematics

#186

Post by k__ » Mon, Mar 07, 2022, 11:41 PM UTC

auggierose wrote: post #135

Koshkin wrote: post #82

Earlier quoted context omitted.

I think one can get far enough by applying known mathematical facts. (Proof: elementary school math is useful.)

The problem is, usually you need to combine known mathematical facts to solve a problem. Now, how do you know that you combined them properly? Yup. You need to know what a proof is.

This.

Applying mathematical facts isn't math, it's calculation at best.

That's what most programmers already do. They throw things together in the hope they work without proof they got all edge cases covered.

And it works pretty good, because often the edge cases aren't that bad.

Re: So you want to study mathematics

#187

Post by jltsiren » Mon, Mar 07, 2022, 11:47 PM UTC

eftychis wrote: post #75

jll29 wrote: post #62

Earlier quoted context omitted.

Calculus and linear algebra seem to _totally_ dominate the curriculum in most (all?) countrie. What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO. And I fully agree, in-depth knowledge of probability theory as well as descr…

Anything besides a cursory layperson's outlook on a lot of these topics (besides basic logic and history/philosophy of math -- although not sure how you would teach the last two/what you have in mind as curriculum) requires calculus and/or linear algebra. There is a reason they say you can never learn too much linear algebra. And yes probability and statistics are fundamental. I was shocked a bit when I learned it wa…

In my experience, mathematics can be roughly divided in two: topics that use real numbers all the time, and topics that rarely need them. In the former, almost everything builds on calculus and/or linear algebra. In the latter, they are just two topics among many others.

The true foundational classes in the typical undergraduate mathematics curriculum are logic and abstract algebra. People rarely start with them, because the usual way of teaching mathematics is applications before foundations. You learn linear algebra before abstract algebra, proofs before formal logic, and axiomatic probability before measure theory.

And there is definitely such thing as too much linear algebra. Once upon a time, I wanted a decent mathematical background for theoretical CS and continued (at least) until the first graduate-level class in most major topics. Graduate linear algebra was "foundations without applications" for me, as I've never worked on anything building on it.

Re: So you want to study mathematics

#188

Post by com2kid » Mon, Mar 07, 2022, 11:52 PM UTC

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.…

The ideas behind Calculus are much more useful IMHO. The idea of rate of change, or change over time, converging and diverging series, these all have practical application to the real world.

And the basics of physics makes next to no sense without calculus, and even parts of intro chemistry make more sense after having taken calculus.

I remember in one economics class, which didn't have calc as a pre-req, the prof said "alright, for those of you who have taken calculus, this is an integral, you can now leave the classroom and come back tomorrow. Everyone else has to stay."

The ideas of calculus just seem so fundamental to me. It is sad that the American schooling system is so slow, and expectations so low, that it isn't taught to everyone. Meanwhile in other countries, everyone, artists to engineers, learns calculus.

Re: So you want to study mathematics

#189

Post by Koshkin » Mon, Mar 07, 2022, 11:52 PM UTC

criddell wrote: post #53

ghufran_syed wrote: post #16

I went from only having done high school math 10 years ago to completing an MS in math and statistics at my local state university while working in an unrelated field. I would recommend NOT starting with calculus if you haven’t done it, instead, just learn how to do proofs - I used Chartrand “Mathematical proofs” - You don’t need to know any math beyond algebra in order to do that most of this book. If you need to re…

I looked up the Chartrand Mathematical Proofs book and it's been a while since I had to buy a textbook, but $175 for hardcover and $75 for paperback or ebook? That's nuts. If I were a student today, I'd pirate that and feel absolutely no remorse for doing so.

Maybe, but this is not so bad, if you realize that the time and the effort you would spend on learning the material from such textbook is in some ways incomparable with the price of a couple of dinners at a restaurant.

Re: So you want to study mathematics

#190

Post by narrator » Mon, Mar 07, 2022, 11:53 PM UTC

macrolime wrote: post #162

narrator wrote: post #128

Earlier quoted context omitted.

If you just want to learn math for purely practical reasons, Khan Academy ( http://www.khanacademy.org ) is great. It might have added more lessons, but when I went through it, it went up through 1st year college calculus. The thing that makes it VASTLY better than most self study math programs or books is that there are hundreds of exercises that you can do, and see if you got the right answer. If you didn't, it wil…

Where are there exercises? I just tried clicking on some linear algebra topic, but it seems there are just videos https://www.khanacademy.org/math/linear-algebra/alternate-ba...

They don't have exercise for some of the more advanced college level topics. You can see the exercises on the right side of the screen under "practice" for some of the more introductory topics:

https://www.khanacademy.org/math/linear-algebra/vectors-and-...