So you want to study mathematics • Hacker News Board

Re: So you want to study mathematics

#321

Post by caddemon » Tue, Mar 08, 2022, 1:41 PM UTC

dwohnitmok wrote: post #80

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…

> meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? Honestly all of those feel more niche than calculus. I agree with you and joatmon-snoo on the usefulness of statistics and would probably support bumping calculus in favor of statistics, but meta-mathematics, topology, logic (which bleeds into meta-mathematics), co…

I think the problem is that students don't know these fields exist unless they took an especially strong liking to the standard school math curriculum. Going into college I couldn't have told you what most of the fields mentioned were about even broadly. Truly I had no idea what math was, because we barely talked about proofs through high school. The only reason I swung back around to learning some of it is because my upper level college CS classes introduced me to ideas that then made me want to revisit my math curriculum.

There ought to be some way to encourage late high school/early college students to "survey" the field without necessarily taking full courses in these topics. This could also give some earlier understanding in how the different fields relate - for example you could present a toy graph theory problem within linear algebra as a matrix problem, later presenting the same problem in graph theory section and walk through it using graph representation. I think high school courses struggle with memorability precisely because most units are taught basically in a vacuum.

Regardless though, the point of the survey course wouldn't be to remember details so much as to find topics of interest for further study/be generally aware of their existence in case a relevant problem comes up in the future.

Re: So you want to study mathematics

#322

Post by Trinotom » Tue, Mar 08, 2022, 1:50 PM UTC

torbTurret wrote: post #289

geebee wrote: post #151

Earlier quoted context omitted.

I'm really wary of this. House wiring would be more useful than Shakespeare. That's not to knock house wiring, or statistics. I'd love to know more about both. But don't deny yourself an understanding of the meaning of limits. Almost all mathematics before calculus leaves you with a misimpression that neat formulas exist to solve problems. In reality, you've learned to draw straight lines with a ruler, and maybe a fe…

House wiring and Shakespeare are completely different subjects. (I’d expect someone professing Shakespeare to avoid such a straw man, but I see your point!) Most people don’t need mathematical statistics, but far more will find meaning/interest in the immediate applications of lighter statistics than the another-math-class-full-of-equations that calculus feels like to many. Anecdotally, most universities are scrambli…

We dont need proofs, calculus, reasons, process, principles and theorems actually. Just results and facts are in demand for our lives. But it may make us subordinate to people who have invented theories of statistics or calculus, because we have no choice but refer to and based on their theories, I mean, depend on them even if the theories are absolutely correct or truth of the universe.

Re: So you want to study mathematics

#323

Post by caddemon » Tue, Mar 08, 2022, 1:51 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…

I think a layperson's outlook on these topics in high school could be very interesting though. Use this outlook to help motivate learning calculus and linear algebra. With a mix of engineering survey and math survey there are plenty of kids out there that would find parts of the course that stimulate their natural interest. Then the "calculus has many practical applications" thing would feel way less contrived. I mean of course I know now that it is true, but in high school I didn't feel it because I didn't have enough broader context.

Re: So you want to study mathematics

#324

Post by wizzwizz4 » Tue, Mar 08, 2022, 1:59 PM UTC

ChrisLomont wrote: post #305

wizzwizz4 wrote: post #140

Earlier quoted context omitted.

Is that such a bad thing? I'm half-convinced that the “rank of a matrix” is just an artefact of a particular algorithm for inverting matrices. And distributions aren't everything; I can look up any distribution I want on Wikipedia, just as soon as I need it, but a proper foundation in what statistics means is much harder to come by. (I have enough of a foundation to know when it's being taught very wrong, but not eno…

>I'm half-convinced that the “rank of a matrix” is just an artefact ... The rank of a matrix is fundamental to understanding linear transformations, since it given you knowledge about the dimensions of the "output space". It becomes more fundamental if the person goes on to study deeper math. It tells you how to compute the size of a basis for the target space. The uses go on and on. >And distributions aren't everyth…

> The rank of a matrix is fundamental to understanding linear transformations, since it given you knowledge about the dimensions of the "output space".

That's more to do with the space of the coefficients, I think.

> Saying one can look fundamental stuff up elsewhere means the book is lacking.

A list of distributions is not fundamental to statistics. Discovering a new distribution doesn't meaningfully expand the fundamentals of the field of statistics: all the basic principles are the same. Anyone familiar with those basic principles can pick up a new distribution quite easily – and can probably derive new distributions when they're needed.

Re: So you want to study mathematics

#325

Post by caddemon » Tue, Mar 08, 2022, 2:01 PM UTC

bee_rider wrote: post #200

joatmon-snoo wrote: post #49

Earlier quoted context omitted.

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.…

I think I'd rather we lose geometry than calculus. Geometry made more sense as an intro to the idea of 'proving stuff' when everybody's dad had a workshop, compass, etc.

Agreed, also the way my high school taught it made proofs seem like a boring subset of Geometry. The unit was poorly connected to other units and the problems I found by far the easiest in the course as we got very formulaic ones. It's entirely possible my teacher was just awful but I definitely walked away with some misconceptions about what proofs are. Perhaps it would have been more appropriate at an earlier stage of development.

Re: So you want to study mathematics

#326

Post by bick_nyers » Tue, Mar 08, 2022, 2:31 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 without calculus relies pretty heavily on memorization to get the right answers. I'm remembering my AP Statistics class (high school) where most of the time students got lower grades because they looked up numbers in a table wrong, or used the wrong table. Never mind when you start using regression models and need to start understanding axioms of the models you are using like homoscedasticity. There's some intuition there, but generally speaking, calculus is where most people "mature" into mathematics. I would argue that most people (NOT engineers/doctors/etc.) don't really get the point of math until around calculus where it starts to really open up for them. While statistics would certainly be useful, and is a must for modern day ever-connected society, it would just be another math subject that needs to be memorized like trigonometry. Some people mature in math a lot younger, but it entirely depends on the quality of their math teachers when growing up (and the curriculum/testing focus in which they are subjected to).

Re: So you want to study mathematics

#327

Post by bick_nyers » Tue, Mar 08, 2022, 2:33 PM UTC

bee_rider wrote: post #200

joatmon-snoo wrote: post #49

Earlier quoted context omitted.

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.…

I think I'd rather we lose geometry than calculus. Geometry made more sense as an intro to the idea of 'proving stuff' when everybody's dad had a workshop, compass, etc.

I never thought about it like that, thanks for that insight

Re: So you want to study mathematics

#328

Post by bick_nyers » Tue, Mar 08, 2022, 2:40 PM UTC

thaumasiotes wrote: post #231

joatmon-snoo wrote: post #49

Earlier quoted context omitted.

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.…

> 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. I think math people may have a view of this question that is skewed in an interesting way. Statistics is very useful and its common techniques are not difficult to apply. But they seem to be very difficult…

My thoughts exactly. What you want to teach ultimately before anything else is mathematical maturity, which in my opinion, calculus is currently doing the best job at providing.

Re: So you want to study mathematics

#329

Post by bick_nyers » Tue, Mar 08, 2022, 2:47 PM UTC

auggierose wrote: post #103

whatshisface wrote: post #77

Earlier quoted context omitted.

Geometry uses a deduction-only basket of proof techniques that don't prepare students for proofs done afterwards. I would like to see it replaced by elementary number theory which naturally uses a lot of induction/recursion and has some uses for proof by contradiction.

Geometry has really all that is needed for proofs: * Axioms * Substitution * Modus Ponens * Universal Quantification Induction or proof by contradiction are just special cases of this. But yeah, geometry for introducing proofs is difficult, because it is so easy to confuse visual intuition with proof. At the very least, you need a capable teacher who knows the difference. But nobody expects children to understand it…

I got a math degree, but I definitely failed at understanding that difference when learning geometry in high school. My teacher was good, but not good enough in separating that, and with classroom sizes getting larger by the decade, I think it's not the best approach to require that kind of tip-toeing. I'm not sure what would be better though, I've thought about maybe combinatorics could be a good replacement, but I also don't have the brain of a teenager anymore and I don't teach them either, so I don't know how far of a reach that would be.

Re: So you want to study mathematics

#330

Post by susanrigetti » Tue, Mar 08, 2022, 2:47 PM UTC

mrjangles wrote: post #271

susanrigetti wrote: post #184

Earlier quoted context omitted.

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…

It's just that it is very much a "mathematics for engineers" style course. I think very few of the subjects outlined there give you a flavor for what "real" mathematics is really all about at all (except for algebra, which you do mention). Apart from the applied stuff you mention, the real core of a mathematics education involves, I think, 4 main areas with significant overlap Group A: number theory, graph theory, co…

I suggest taking another look at the list and comparing it to the required courses of the undergraduate math majors at the top 20 universities in the USA.

Real analysis, complex analysis, topology, and number theory are there (topology and number theory are both listed as electives since most math programs categorize them as such). Graph theory, functional analysis, differential geometry, probability, and statistics are almost always either electives or graduate courses.

It’s funny, because most of the things you mention as “real math” are things that many math undergraduates don’t learn (not until graduate school at least) but that physics students learn as undergraduates (differential geometry, measure theory, functional analysis, etc.).