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So you want to study mathematics

susanrigetti.com

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Re: So you want to study mathematics

#71

Post by jerry1979 »

itcrowd wrote: post #60

Susan, I greatly appreciate this list and will definitely come back to use it as a reference if I need a book recommendation. (I don't think I'm the target audience, although who knows what the future brings..) That being said, I think you are missing out on an opportunity to reach a wider audience. It bugs me a bit that the requirements seem very American-centric. What I mean is the following bit: > A high school ed…

Do we have good universal descriptors for math levels? I'm a big fan of accessibility, and I think your idea about tweaking language to reach a wider audience could be a big win for increasing the article's impact.

To update the article to include your recommendations, the author would probably need some kind of "cross-walk" which would map the American perspective to a more universally understood framework. Would you happen to know what "pre-calculus's" opposite number would be in the universal framework?

Re: So you want to study mathematics

#72

Post by dang »

pphysch wrote: post #33
susanrigetti wrote: post #27

Earlier quoted context omitted.

You cut out the middle of that paragraph, which says: "Sadly, there is all sorts of baggage around learning it (at least in the US educational system) that is completely unnecessary and awful and prevents many people from experiencing the pure joy of mathematics. One of the lies I have heard so many people repeat is that everyone is either a “math person” or a "language person” — such a profoundly ignorant and damagi…

I'm not sure what your point is. Are you implying that you are not contributing to the mystification and idealization of mathematics? In other words, I do not see how you are dealing with the "baggage" of learning mathematics beyond name-dropping it. In my opinion, the mysticism is the baggage. And then the rest of the blogpost reads like a conventional curriculum within the conventional academic regime with which we…

Please don't post in the cross-examining style. We want curious conversation here.

This is in the site guidelines: https://news.ycombinator.com/newsguidelines.html.

Re: So you want to study mathematics

#73

Post by dwohnitmok »

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.…

FWIW (and I know you're not making this claim) I don't believe Math 55 is a good example of how mathematics should be taught at large, nor do I think calling it an intro class accurately conveys what it is. It's effectively the compression of an entire undergraduate degree into a single freshman course. I say "effectively" because the material covered depends heavily on who's teaching it, but certainly anyone coming out of Math 55 can pick up any undergraduate math content trivially. For example, undergraduates who have taken it are generally explicitly prohibited in course descriptions from taking further undergraduate mathematics classes (because it would be free credit for retreading the same ground) and can only take graduate courses from then on out.

It's strongly self-selecting and as a result can afford to cover a truly insane amount of ground. The overwhelming majority of people who take it drop out (the usual dropout rate from people who take it in the first week is probably > 90%), but the people who stay almost all get As. And you will need to be almost entirely self-motivated because a lot (maybe most) of your waking hours will be thinking about math.

There's a very small minority of students for whom this is an optimal way of learning. For most students this is the quickest way to make them run screaming away from mathematics even faster than they already do.

Re: So you want to study mathematics

#74

Post by nyc111 »

I don't agree with this article, it as off-putting as the usual math eduacation it criticizes. I wonder how one can propose a curriculum to study math and not mention Euclid. One learns more mathematics from this article https://mathshistory.st-andrews.ac.uk/Extras/Russell_Euclid/ by B. Russell where he harshly criticizes Euclid than 2 years of calculus. Newton did not know calculus but he knew Euclid's Book 5, the book about ratios and proportions. Euclid's 5th Book must be the starting point for the study of math. When we say "math is the language of nature" we really mean that nature is proportional. Ratios and proportions are fundamental.

Re: So you want to study mathematics

#75

Post by eftychis »

jll29 wrote: post #62
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.…

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 was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at.

Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.

Re: So you want to study mathematics

#76

Post by psyklic »

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.

Having a deep (proof-based) understanding leads to more "real world" insights. So, there may not be a clear distinction.

Re: So you want to study mathematics

#77

Post by whatshisface »

dhosek wrote: post #66
auggierose wrote: post #56

Earlier quoted context omitted.

Proofs are indeed very practical. For example, I believe Leslie Lamport mentioned somewhere that he only came up with the final version of Paxos once he tried to prove it, and noticed that some condition he assumed wasn't necessary at all.

The reasoning behind having geometry be the standard high school sophomore math class is that that’s the age where kids would be ready to do proofs. Except that curriculum designers seem to have forgotten this and except in honors classes, most sophomores don’t get taught proofs in geometry and instead get a set of inert rules about shapes that they have no use for.

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.

Re: So you want to study mathematics

#78

Post by Bostonian »

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.…

I wonder how much statistics someone can understand without calculus. For example, how do you explain what a continuous probability density function (such as the Gaussian) is without calculus?

Re: So you want to study mathematics

#79

Post by jeffreyrogers »

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.

Re: So you want to study mathematics

#80

Post by dwohnitmok »

jll29 wrote: post #62
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.…

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), combinatorics (which is kind of covered by stats), number theory, discrete mathematics, and graph theory are all much less useful even in adjacent STEM fields (discrete mathematics and graph theory matter more in CS, but far less for day-to-day programming). History of mathematics is effectively an entirely separate discipline and philosophy of mathematics has meta-mathematics/mathematical logic as a prerequisite.

Calculus unlocks much of physics and engineering (and lots of stats!). Large cardinal theory does not unlock any other field to the best of my understanding.

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