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So you want to study mathematics

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Re: So you want to study mathematics

#231

Post by thaumasiotes »

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.…

> 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 to apply correctly. We have entire academic fields that are built mostly on the spurious application of statistical methods in contexts that make the whole project invalid.

And this sort of "techniques without theory" approach is what's being advocated for upthread and represents a failure mode that math people are unlikely to consider -- because they know that part of being able to apply a technique is being able to tell whether or not applying it is valid. Math people are unlikely to fall into the trap this approach sets. But the same people who want this approach are also likely to end up being hurt by it.

Re: So you want to study mathematics

#232

Post by syntheweave »

brimble wrote: post #209
pnin wrote: post #176

Earlier quoted context omitted.

I have some sympathy for this sentiment, and there is no doubt that the producers of mathematics could and probably should spend more time making life easier for the users and consumers of mathematics. This is true even /within/ the discipline, for very theoretical stuff. In the end this takes vastly more work than people expect. That's not an excuse, but it's true. However, I also have a fundamental objection. I don…

I do not have the right background to be an authority on this, but from going through school myself and paying a fair amount of attention to math reformers over the years, it seems like they usually believe: 1) The problem with math in school is that there's not enough "real math". 2) Relatedly, insufficient exposure to "real math" in compulsory schooling is also (a major part of) why people think they don't like mat…

I think it's a kind of rope that frays on both ends for systemic reasons:

1. Students in math courses and their parents grow to prefer(through the overall institutional constraints) to have a simple exercise that guarantees them credit - while actually doing math is a matter of crossing the Rubicon into tough puzzle-solving, and it needs some guidance for unexceptionable students to start enjoying.

2. Math teachers, particularly in the lower grades where qualifications are lower, have a harder time teaching concepts than they do exercises. And they are also incentivized to hand out a grade, preferably one that satisfies the parents.

So no matter how the high level is set up, everything converges into giving the kids a worksheet to "plug 'n chug." Which is just a confusing, badly paced grind, and therefore an easy reason to hate math. Either you get it completely and are just sitting there chugging through the problem set, or you have no idea what's going on and it's due tomorrow so your grade rests on something you feel defeated by.

I actually think that for the parts that are currently treated as rote memorization work, the curriculum should lean into it and treat it like learning the alphabet, with worksheets where you literally fill in the dotted lines repetitively; hand them out to everyone as a portion of the homework. And then the logic and critical thinking aspects need to proceed like a philosophy course, with interaction through a step by step process, not "get the answer in the back of the book". This element is something I've long thought could be automated in some degree with computer systems that let you play with the concepts, and therefore correct your thinking.

Re: So you want to study mathematics

#233

Post by BeetleB »

andrepd wrote: post #157
dr_dshiv wrote: post #145

If doing math is essential to conceptual understanding and application, could the interface of math and physics be made more human-centered? For instance, the shift from Roman numerals to Arabic numerals made doing math easier. Based on your experience, might it be possible to increase accessibility by revising some of the arcane conventions of math and physics? See Brett Victor’s 2011 proposal: http://worrydream.com…

Why is it that every time any subject about mathematics comes up there is always a complaint about notation? Your link doesn't even exactly talk about notation, but about pedagogy. Can you be more specific about which notation your consider "arcane"?

> Why is it that every time any subject about mathematics comes up there is always a complaint about notation?

The assumption that there is a much better notation is one I tend to see only with the HN crowd. Outside of this group, even people who dislike the notation and/or struggle with math do not claim that a better/simpler one obviously exists.

Re: So you want to study mathematics

#234

Post by JadeNB »

mmmpop wrote: post #208
ghaff wrote: post #172

Earlier quoted context omitted.

For tests and the like, when I was an undergrad, it was fairly common to be able to bring in a one-page "cheat sheet." The idea was that you probably could derive a lot of the stuff given time and maybe references, but you almost certainly couldn't in a 1 hour exam in addition to solving the actual problems on the test.

>> 1 hour exam I'm trying to remember when I had one of those last! I actually don't love the idea of cheat-sheets but maaaybe if it's a cumulative final I could see it being helpful? If you're taking chapter exams on some material, I think you ought to have worked enough problems so that formulas/constants are drilled into your head. But I guess too, where do you draw the line at engineering appendices? I sure don't…

> If you're taking chapter exams on some material, I think you ought to have worked enough problems so that formulas/constants are drilled into your head.

Well, sure, but, if you don't, then what's the point of punishing you? As a teacher, frankly, I'd be happy to have my students bring in any static resources they wanted to consult—I say 'static' to emphasise not, e.g., consulting a cheating site, although it's fine with me if they've pre-compiled solutions in advance to any problems they think might be interesting or important—except that (1) I think that would encourage bad study habits, and, more importantly, (2) it would be unwieldy in a packed classroom to try to have adjacent people juggling multiple textbooks, notebooks, etc.

In fact, I loved the freedom to give extended-time, fully open-book, open-note exams during the fully remote classes. I wish I could still do that; if cheating weren't so endemic under those conditions, then I would.

Re: So you want to study mathematics

#235

Post by BeetleB »

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.

> This felt like it was written by a physicist or engineer.

I just compared it to my undergrad's math curriculum and it matches up pretty well. Everything you mentioned is an elective.

Re: So you want to study mathematics

#236

Post by BeetleB »

kurthr wrote: post #110
potbelly83 wrote: post #85

As a math PhD I have to say the only way you're going to learn mathematics is if you actually have a pressing need to do so. i.e. You have a project at work that needs some math, you have a hobby that needs some math. In this case you just learn what you need. Just learning math for its own sake outside of a University STEM track is just too hard (I wouldn't be able to do it and I've tried).

I hesitate to contradict someone who has gone through the whole Math PhD process, but I have to say that the best mathematicians that I know treat the problems they're working on as games, or riddles to be solved... and they've taught their kids (and others) this same method of thinking about these problems. There's a huge mental tool set, and often a lot of grinding to get to a solution, but it's just a game (and th…

I don't think you're contradicting the GP. Solving problems for fun still counts as "pressing need".

The alternative is something like "Oh, I should learn abstract algebra because it's a fundamental course in all curricula". You'll learn it, and then forget it.

Re: So you want to study mathematics

#237

Post by BeetleB »

mbustamanter wrote: post #13

> My goal here is to provide a roadmap for anyone interested in understanding mathematics at an advanced level. Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics. NO, NO, NO. There is no real way to go up to the real deal without having understood elementary Functional Analysis, which the article doesn't even mention. FA is roughly…

My undergrad institution does not require FA to graduate. It doesn't even list as an elective.

There's a very good reason for this: FA is not all that useful if you're more on the algebra side of things.

Re: So you want to study mathematics

#238

Post by BeetleB »

wanderingmind wrote: post #159

No recommendation on probability. Thats strange given that the author is a physicist and fundamentals of modern physics rests on probability. My recommendation is the classic "Probability Theory, The logic of Science by E.T.Jaynes" which is a Bayesian formulation.

You may be surprised to discover that many/most physics undergrad curricula do not mandate a course on probability.

Re: So you want to study mathematics

#239

Post by kjasdfj3 »

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.…

> 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'm questioning your math pedagogy.

Calculus is the single most important math anyone, of any field, can learn as it's the first "practical math" you actually learn. Life behaves like calculus and in order to think about real life you need the concept of limits, derivatives, integrals, differentials, etc. It's patently absurd to say this should be replaced by statistics, which done to any rigor requires up to 2 years worth of calculus (through diff eq.) to even appreciate.

I'm shocked that you're a math major and didn't take away the biggest thing from learning analysis - the ability to think clearly through a problem and prove it correctly. While you may not be asked to vomit cantor's diagonalization onto paper for an interview the ability to think about problems you learned from doing these proofs translates to so many different fields, jobs, and life skills that I take the complete opposite view. If you want to understand anything you need to learn how to proof. I don't care if you're a nurse or an accountant. A rigorous proof based math course will change your life.

If by "learn statistics instead of calculus" you mean being able to mindlessly vomit today's new machine learning paradigm without understanding a thing then I think I can understand where you are coming from. Otherwise, I think this is some absurd parody of someone who studied math.

Re: So you want to study mathematics

#240

Post by kjasdfj3 »

wbsss4412 wrote: post #169
zozbot234 wrote: post #150

Earlier quoted context omitted.

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

You can certainly make significant headway with some basic combinatorics and basic tables for things like the normal distribution, no calculus needed. This is in fact how most people currently learn statistics.

What happens when you dont have tables?

In the land of the blind the guy with one eye walks everyone off a cliff.

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