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So you want to study mathematics

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Re: So you want to study mathematics

#151

Post by geebee »

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'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 few curves with a compass. Before Calculus, you might actually believe that numbers that can be expressed as the ratio of two integers are typical, and that numbers like pi and the square root of two are "irrational" rarities (and until calculus, you probably don't know about Euler's constant unless it was introduced in precalc as another one of those odd and rare numbers).

Look out at nature, where are the triangles, rectangles, and circles? Maybe a wasp nest? Nah, not really. Try to draw a cloud, a tree, a tiger, or a human face. How useful is that straight line or compass? How useful is a line at all, other than to hint at something you can't actually draw, maybe by implying it exists as an ever vanishing limit from above and below? Math required calculus the instant humans decided to describe the world as it is, rather than by the limits of what we impose on it.

Also - in stats, how do you know what the area is under the probability density function?

Re: So you want to study mathematics

#152

Post by l33t2328 »

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…

Spivak is a better analysis book than Abbot.

Re: So you want to study mathematics

#154

Post by dwohnitmok »

C-x_C-f wrote: post #142
dwohnitmok wrote: post #73

Earlier quoted context omitted.

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…

> 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. Not at all, you are only prohibited from taking "freshman courses"[1]—that's what 55 is supposed to cover (definitely not all of the undergra…

I wonder if the pamphlet's changed or I'm misremembering. I distinctly remember Math 123 and Math 114 both explicitly excluding Math 55 (It's also worth stating that Math 122 and Math 113 are definitely not freshman courses). But regardless I'd be very surprised to learn a Math 55 student took Math 114 or Math 123. I would also be surprised (although less so) to learn of a Math 55 student in the 130 series.

The Math 140 series have only become more serious courses in the last 10 - 15 years or so IIRC. The 240 series was generally where to go for serious mathematical logic courses (and generally is where you would go after e.g. a first 140 series course in set theory anyways).

Re: So you want to study mathematics

#155

Post by SamReidHughes »

geebee wrote: post #151
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'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…

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.

Re: So you want to study mathematics

#156

Post by ghaff »

JadeNB wrote: post #125
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…

> If anything should be dropped from a highschool level its all of the insane memorization you have to do for some of the lower level math classes - I found that totally useless. You then learn some basic calculus and realize "I just wasted so much of my life" and never need to memorize those things again. What sort of memorizations do you have in mind that you no longer need to memorize once you know calculus?

Pretty much all the classic Newtonian formulae you memorize in high school physics classes can be derived (relatively) easily once you bring calculus in. A general problem with high school classes in particular--still somewhat true in certain university classes but much fewer--is that there are dependencies across classes. So you end up with a lot of "Memorize this thing because you don't have an advanced enough background to understand why $XYZ is the case."

Re: So you want to study mathematics

#157

Post by andrepd »

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"?

Re: So you want to study mathematics

#158

Post by C-x_C-f »

The website mentions some good courses, personally I love Richard Borcherds' YouTube channel[1] for both undergraduate and graduate courses. No frills, exceptionally clear, (mostly) bite-sized lectures that cover a good range of material (especially in geometry).

Something that might interest HN's demographic is Kevin Buzzard's Xena Project[2], centered around proof systems (in Lean). The natural numbers game [3] is particularly fun IMHO. I don't know if it counts as learning materials per se but it's certainly instructive.

[1] https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ/pla...

[2] https://xenaproject.wordpress.com/

[3] https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g...

Re: So you want to study mathematics

#159

Post by wanderingmind »

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.

Re: So you want to study mathematics

#160

Post by hardwaregeek »

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 dunno, at that point, are you really doing math? Math is proof. Now, granted, nobody expects you to know exactly how everything is proved. But there is an expectation that you can prove most of the stuff. Otherwise, it's very easy to stray from the truth into the plausible-sounding, but incorrect.

That said, you're absolutely correct that more justification and motivation is important. So much of math can be taught with problems from physics, computer science, etc. Perhaps a good book for you would be Concrete Mathematics by Knuth? I haven't read it but people swear by it.

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