my thinking matured a lot, esp. my proofs, once i took a logic course, which suggests that maybe mathematicians in my hood were not as plain about what we were doing as they could have been.
but abstract algebra was always more elusive for me (even though really interesting), despite having lots of proof-theoretic niceties in its usual u.g. presentation, because i didn't have so many of those intuitions and had trouble learning how to visualize it or play with the structures. never had much of a feel for numbers, compared to many math majors, when younger, which i think would have made a difference. number theory as a gentleman's pastime, say, still has mostly zero attraction for me.
― j., Sunday, 10 November 2013 03:07 (ten years ago) link
sometimes i wish i could just chill and enjoy math instead of stressing about mathematically illiteracy
let's just like, do some problems
"A farmer has some rabbits and some cages. When he puts 2 rabbits in each cage, there are 2 rabbits left over. When he puts 3 rabbits in each cage, there are 16 cages (but no rabbits) left over. How many rabbits and how many cages are there?"
― the late great, Sunday, 10 November 2013 03:08 (ten years ago) link
lol i swear the cover of that book practically reads like 'the art of fucking shit up and guessing about things' to me
― j., Sunday, 10 November 2013 03:09 (ten years ago) link
number theory is ridic ... as far as algebra goes i learned groups rings and fields to pass a test and promptly forgot everything. i know nothing about topology or real analysis.
this course was about as far as i got in math before i gave up (i passed!)
http://www.amazon.com/Foundations-Higher-Mathematics-Peter-Fletcher/dp/053495166X
― the late great, Sunday, 10 November 2013 03:13 (ten years ago) link
i just tried to solve that rabbits and cages problem and got six cages and eighteen rabbits
hm
― the late great, Sunday, 10 November 2013 03:20 (ten years ago) link
ahem
http://math.arizona.edu/~savitt/GTM.html
― j., Sunday, 10 November 2013 03:21 (ten years ago) link
okay, fixed
― the late great, Sunday, 10 November 2013 03:22 (ten years ago) link
apparently i am
You are William S. Massey's A Basic Course in Algebraic Topology.You are intended to serve as a textbook for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery. Wherever possible, the geometric motivation behind the various concepts is emphasized.
You are intended to serve as a textbook for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery. Wherever possible, the geometric motivation behind the various concepts is emphasized.
so this test has really revealed something about my self to me
― j., Sunday, 10 November 2013 03:23 (ten years ago) link
Awesome. I have a copy of that book somewhere that I can give if you want, if I can find it and you are in NYC.
Tried to do that caged rabbit problem in my head but suspect it's broken meaning non-integral solutions. But maybe it's my brane that's broken will solve properly when I get home.
― I Wanna Be Blecch (James Redd and the Blecchs), Sunday, 10 November 2013 04:09 (ten years ago) link
nah there's a simple integer answer
― the late great, Sunday, 10 November 2013 04:09 (ten years ago) link
you can do it with just a system of linear equations but wording is so funny it took me a few tries to get the right expressions
― the late great, Sunday, 10 November 2013 04:10 (ten years ago) link
The sides of triangle ABC have lengths 6, 8 and 10. A circle with center P and radius 1 rolls clockwise around the inside of triangle ABC, always remaining tangent to at least one side of the triangle. When P first returns to its original position, what distance has P traveled?
^^ this one is grate
― the late great, Sunday, 10 November 2013 04:13 (ten years ago) link
and everyone saw this right?
http://sciencevsmagic.net/geo/
― the late great, Sunday, 10 November 2013 04:14 (ten years ago) link
OK, took out a pen and the answer is obvious. Now to review why my mental meanderings failed and see if I learned something.
― I Wanna Be Blecch (James Redd and the Blecchs), Sunday, 10 November 2013 04:31 (ten years ago) link
See the error of my ways too, which I may explain to y'all in a bit.
― I Wanna Be Blecch (James Redd and the Blecchs), Sunday, 10 November 2013 04:52 (ten years ago) link
Here's where I went RONGWanted to do it without pencil and paper and without algebra, so I decided to1) figure out what number rabbits had to be multiple of 2) what number cages had to be a multiple of 3) try various values of 1 and 2 until I was close and then4) perturb to get exact answer
Got to 3,but then 4 wouldn't quite work. Thought it might be because I perturbed in wrong direction.bIt turned out I had mixed up 1 and 2. When I wrote down the two equations I was able to solve it immediately.
Now I remember the potential problem with word problems. For the "smart" kids, they see the same word problems over and over and just know immediately how to translate them into mathematical terms and solve it, it is too easy for them. For the other kids or the old geezers like some of us, the mathematical content is elusive. In my experiment I went into the transporter beam and was split into both smart kid and dumb kid, and hopefully learned something from it.
― I Wanna Be Blecch (James Redd and the Blecchs), Sunday, 10 November 2013 05:28 (ten years ago) link
Can you get to that?
― Pazz & Jop 1280 (James Redd and the Blecchs), Sunday, 10 November 2013 05:31 (ten years ago) link
vahd33m's last problem reminds me of PSAT (or SAT?) controversy in the 80s in which one of the Jungreis brothers, must have been Doug, challenged the testing authorities on their wrong answer to a similar problem.
― Pazz & Jop 1280 (James Redd and the Blecchs), Sunday, 10 November 2013 05:43 (ten years ago) link
Somehow that is one of those rare bits of information from Ye Olden Times that has not been caught up in the meshes of the undiscriminating internet.
― Pazz & Jop 1280 (James Redd and the Blecchs), Sunday, 10 November 2013 05:49 (ten years ago) link
Here is a book of high school contest problems from way back when: http://www.math.nyu.edu/cmt/pdfs/NYCIML.ProblemBook.pdf
― Pazz & Jop 1280 (James Redd and the Blecchs), Sunday, 10 November 2013 14:21 (ten years ago) link
Another, related book: http://www.mathpropress.com/books/ARML/ You could also look at some of it in google books.
― Pazz & Jop 1280 (James Redd and the Blecchs), Sunday, 10 November 2013 14:33 (ten years ago) link
http://www.mathpropress.com/books/index.html
― Pazz & Jop 1280 (James Redd and the Blecchs), Sunday, 10 November 2013 14:35 (ten years ago) link
Looks like you can't really get it nowadays.
― Pazz & Jop 1280 (James Redd and the Blecchs), Sunday, 10 November 2013 14:38 (ten years ago) link
Isn't that what computer science freshman learn in their Discrete Math course, flopson, iirc?
― I Wanna Be Blecch (James Redd and the Blecchs), Saturday, November 9, 2013 9:21 PM (Yesterday) Bookmark Flag Post Permalink
well how does that help high schoolers?
besides logic, teaching a critical thinking course now that's heavy on inductive logic and probability, for the first time ever, and already (not being super deep into the material) i'm struck by how useful it could be if more people learned this stuff early on. i had a science-heavy math education, and i've spent some time thinking about how science works, but i can see that there's a lot about it that's overly opaque to me because my knowledge of probability/statistics is shallow. especially when it comes to the social sciences.
― j., Saturday, November 9, 2013 9:41 PM (Yesterday) Bookmark Flag Post Permalink
isn't the typical college-prep math track basically set up to produce calculus students (then engineers, and the few others who need calculus, like physicists)?
― j., Saturday, November 9, 2013 9:51 PM (Yesterday) Bookmark Flag Post Permalink
yeah i guess what i meant was that you spend most of high school math doing pre-cal + geometry, but don't even get to the punchline of the former unless you take calculus in university, by which point your relationship to math is already largely determined. i enjoyed geometry but that's not most people's experience, particularly due to the focus on tedious things like trigonometry. i think those topics i mentioned have some fun, big results that come very quickly and can be derived and made intuitive with elementary methods accessible to hs students, and broadening the amount of topics you see might make more people likely to realize that they like math, and would give people a better idea of what it's all about. i don't have very strong feelings about pedagogical approaches to math and i know that it's very hard to get people to learn even the simplest stuff. but i don't think my suggestion necessarily means students should learn more or harder material, if anything it's more about giving a practical layman's set of tools for people not continuing calculus
― flopson, Sunday, 10 November 2013 22:49 (ten years ago) link
I’m fascinated by this debate. Excuse my rambling:
Calculus is an opportunity to survey mathematics. It’s possible that a student’s calculus sequence will be their first, and only exposure to foundations (e.g. proof by Riemann sums), abstract algebra (e.g. vectors and vector spaces), combinatorics (e.g. infinite series and sequences), geometry (e.g. differential and topology), and a variety of philosophical and applied areas of mathematics. If you care about mathematics as a subject of humanity, there’s no better introduction to the themes, relationships, and problems within mathematics. In a perfect world, we’d rename the courses “An Introduction to Mathematics.”
Nonetheless, I don’t think the typical calculus sequence is useful for non-hobbyists (e.g. “engineers”)—it’s far too comprehensive and not enough time is spent on useful knowledge (e.g. dynamics, differential equations, and numerical approximation). I also don’t think it’s useful for young mathematicians—too much time is spent on sharpening basic knowledge (e.g. algebra, counting, geometry, sentential calculus) rather than rigor. For example, my undergraduate track:
* Introduction to Calculus* Intermediate Calculus* Multivariable Calculus* Functions of a Complex Variable (i.e. complex analysis)* Introduction to Analysis* Functions of a Real Variable (i.e. elementary real analysis)
I also took a differential equations and Fourier analysis course. I suspect most people had a similar experience.
In retrospect, I feel like the calculus sequence could’ve been reduced into one course—using Spivak—better preparing me for the jump into analysis and freeing my schedule to study non-analysis subjects.
Incidentally, I feel like algebra has the reverse problem. In my experience, most students enter Ph.D programs with two courses in algebra: linear and an abstract algebra course (covering groups, rings, fields). If you replaced a calculus course from the standard calculus sequence with an algebra course, you could then require an upper-level algebra course in the junior or senior year (e.g. commutative algebra) that’d better prepare everyone.
― Allen (etaeoe), Monday, 11 November 2013 00:43 (ten years ago) link
Also: I’ll join the interested in Homotopy Type Theory choir. While I’ve spent the past while working in compilers, I’ve been studying PLT topics wherever necessary (mostly semantics and types). I enjoyed Types and Programming Languages, but I’ve been looking for something a bit wilder. If there’s enough interest, maybe we can use this tread as an informal book club/Agda help desk.
― Allen (etaeoe), Monday, 11 November 2013 00:50 (ten years ago) link
My interest in mathematics is very much related to computer science. I didn't post here earlier, because I thought this thread would only be mathematics.
But I see some may appreciate these drawings that use Turing machines: http://maximecb.github.io/Turing-Drawings/
― c21m50nh3x460n, Monday, 11 November 2013 00:56 (ten years ago) link
Also recently bought a copy of Best Writing on Mathematics 2012 which is very nice and pitched at a similar level, the article on Math and Music was very interesting, written by a guy who is both a practicing musician and research mathematician.
I haven’t read it in its entirety, but I’ve liked everything I’ve read from Paul Hudak’s Haskell School of Music:
http://www.cs.yale.edu/homes/hudak/Papers/HSoM.pdf
(However, I didn’t know anything about music theory. So, the relationship between music and, say, lazy evaluation and typing might be overwrought.)
― Allen (etaeoe), Monday, 11 November 2013 00:57 (ten years ago) link
Sort of agree on calc. Way over done on specific tricks, and not enough emphasis on intuitions and meaning. All the exercises that are just about remembering a zillion tricks and identities for symbolic manipulation and simplification are just really about rote memorization and speed. Calc is a pretty specialized subject and treating it as "genuine advanced math" is pretty maddening compared to what gets left out.
Math is really used due to the history of standards bodies and testing, etc. as much as a screen/weeder as actually taught to teach math, and the emphasis on calc is the biggest symptom of this (but all the trig identities actually similar -- I mean we should really start with the unit circle, euler's identity, etc. and build trig on that.
also hate the way linear algebra is taught, determinants first.
its all derived from teaching things you can test in a certain way.
― lollercoaster of rove (s.clover), Monday, 11 November 2013 01:06 (ten years ago) link
I'm in the middle of taking a linear algebra course and we haven't got to determinants yet. The text is sort of unorthadox in its presentation, though, according to the instructor.
― brimstead, Monday, 11 November 2013 01:23 (ten years ago) link
which one is it?
― flopson, Monday, 11 November 2013 01:26 (ten years ago) link
axler? axler is gr8
― lollercoaster of rove (s.clover), Monday, 11 November 2013 01:26 (ten years ago) link
Bretscher
― brimstead, Monday, 11 November 2013 01:29 (ten years ago) link
wow the amazon reviews for it are beyond vicious. but it all seems to be from frustrated undergrads...
― lollercoaster of rove (s.clover), Monday, 11 November 2013 01:34 (ten years ago) link
Which text is it? If it was graduate course I might have a guess
― Pazz & Jop 1280 (James Redd and the Blecchs), Monday, 11 November 2013 01:36 (ten years ago) link
(Xp obv)
― Pazz & Jop 1280 (James Redd and the Blecchs), Monday, 11 November 2013 01:42 (ten years ago) link
Dont't Axler was around yet in my time. I liked Valenza, Halmos and, for the more applied approach, that old warhorse Strang.
― Pazz & Jop 1280 (James Redd and the Blecchs), Monday, 11 November 2013 01:48 (ten years ago) link
Don't think
― Pazz & Jop 1280 (James Redd and the Blecchs), Monday, 11 November 2013 01:49 (ten years ago) link
Used copies of third edition of Strang pretty cheap!
― Pazz & Jop 1280 (James Redd and the Blecchs), Monday, 11 November 2013 01:51 (ten years ago) link
Not knowing any alternative, I will say this: Thank god i have a good instructor. The instructional portion of the text compliments the lectures more than anything.
― brimstead, Monday, 11 November 2013 01:57 (ten years ago) link
or more than vice versa, i should say
mosly taught myself linear algebra reading wikipedia
― flopson, Monday, 11 November 2013 02:21 (ten years ago) link
man, figures that searching for 'homotopy' to find this thread on ilx would not be a sure bet
...
http://www.theatlantic.com/education/archive/2013/11/the-stereotypes-about-math-that-hold-americans-back/281303/
― j., Wednesday, 13 November 2013 16:27 (ten years ago) link
i like that article but i'm not sure i agree with this
"The U.S. does not need fast procedure executors anymore. We need people who are confident with mathematics ..."
if you are in a college-level math class it is hard to feel #2 without #1 under your belt
― the late great, Wednesday, 13 November 2013 17:11 (ten years ago) link
i am giving students an 'all proofs' exam next week and they are also of that mind
― j., Wednesday, 13 November 2013 20:17 (ten years ago) link
i love jo boaler though, she fights the good fight
really bummed that she was gone the year i was at stanford
― the late great, Wednesday, 13 November 2013 20:25 (ten years ago) link
i took half of jo boaler's MOOC this summer, rly enjoyed it and it informed my practice a lot
― shiny trippy people holding bandz (m bison), Thursday, 14 November 2013 00:16 (ten years ago) link
Famous Mathematician's Daughter Weighs In On Topic Of Interest To Ilxors
― Pazz & Jop 1280 (James Redd and the Blecchs), Thursday, 14 November 2013 15:59 (ten years ago) link
assumed that was going to be about tipping tbh
― famous for hits! (seandalai), Thursday, 14 November 2013 16:03 (ten years ago) link
― shiny trippy people holding bandz (m bison), Wednesday, November 13, 2013 7:16 PM (Yesterday) Bookmark Flag Post Permalink
what was it about?
― flopson, Thursday, 14 November 2013 23:17 (ten years ago) link