ℝolliℵg M∀th Thr∑a∂

Here is a book of high school contest problems from way back when: http://www.math.nyu.edu/cmt/pdfs/NYCIML.ProblemBook.pdf

Another, related book: http://www.mathpropress.com/books/ARML/ You could also look at some of it in google books.

http://www.mathpropress.com/books/index.html

Looks like you can't really get it nowadays.

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

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.

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.

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/

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.)

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.

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.

which one is it?

axler? axler is gr8

Bretscher

wow the amazon reviews for it are beyond vicious. but it all seems to be from frustrated undergrads...

Which text is it? If it was graduate course I might have a guess

(Xp obv)

Dont't Axler was around yet in my time. I liked Valenza, Halmos and, for the more applied approach, that old warhorse Strang.

Don't think

Used copies of third edition of Strang pretty cheap!

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.

or more than vice versa, i should say

mosly taught myself linear algebra reading wikipedia

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/

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

i am giving students an 'all proofs' exam next week and they are also of that mind

i love jo boaler though, she fights the good fight

really bummed that she was gone the year i was at stanford

i took half of jo boaler's MOOC this summer, rly enjoyed it and it informed my practice a lot

Famous Mathematician's Daughter Weighs In On Topic Of Interest To Ilxors

assumed that was going to be about tipping tbh

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), Wednesday, November 13, 2013 7:16 PM (Yesterday) Bookmark Flag Post Permalink

what was it about?

fixed v growth mindsets, beliefs about innate abilities affecting learning outcomes

a little about stereotype threat as well

hey sterl, can you answer a dumb question for me?

if one of the problems with russell's resort to types is that it is essentially a kludge, what is it about HTT that improves on the kludginess?

Hmm. Apols if this is somewhat rambly. The brief summary is the "point" of type theory is now very different from why Russell was interested in types, and much more about taking types as a foundational setting for logic/computation/topology. However, this doesn't mean that we still don't need a "hack" (in the form of a cumulative hierarchy of universes -- aka a predicative [as opposed to impredicative] logic) to avoid russell-style paradoxes. However, the current approach to dealing with russell-style paradoxes in this setting is arguably not too terrible.

So I guess the argt w/ russell is that types are just a device to prevent your language from saying paradoxical things, but you can see them as imposed from without, semi-arbitrarily.

We don't need to go all the way to HTT, but really should talk about Martin-Löf type theory (HTT is basically just a full model of MLTT in topological terms plus one new axiom [ univalence ] and one new construction [higher inductive types]).

If you haven't read Martin-Löf's three lectures, I'd start there. They're a good motivation of what he's up to in v. philosophical terms http://www.ae-info.org/attach/User/Martin-L%C3%B6f_Per/OtherInformation/article.pdf

The gist is that we don't start with some system and then impose a typing discipline, but instead types can be taken as foundational, via curry-howard.

So in a fully constructivist standpoint we can say types _are_ propositions, terms _are_ proofs, and so we don't have an "object language" and a "meta language" the same way we tack eg first order logic onto set theory. Now types aren't a "kludge" but your foundational syntactic objects, and the fact that your terms have goofy binders like "lambda" becomes the hack. And via Lambek, we don't need that! we have another correspondence/isomorphism that also lets us interpret our language of types into cartesian closed categories, and in a sense MLTT can be (though here my maths falls down a bit) the "internal logic of a topos". (I really need to tackle topos theory at some point but it scary). Here we read types as objects and terms as morphisms.

So this brings us to HoTT that gives us still another isomorphism -- types are propositions are objects in a CCC are also _spaces_. terms are proofs are morphisms are _points_.

But in a sense that doesn't get to what's important about MLTT, which the lectures do. MLTT distinguishes between two notions of equality -- judgmental equality in the meta-language, and which is direct and obvious, and "internal" or propositional equality, which is a proof-theoretic concept.

In this framework you still need to distinguish between levels of universes to prevent paradoxes, and the _hierarchy_ of your universes still needs to come from some external definition, and you still arguably would like "type-in-type" or "universe in universe" to make constructions easier, and doing so still leads to paradoxes. But that's sort of secondary to most of the "important" things type theory does, and universe-bookkeeping is seen as a sort of chore that should be mechanically automated by various tools, just like working in a typed logic is less of a pain if your tools can infer types for you.

Also, arguably, the notion of a "cumulative hierarchy of universes" is itself fairly natural in its own way though, if that helps, in that the construction is iterative and straightforward and can be done "on demand" rather than all at once.

ooh someone put lawvere's "Sets for Mathematics" online http://patryshev.com/books/Sets%20for%20Mathematics.pdf

got a riddle for yall

there are four coins on a turntable, arranged in a square formation. you're blindfolded. your task is to have all the coins facing the same way (that is, all face up or all face down.) you're allowed to make a move of the following sort: you flip any number of coins, and then ask if you're done. if you're done, you're done. otherwise, i randomly spin the turntable (some integer multiple of 90 degrees.)

Interesting.

So, if the base case is verified (i.e. we’re told that the coins aren’t already in a correct position—all H or all T):

   A
B     C
   D

I’ll flip A and D, or B and C. Either:

• I’ve won.
• I have an odd number of Hs.
• I have two bordering Hs.

• I’ve won.
• I have an odd number of Hs.
• I have two opposite Hs (e.g. B and C)

• I’ve won.
• I have an odd number of Hs.

• I’ve won.
• I have two Hs and two Ts.

• I’ve won.
• I have two bordering Hs.

• I’ve won.
• I have two opposite Hs.

• I’ve won.

nice

SPOILER ALERT

y'all wanna see some hardcore bullshit?

http://lmgtfy.com/?q=critical+mathematics

the phrase 'i want to argue...' always a red flag in an academic paper.

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CDMQFjAB&url=http%3A%2F%2Fpeople.exeter.ac.uk%2FPErnest%2Fpome25%2FMarilyn%2520Frankenstein%2520%2520Critical%2520Mathematics%2520Education.doc&ei=E5WOUv79GIiLqAH3yYHYCw&usg=AFQjCNFu31eLETOh7_RmT9x-HgowEtwJPQ&sig2=kyUOwZYChMpCyTHTUff7xw&bvm=bv.56988011,d.aWM&cad=rja

but the idea that math (not just math education) has a socio-ethical dimension is not preposterous. despite how easy it probably is to sound preposterous when you ahem try to 'problematize' that dimension.

Liberatory social change requires an understanding of the technical knowledge that is too often used to obscure economic and social realities. When we develop specific strategies for an emancipatory education, it is vital that we include such mathematical literacy. Statistics is usually abandoned to “experts” because it is thought too difficult for most people to understand. Since this knowledge is also considered value-free, it is rarely questioned.

^ this is otm. imo the thing about applying "knowledge is socially constructed" critical theory to math is, like, you lose the power of math to challenge speak truth to power if you don't believe in its objectivity in the first place. like if someone uses a study drawn from a small sample to justify your oppression and you can find one with a larger sample size and say, "weak law of large numbers" or whatever analogous result for a consistent estimator. but if you're a critical theorist do you just shrug and say, ah it's all subjective?

just because something is socially constructed doesn't mean that it's not objective /spinoza

Mathematics is traditionally seen as the most neutral of disciplines, the furthest removed from the arguments and controversy of politics and social life. However, critical mathematics challenges these assumptions of neutrality and actively attacks the idea that mathematics is pure, objective, and value neutral.

was just going by this

i just think there's a tension bw 'think critically about mathematics' and 'use mathematics as a tool to think critically about other things.' many philosophers seem to have a similar attitude towards language, like they're suspicious of it, they don't trust it, it's unstable. i think mathematicians and people who use math are probably very grateful that they don't have this suspicion.

the same page suggests a pedagogical approach based on "criticality towards received opinion" but indeed mathematics is the field where you can be most critical of received wisdom, because everything is proved rigorously, you don't have to take anything for granted

good mathematicians are very conscious of its limitations

in application, sure. but do you think "good mathematicians" are suspicious of the truth of basic results? a friend of mine became close with an old logic/foundations prof called Pf3nder who has all these crazy sounding theories, like he doesn't believe in infinite sets or the existence quantifier and claims to have a disproof of the incompleteness theorem, but according to friend his views were very far from typical and that he is not taken seriously among the faculty where he used to teach.

what is truth

- Bob Marley