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A Mathematician’s Lament (2002) [pdf]

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Re: A Mathematician’s Lament (2002) [pdf]

#71

Post by auggierose »

codyb wrote: post #19
bridgeyman wrote: post #4

I read this a couple weeks ago and decided to try and find a math book that incorporates some history in it. I found Journey through Genius: The Great Theorems of Mathematics with some great reviews. I am working on an iPad app that pairs up people to mentor each other through books like this. It has video chat and a shared whiteboard, so it is ideally suited for discussing math. If anyone is interested in reading th…

To add to the suggestions here, "Number" by Tobias Dantzig is absolutely a wonderful history of mathematics. I mean, shoot, it's actually got a quote from Einstein: "This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of math…

if you are in a hurry:

http://www.engineering108.com/Data/Engineering/Maths/Number_...

Re: A Mathematician’s Lament (2002) [pdf]

#72

Post by dnautics »

csense wrote: post #56
dnautics wrote: post #20

As an erstwhile math major (I couldn't hack the honors basic algebra class - the difference between a euclidean domain and a principal ideal domain got too confusing; but I rocked proofs in analysis) I have to say that the author is confusing Mathematics (which is an art) and Arithmetic (which is a skill). Part of what make the opening farce absurd is that musical skills and painting are not terribly necessary as a m…

> The difference between a euclidean domain and a principal ideal domain A Euclidean domain is an integral domain where the Euclidean algorithm works. For the Euclidean algorithm to work, you need to be able to divide two elements and produce a remainder that's smaller than the divisor. So an ED requires a notion of "smallness" (the Euclidean norm) which interacts with division in a way that makes the Euclidean algor…

yes, I know what the technical definition is, but I never felt like I understood in a deep, mechanistic way, why all EDs are PIDs, beyond the constructive proof, and while I could prove that some wierd Q[some element] was a PID, but !ED, I never felt like I truly understood why. Mathematical taxonomy always wierded me out, maybe I took the Rutherford quote "all science is either physics or stamp collecting" too seriously.

Re: A Mathematician’s Lament (2002) [pdf]

#73

Post by dnautics »

saraid216 wrote: post #55
dnautics wrote: post #48

Earlier quoted context omitted.

in an ideal world, I would agree with you. We do however live in the real world.

Please give one example applicable to a reasonable majority of human beings on the planet at this current time where "understanding sums and compounding processes" are a matter of life and death.

did you not read what I wrote? "or quality of life [or quality of death]". Nor did I proclaim that the life-or-death situation was applicable to a majority of people.

Re: A Mathematician’s Lament (2002) [pdf]

#74

Post by nullc »

ctl wrote: post #57

I love this article, but: what can a practicing math teacher take away from it? How can you apply this stuff if you still have to teach a standard curriculum? I'm really asking -- my friend is about to start as a high-school math teacher. I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Loc…

One thing you could do is take an interesting piece of subject matter from later curriculum (like next years or later in the current year) and present it as a puzzle for the students to explore with the inquisitive techniques presented in the paper.

Award participation credit, etc as relevant to help keep people engaged who need it. The fact that it's "future" material can also help students who need the extra goals pay attention.

Most likely you can't actually cover the required curriculum like this— as you note, a lot of things do not lend themselves to compact discovery. (It's all ashame, it's not like the students actually retain into adulthood all those procedures that they don't really understand in any case :( ) But maybe you can still inspire people with a few things which do lend themselves to compact discovery, and that inspiration may also make the rest of the subject more accessible to them. "This things have a reason and a pattern to them, even if I don't know what it is right now."

I had some challenges in math in school because I studied calculus, analysis, linear algebra, discrete math, etc. on my own and would derive solutions— sometimes the same as they wanted me to memorize, sometimes not— on my own instead of memorizing the fixed routines, and this was unwelcome. It would be nice if more teachers made an effort to at least not penalize students that were independently interested.

Re: A Mathematician’s Lament (2002) [pdf]

#75

Post by abecedarius »

whiddershins wrote: post #54

This is how I feel about doing most coding tutorials versus the MIT intro to CS through python, or project euler. When coding is presented as: - here is a problem - how would you solve this problem? - here are some hints to get you started it is incredibly fun for me. when it is presented as: - follow along - look what you did! it can be a bit dry.

Hi whiddershins, here's a little present for you then: https://github.com/darius/regexercise

(It's not quite finished and I'd love to get feedback.)

Re: A Mathematician’s Lament (2002) [pdf]

#76

Post by nileshtrivedi »

john_b wrote: post #40
nsphere wrote: post #5

"There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could al…

I think that language (at least English) fails us here. The word "creative" can be used to mean both "relating to or involving the imagination or original ideas, especially in the production of an artistic work" [1] or "resulting from originality of thought, expression, etc.; imaginative" [2]. It seems that most people identify the word more strongly with the artistic sense of the word, in which creativity is a proxy…

The way I read the essay, it seems to me that math is compatible with [1]. You don't have to follow any given rules. Make up your own ideas and see what they lead to. What if square roots of negative numbers exist? What if the sum of angles of a triangle is not 180 degrees? What if a proposition can be both true and false at the same time?

Re: A Mathematician’s Lament (2002) [pdf]

#77

Post by westoncb »

bridgeyman wrote: post #4

I read this a couple weeks ago and decided to try and find a math book that incorporates some history in it. I found Journey through Genius: The Great Theorems of Mathematics with some great reviews. I am working on an iPad app that pairs up people to mentor each other through books like this. It has video chat and a shared whiteboard, so it is ideally suited for discussing math. If anyone is interested in reading th…

I've spent a lot of time looking for books that'd teach some mathematical ideas while keeping the original progression of motivations/concepts in tact.

I'd first recommend "Men of Mathematics" by E.T. Bell. It's a collection of short biographies on 20 or so Mathematicians, also discussing a few of the most salient points of each's work. It's an enjoyable introduction, useful for getting a broad view of what math is made of and how mathematicians think. Bell was a serious mathematician himself (not of the rank of anyone he's writing about, of course), as well as a sci-fi author apparently :)

edit: could also try "Mathematics and the Imagination" as an alternate introduction.

After that would be "What is Mathematics?," by Richard Courant and Herbert Robbins. This one's a bit tougher, and I have to admit I had the experience of being perplexed at the selection of topics, and that it didn't tell me immediately what mathematics is -- but! Without too much time passing, I now appreciate the selection and think it could be read profitably by trusting that the selection is good and trying to answer the question why that's the case while reading.

At the moment I'm trying my second book from E.T. Bell, The Development of Mathematics, and like it quite a lot so far, though it assumes a little more math knowledge. This one's probably great if you did a mathematics undergrad, or similar, but would like to see the various topics related and given context.

Another I believe worth checking out, if none of the others fits exactly, is William Kingdon Clifford's "Common Sense of the Exact Sciences." I've only skimmed sections in this one, but it looks extremely promising; and from what I've read about it and about Clifford, I think it could be an important piece of pedagogy along the lines of what Lockhart's into. Not too long and pretty accessible I think.

Re: A Mathematician’s Lament (2002) [pdf]

#78

Post by abecedarius »

lmartel wrote: post #52

I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it. I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break i…

http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755... (by guess who) looks like the best starting point since this resonated with you. (I've dipped into it but not seriously tackled it yet.)

It's hard to keep at it, learning on your own. I sometimes find problems where the usual solutions or explanations feel kind of ugly, and try to make them cleaner, like rewriting someone's code. (A couple days ago it was Snell's law: this optics tutorial http://www.bigshotcamera.com/learn/imaging-lens/refraction linked from HN just dropped this formula down, and to most kids it's going to be magic. Can you formulate the law of refraction in a more elementary way and derive it from some simple assumptions? I did come up with a version that never mentions sines, but I'm not really satisfied and it's gone back on the to-do list to try to take it further. See: hard to keep at it.)

More generally, this kind of work can come up all the time when programming if you say, "No, I'm not going to look up the algorithm, I'll work one out for myself and then see what's been done." Occasionally you find something kind of new that way, besides often deepening your appreciation of the usual solutions. For example, last week I found a new way to avoid the epsilon-loops in Thompson's regular-expression search algorithm -- new to me, at least. This has minor significance and came out of a ridiculous amount of work rediscovering things taught in automata-theory classes, but Lockhart wasn't kidding: it's a rush when you figure it out.

Re: A Mathematician’s Lament (2002) [pdf]

#79

Post by nileshtrivedi »

gizmo686 wrote: post #70
ctl wrote: post #57

I love this article, but: what can a practicing math teacher take away from it? How can you apply this stuff if you still have to teach a standard curriculum? I'm really asking -- my friend is about to start as a high-school math teacher. I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Loc…

I was a student from a small private school that literally wrote their own math book, so I have no idea how generally applicable this is. As you suggest, a common technique my teachers employed is setting us loose on problems we did not yet have the tools to easily solve (but which were within reach). We would typically work in small groups, and if necessary the teacher could speed up progress by dropping us hints. W…

Would it be possible to get that math book they wrote and perhaps the teachers' notes to go with it?

Re: A Mathematician’s Lament (2002) [pdf]

#80

Post by westoncb »

lmartel wrote: post #52

I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it. I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break i…

Hey Imartel -- I've been in a similar situation. I think my first starting point was "Mathematics and the Imagination" or "Gödel's Proof." Mathematics and the Imagination is a good high level overview, and would provide some foundational notions that'll reappear repeatedly -- but, it won't given you any practice in mathematical methods. For that I would recommend "What is Mathematics?" by Courant and Robbins. Can be pretty challenging, but you'll actually get somewhere if you put effort into it. If you haven't had much experience with proofs, it's worth focusing briefly on the process of proving explicitly as a preliminary. That's actually a good thing to do with another person or in a class -- can be pretty difficult to get some of the subtleties involved, and to know when you've done things correctly or not.

I mention "Gödel's Proof" because it's the first thing I came across that informed me I am in fact interested in mathematical systems. I'm more interested in architectures than problem solving though. If you suspect that might be the case for yourself, might check it out -- it's about 100 pages.

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