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Too Much Calculus – Gilbert Strang (2001) [pdf]

www-math.mit.edu

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Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#121

Post by xamuel »

mturmon wrote: post #78
xamuel wrote: post #10

(Speaking from my experience as a calc teacher at a big U.S. state U.; mileage may vary elsewhere) Most non-STEM students won't see the 'Calculus I, Calculus II, Calculus III' that starts the essay so dramatically. Increasingly, liberal arts students are able to get by with no more than a "math appreciation" course and maybe what amounts to high school algebra. The "math appreciation" course might actually touch on v…

You sound like exactly the constituency Gil Strang's corrective essay is addressed to. I don't see any of the points you mention as particularly strong. There would be nothing wrong with presenting a more hands-on linear algebra course as a first course for freshmen. (Calling it "matrix arithmetic" does not do the notion service.) The material seems very immediate, and you can use a REPL like octave or matlab to rein…

>a more hands-on linear algebra course as a first course

It is ideal (and indeed, not uncommon) to have linear algebra and calc 1 in parallel --- for special honors classes. It wouldn't be realistic to do it for the general freshman class. Do you realize just how many students that includes, in a big state U.? "a REPL like octave or matlab" would be wonderful, amazing, if it weren't about as realistic as the space elevator. Do you know how many endless headaches there are with software-augmented courses, even when it's just software you run in the browser? I applaud your ambitiousness but let's just say I don't want to be in your shoes when you've just told a thousand 18-year-olds to install Matlab!

>the notion that calculus has had more impact on history/society

Linear algebra is hard to even put your hands on, historically. There was no Isaac Newton figure putting the finishing touches on an amazing breakthrough "Fundamental Theorem of Linear Algebra". It is the book-keeping field of mathematics. "All the numerical modeling that has been done using matrices" almost entirely arises by discretizing differential equations so linear algebra doesn't win any points there.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#122

Post by tel »

romaniv wrote: post #115
peterfirefly wrote: post #84

Earlier quoted context omitted.

But statistics and probability is really just integration. It's either that or stamp collection. And optimization is really just differentiation (+ some geometry). I'd say the problem is too little calculus and not too much.

Statistic teaches you application and helps to develop correct intuition. Can's speak for everyone, but my college algebra courses were 100% abstract stuff detached from reality. All professors had the attitude that once you learn the theory you could apply it to anything. Problem is, it's hard to remember theory if you don't map to something tangible, and it's hard to apply it to reality when you're just used to cru…

Statistics is, classically, more developed in the realm of "applied math" than "pure math" while linear algebra is, again classically, more a "pure math" discipline. The difference in rhetoric you cite falls out of this exactly.

In practice, both have wonderful presence in math both applied and pure. Linear algebra is one form of abstract algebra which is easily one of the most important mathematical disciplines. It's also the characterization and method of manipulating large arrays of numbers in interesting and practical ways. Statistics is the mechanism of interpreting live data under knowledge of uncertainty toward generalization. It's also probably well characterized as a particularly well-behavior substudy of measure theory... the pure mathematical notion of generalized "substance"

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#123

Post by blt »

SixSigma wrote: post #3

I watched Prof. Strang's Linear Algebra courseware, great stuff. He is a wonderful teacher. http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-... As someone who hadn't studied it for 20 years, it was invaluable to revisit the subject as an adult with the ability to reflect "ah, I can use that for... " instead of "Ok but when am I ever going to use this"

Incredible. I learned LA in school but it didn't click 100% until I watched Strang's lectures. I think they are ideal for someone who's already reasonably comfortable with matrices and vectors, but wants to see that higher connection and beauty. His lecture deriving the determinant from a few simple properties is magical to me. His emphasis on the fundamental subspaces of a matrix, and the geometric interpretation over the equation-solving one, are ideal for anyone who thinks visually/spatially.

That was the first online lecture series I ever watched, and I've been hooked ever since. I am overflowing with gratitude that so many professors at the world's best schools have put their lectures online. It is so good for a student like me, who went to a liberal arts school that didn't offer many advanced math or CS courses, and learns better from lectures than books. Strang's Linear Algebra course was one of the first way back in 2002. You're right, he is a wonderful teacher.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#124

Post by UUMMUU »

ekidd wrote: post #7

As a programmer, I've found far more uses for linear algebra than for calculus. It shows up in robotics, in machine learning, in game development, and in many other cool subfields. I didn't get very much out of my university linear algebra course. (It was pure theory with nearly zero applications.) I don't think I really developed a good intuition for linear algebra—and why it's so useful—until I read Jim Hefferon's…

ekidd - thanks for the link!

Also, I know it's not free (the coursera course is free) but it's still a great read and seriously applicable http://codingthematrix.com/

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#125

Post by mafribe »

pmalynin wrote: post #83
xamuel wrote: post #10

(Speaking from my experience as a calc teacher at a big U.S. state U.; mileage may vary elsewhere) Most non-STEM students won't see the 'Calculus I, Calculus II, Calculus III' that starts the essay so dramatically. Increasingly, liberal arts students are able to get by with no more than a "math appreciation" course and maybe what amounts to high school algebra. The "math appreciation" course might actually touch on v…

I must disagree. In my experience Linear Algebra has been so much easier than Calculus because even though at times it can be very abstract it is still more familiar as most of the time you're just solving equations. My math professors have done quite a good job in connecting the two subjects. For example, differentiation and integration of polynomials can be expressed as a linear algebra problem, etc. etc. "The fund…

In what sense is the fundamental theory of algebra (FTA) a theorem of linear algebra? It's rarely proven in linear algebra courses. Most proofs of the FTA involve concepts from the analytical side of mathematics.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#126

Post by SixSigma »

blt wrote: post #123
SixSigma wrote: post #3

I watched Prof. Strang's Linear Algebra courseware, great stuff. He is a wonderful teacher. http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-... As someone who hadn't studied it for 20 years, it was invaluable to revisit the subject as an adult with the ability to reflect "ah, I can use that for... " instead of "Ok but when am I ever going to use this"

Incredible. I learned LA in school but it didn't click 100% until I watched Strang's lectures. I think they are ideal for someone who's already reasonably comfortable with matrices and vectors, but wants to see that higher connection and beauty. His lecture deriving the determinant from a few simple properties is magical to me. His emphasis on the fundamental subspaces of a matrix, and the geometric interpretation ov…

Well, I was going to say "you should also check out Walter Lewin's Physics lectures, they are equally stunning"

But it seems MIT took them down

http://www.reddit.com/r/Physics/comments/2ope15/mit_indefini...

This borders on criminal and is certainly a tragedy.

I'm quite struck and emotional about it.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#127

Post by mafribe »

cageface wrote: post #11

I really can't agree with this strongly enough. I had enough Calculus and Differential Equations courses to kill a horse in college (Chemistry major) and I've never used that knowledge once in my entire programming career. But I find uses for linear algebra and matrices in coding all the time. Thank god Strang's excellent course is available online for free.

In order to understand quantum mechanics, the foundation of modern chemistry, properly, don't you need a strong appreciation of calculus?

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#128

Post by mturmon »

xamuel wrote: post #121
mturmon wrote: post #78

Earlier quoted context omitted.

You sound like exactly the constituency Gil Strang's corrective essay is addressed to. I don't see any of the points you mention as particularly strong. There would be nothing wrong with presenting a more hands-on linear algebra course as a first course for freshmen. (Calling it "matrix arithmetic" does not do the notion service.) The material seems very immediate, and you can use a REPL like octave or matlab to rein…

>a more hands-on linear algebra course as a first course It is ideal (and indeed, not uncommon) to have linear algebra and calc 1 in parallel --- for special honors classes. It wouldn't be realistic to do it for the general freshman class. Do you realize just how many students that includes, in a big state U.? "a REPL like octave or matlab" would be wonderful, amazing, if it weren't about as realistic as the space el…

No, linear algebra does not have an Isaac Newton figure. The historical development was different.

The different history does not bear on the ultimate impact and general utility of the tools offered by calculus or by linear algebra.

Eigenvalues and orthogonality are not just "book-keeping". Why give forth empty slurs like this?

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#129

Post by xamuel »

mturmon wrote: post #128
xamuel wrote: post #121

Earlier quoted context omitted.

>a more hands-on linear algebra course as a first course It is ideal (and indeed, not uncommon) to have linear algebra and calc 1 in parallel --- for special honors classes. It wouldn't be realistic to do it for the general freshman class. Do you realize just how many students that includes, in a big state U.? "a REPL like octave or matlab" would be wonderful, amazing, if it weren't about as realistic as the space el…

No, linear algebra does not have an Isaac Newton figure. The historical development was different. The different history does not bear on the ultimate impact and general utility of the tools offered by calculus or by linear algebra. Eigenvalues and orthogonality are not just "book-keeping". Why give forth empty slurs like this?

Wasn't meant as a slur, I love linear algebra. Especially graduate-level, when you learn about the abstract (commutative diagram) approach to tensor products! :D Current-day impact doesn't retroactively grant historical value, though: the smartphone has not had a big impact on history (yet).

If you can do a decent treatment of eigenvectors in a non-honors freshman course, my hat goes off to you. Even in a 2nd-year course where students are more mature, that's hard (first time I took LA, eigenvectors were rushed at breakneck speed in the very last lecture). And while they're very important, eigenvectors are still a shadow of the historico-cultural impact of calculus.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#130

Post by romaniv »

romaniv wrote: post #115
peterfirefly wrote: post #84

Earlier quoted context omitted.

But statistics and probability is really just integration. It's either that or stamp collection. And optimization is really just differentiation (+ some geometry). I'd say the problem is too little calculus and not too much.

Statistic teaches you application and helps to develop correct intuition. Can's speak for everyone, but my college algebra courses were 100% abstract stuff detached from reality. All professors had the attitude that once you learn the theory you could apply it to anything. Problem is, it's hard to remember theory if you don't map to something tangible, and it's hard to apply it to reality when you're just used to cru…

my college algebra courses

Sorry, I meant my college calculus courses, although incidentally the statement applies to both.

I think colleges would be much better off teaching introductory math as "applied". I got significantly more out of statistics, because that class taught us how to spot problems where that branch of mathematics is applicable. I really liked how Udacity used real-life orbit calculations from various space missions as assignments. It gives the whole thing some context.

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