Rogue Math

[roguemath-blog]

Pining for a Bygone Era

There was once a time when mathematics textbooks were intended to educate, rather than obfuscate.  The modern math text offers no advice, and is little more than an assault upon the senses.  Here is what a good textbook looks like:

(The full text is in the public domain and can be found at https://archive.org/details/CalculusMadeEasy)

[raginrayguns]

the secret tedious math behind famous ppl’s beautifully simple understandings of science

Stephen Wolfram:

I always found it incredible. [Feynman] would start with some problem, and fill up pages with calculations. And at the end of it, he would actually get the right answer!

But he usually wasn’t satisfied with that. Once he’d got the answer, he’d go back and try to figure out why it was obvious.

And often he’d come up with one of those classic Feynman straightforward-sounding explanations. And he’d never tell people about all the calculations behind it. Sometimes it was kind of a game for him: having people be flabbergasted by his seemingly instant physical intuition. Not knowing that really it was based on some long, hard calculation he’d done.

ET Jaynes:

Feller’s perception was so keen that in virtually every problem he was able to see a clever trick; and then gave only the clever trick. So his readers get the impression that

(1) probability theory has no systematic methods; it is a collection of isolated, unrelated clever tricks, each of which works on one problem but not on th enext one; (2) Feller was possessed of superhuman cleverness; (3) only a person with such cleverness can hope to find new useful results in probability theory.

Indeed, clever tricks do have an aesthetic quality that we all appreciate at once. But we doubt whether Feller, or anyone else, was able to see those tricks on first looking at the problem.

Allen Knutson:

More specifically, one thing I learned from Terry [Tao] that I was not taught in school is the importance of bad proofs. I would say “I think this is true”, work on it, see that there was no nice proof, and give up. Terry would say “Here’s a criterion that eliminates most of the problem. Then in what’s left, here’s a worse one that handles most of the detritus. One or two more epicycles. At that point it comes down to fourteen cases, and I checked them.” Yuck. But we would know it was true, and we would move on. (Usually these would get cleaned up a fair bit before publication.)

and then this last one is about songwriting.

Jeff Mangum:

Yeah, usually I create tunes that are fragmented. I think the biggest obstacle for people with their creativity is that they feel they have to sit down and create this finished, polished product. Especially nowadays, it’s so easy to have a library of two thousand CDs, books and records. So many things. We’re used to having all of these finished works of art in our life that seem to arise out of nothing.  I think that so much of the creative process is a fragmentary one, and then it’s about just allowing your intuition to put it together for you. It’s funny how you create something and you think you’re going in a million different directions, and then the thing you end up with is the thing that you wanted to create your whole life, but you’re just as surprised by it as anybody else.

[thousandmaths]

 I’ll add Buckminister Fuller to this list:

When I’m working on a problem, I never think about beauty. I think only how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong.

I don’t have as good a link for that OP has for the others, sadly :(

——

When I first saw this post (probably about two years ago) I saved it in my drafts because the OP’s point is one that intellectually I have always known, but have had a very hard time putting it into practice. 

But as I was cleaning up my drafts folder recently I rediscovered this post, and realized that I had slightly different connections with all of them. So that is what the rest of the post is about.

Wolfram on Feynman:

I’ve mentioned in various places on this blog (these are the only ones I can find at a moment’s notice) that my favorite professor in undergrad was Winston Ou, whose otherwise excellent website does not have two one of the quotes which I remember most fondly. One I’ve already mentioned before*, “To learn, you must be shameless.” The other is relevant to this Wolfram quote: when his advisor was talking to him about how to learn from reading, he stressed that it is not enough to understand the argument. You must reread and reformulate and reinterpret, until you can answer the question: “Why is this completely obvious?”

In the five years since I first heard this story, I have learned that understanding why something is completely obvious is, in fact, very hard work. It’s not work that I have usually wanted to put in. But I do notice that where I have put in this work, I have progressed much further in my understanding. 

(I might also remark that this aesthetic of Feynman’s is very sympathetic with, but not subsumed by, the Grothendieck approach to mathematics.)

Jaynes on Feller:

I think you would be hard pressed to find a single professional combinatorialist who, upon reading this quote, does not immediately think of Tim Gowers’s The Two Cultures of Mathematics. And while there is no stand-out passage from Gowers that this hypothetical person is likely to remember, here is one that perhaps illustrates the connection best:

Combinatorics appears to many to consist of a large number of isolated problems and results […]. Each result individually may well require enormous ingenuity, but ingenious people exist […] and future generations of combinatorialists will not have the time or inclination to read and admire more than a tiny fraction of their output.

For the unaware, Gowers’ piece was specifically written to refute this argument, and others like it, and is probably the most famous of the many essays on combinatorics as respectable mathematics. One of the things that he suggests, is that there are unifying principles in combinatorics— albeit ones which are more obscured and less amenable to formalization than “many” might be used to. These formal difficulties also lead to communication issues between combinatorics and the rest of the community, with remarkable parallels to those that Jaynes finds in Feller.

Knutson on Tao:

I have very little to say about this quote, except that I love it, and I have it stitched on my heart, and there have been times when I remembered this quote specifically which encouraged me to push through with some less-than-pleasant computations.

Magnum on Magnum:

A podcast called Webcomics Weekly explains the prevalence of bad webcomics (which I think applies equally well for bad fanfiction and bad blogs) as a matter of visibility. It is true even for most professional cartoonists, that the amount of shit they have produced surely outweighs the number of quality cartoons they produced by several pounds, optimistically. So the content itself has not changed, but the public’s access has: the (huge quantity of) sludge which used to be hidden, is now prominently displayed on the internet.

My blog is certainly a product of a similar phenomenon happening with mathematical exposition. Before blogs became such a mainstream thing**, where would a no-name grad student like me go to publish long, rambling thoughts on math?

Now, for however bad you think I am at being interesting or coherent, you should check out the first hundred posts on OTAM… I am certainly much better at it than I once was. This improvement was made possible by practice. Without OTAM, it probably would never have happened. I probably would never had reason to step foot in the writing center, and I probably would be a much worse mathematical expositor than I am today. Hence I would be subjecting colleagues to worse talks, and to worse papers.

——

[ * I mentioned it before, and I mentioned it incorrectly: the actual quote is “The key to learning is shamelessness.”, and it is not original to Ou: it is by “C. P. Chou”, according to his admonitions page. ]

[ ** I’m acutely aware that since I am on tumblr, I am writing as a blogger to bloggers— and so we are more likely to inflate the significance or prominence of blogging, which makes any statement like this one suspect. But in the broad sense, a facebook page is essentially a blog, and that alone is just about enough to put blogging comfortably in the mainstream. ]

[ A spare thought: Magnum’s quote can also be read as a recognition that emerging struggle for the public has always been a problem for the artist: here is everyone producing such “finished, polished” work, and here is me producing… something else. Of course this is not what is going on at all, and on some level you know this, but it is hard to remember when you’re in the trenches. This is as true for mathematicians as it is for other artists. ]

[ambivalencerelations]

Oh my god, someone actually put this into words.

A large portion of my work as a calculus tutor is explaining the mechanics of the mathematics when the intuition was used to explain something in class. To the uninitiated, only seeing intuition and general concepts is especially detrimental to their mathematical development. In the end, they are awed by the general Ideas, but they do not really know how to take actual mathematical steps to a soloution. They may feel as if mathematical thought process is supposed to be a flash of elegant and astounding brilliance, as if extreme intelligence is required to understand mathematics. (Sound like a familiar sentiment, anyone?)

I have watched the firsthand and secondhand dangers of hiding ugly mechanics, and it goes like this: a lack of concrete mechanical understanding creates an air of uncertainty. Uncertainty leads to a lack of confidence in intuition and ability. Lack of confidence in your answers creates confusion, and eventually manifests itself as the loss of understanding. Hiding the nitty gritty from the beginner is a pretty good way to ensure they fail.

Now you may be asking, how does one get to the insight and intuition if you make them spend their time doing the mechanical? I know from experience with mathematics and pottery that struggling with a task does develop your intuition and insight. A well timed hint from an instructor always helps, and i’m definitely not suggesting no intuition is a good thing, but when you experience what doesn’t work, you actually get a detailed peek at the inner workings of the subject material. And that detailed peek is that lower level understanding that builds a trustworthy higher level of understanding.

In Conclusion: “Kinda suckin’ at something is the first step to getting kinda good at something” because failure drags out the small inner workings of a subject and shoves them in your fucking face. When you’re finally intimately acquainted with the low level mechanics, you can actually trust your intuition and have it work out well (most of the time). Hiding the mechanics from someone and just presenting them with the intuition is a good way to give them an unsteady footing that they won’t enjoy.

[roguemath-blog]

UT Austin Pure Calculus

Ok, so on further review of the UT Austin “pure mathematics” curriculum, I have come to the conclusion that is in fact “Pure Calculus”, with a little side of other maths thrown in. Really it’s not even that.  It’s Calculus Computation with one theory (read actual) course.   Very disappointing, but I’ll still work through it.  I’ll be a semester behind though as it is also impossible to begin this major in the Spring. Also, they use a very expensive calculus text “Calculus: Early Transcendentals 7e by James Stewart”.  It’s also very difficult to pirate!  Though I found that the solutions manual is readily available all over the web.   Luckily I did finally get a hold of a pdf of the book.  To be fair, this $260.00 purchase would be the text for about a year and a half.  The downside, of course, is that UT Austin math students are students of one book.  VERY DISAPPOINTING!  

[roguemath-blog]

The First Hurdle!

Ok, so as I’m looking at the UT Austin website, I can’t quite get a listing of the courses offered next spring.  Those are behind a login wall where I have to have an EID. Fortunately, they have a mechanism by which I can get a guest EID.  So I’ve applied for one, under an assumed name of course!  I’m guessing that The beginning of the calculus sequence won’t be there though as that’s more of a fall course.  We’ll see though, it is a big school!

[roguemath-blog]

Here are my proposed courses.  I will be stealing the knowledge contained in 11 courses, here they are scrawled on a spiral bound notebook with course titles.  I noticed that their “pure math” seems a little calculus heavy, also a lot of “computation” under the guise of “mathematics”, but I’ll give it a shot!

[roguemath-blog]

I found a set of “Degree Checklists” at https://cns.utexas.edu/degree-checklists.  This is the front page of the degree plan I will illicitly follow.  All of the math courses are on this page, the second one is just a tally of hours as well as some more general graduation requirements.  Using this and their course listing, I have selected my path! (boxed in)

[roguemath-blog]

Stealing a Degree

So I have been thinking about what I wanted to post, ever since my first rant.  Then I decided, why not steal a degree from a prestigious school?  I’ll take their curriculum, look at their course schedule, and do their homework.

For extra measure, I think I’ll even pirate the textbooks!  Now, as I already have a few degrees, I’m just going to give myself transfer credit for all the gen-ed stuff.  I’ll just be doing undergrad coursework from, ohhhh.... say The University of Texas Austin.

They have a BS in “Pure Mathematics”, so I think I’ll use that as my guide.  Of course, I won’t have an advisor so I’ll have to work out my own “schedule”.  I’ll likely not keep pace with their classes, but I suppose if you take future versions of these, this blog could also help you cheat.

So, let’s see just how pure their pure math is!  I’m heading over to https://www.ma.utexas.edu/ to start my stolen academic career! (NOTE: I have never even set foot on the UT Austin Campus.  That’s what makes this so sneaky.)

[roguemath-blog]

On the Lack of Math in School

This blog stems from several discussions I have had with my colleagues.  We frequently discuss math books for use in our classes, and I often bring up one of my biggest complaints about modern math textbooks.  My big complaint is that they are almost completely devoid of text.  Worse, upon closer inspection, they are almost completely lacking in mathematics as well!

Mathematics textbooks of the past did, oddly enough, contain text and explanations of their subjects.  Most modern students haven’t been exposed to one of these mythical books.  If you look at math textbooks, starting at some point in the 1950′s you’ll see that they start to become a little less wordy, and start to have more pictures and gimmicky little typesetting tricks.  The final stage of this progression is a book which is little more than an anthology of rote calculation problems coupled with pictures of bridges and roller coasters.  

The other thing to bear in mind is that this transformation has been a gradual process.  It started with the elementary school texts.  Once those had been compromised, the affliction spread to high school texts.  What we are presently watching is the slow cancerous death of the college textbooks.  You can try an experiment for yourself.  Grab a college textbook in Calculus and read it.  You can’t!  There aren’t paragraphs or sentences.  There are blocks of text, collections of rules, disjointed examples, and then about 50 symbolic sentences for you to play with.  Who could learn from that?  

Take, for example, this segment of a page.  In spite of years of schooling, not to mention several years of using this material on an almost daily basis, this page has completely destroyed my ability to work with limits.  Just imagine the affect it would produce in a novice!

Inevitably, when I bring this up, I am either met with knowing nods, or told that modern students couldn’t cope with actual textbooks.  This latter attitude is largely brought about by the acknowledgement that these students have never had a chance to really try an actual math textbook out.  It creates what one of my colleagues has termed a “Chicken and Egg” type problem.  I suppose that could be the case, but I also think that I am not ready to concede to the loss of mathematics just yet.

Even worse!  What the courses surrounding these horrific books teach is just simple symbolic calculation.  While symbol shuffling is part of mathematics, it is probably the least important part.  We teach the consequence of the subject as thought it were the subject itself!  This would be akin to teaching English majors 4 years of penmanship and then calling them authors.  It is absurd on its face, and I die a little every time a student receives a degree in a mathematics related field when they are taught under such conditions.

I have decided that there is a distinct “anti-math” attitude in modern education.  I suppose I shouldn't be too surprised in that an actual mathematical education would impart honed skills in logic and critical thinking to students.  If there is one thing the dark cabal that controls everything hates, it is an informed populace.  (Or less conspiratorially, people are becoming too lazy for such endeavors.)  Whatever the cause, I have decided that in the face of the present atmosphere, actually teaching mathematics is a subversive act.  If there is one thing I enjoy, it is subversion!  So to that end, I thought I would start writing and linking articles that do just that.  Next week, I’ll start in with the actual math.  I will try to maintain a poor and irreverent attitude throughout the whole experiment.  I hope you enjoy the ride!

Here’s some further reading for you:

An actual calculus book: https://archive.org/details/cu31924031219318 

A much more complete rant about the present lack of mathematics in mathematics education: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

So, what do you think?  Am I right? Am I over-reacting? Let me know!  I hope you enjoyed this rant, and I hope you enjoy the ones that shall follow.