This'll be a short one. But hopefully something I'll comment on more in the future.
I've found a terrific article by Professor Melvin Henriksen published in The Mathematical Intelligencer in 1993 and republished online at the Topology Atlas by him. Henriksen-if you're unfamiliar with him-is currently Professor Emeritus at Harvey Mudd College and he's one of the few remaining active-sort of-mathematicians in one of my favorite areas of mathematics:point set topology. He's also published quite a bit in algebra,has been very active in historical aspects of mathematics and is currently one of the major overseers of the mammoth virtual site Math Forum supported by Drexel University.
The elitism in mathematics is nothing new-nor,sadly,is it unique to mathematics among academic endeavors. Henriksen-never one to keep his opinions to himself-summed up the situation-and the reasons behind it-beautifully and informatively,in this article:
http://at.yorku.ca/t/o/p/c/10.htm
If anything,the situation has worsened since he wrote this article.
When you mention certain branches of mathematics at some unmentionable universities that believe only the half a dozen places on Earth called "Ivies" are where civilized humans exist and everywhere else is untamed jungle with blood drinking,grunting barbarians with pieces of paper masquerading as educated humans-you actually get audible laughter. They don't even try to be kind about it. Why should they? Anyone who can't see that they're right is a fool and won't get promoted anywhere. At least,not if they have anything to say about it-and sadly,they do.
I'm currently investigating the possible role of general topology in additive number theory. To be honest,if arXiv-simply called "Archive" by most of us-didn't exist,I doubt I could publish it anywhere. This remarkable tool has changed publishing forever through open access and free publishing in mathematics and physics with no formal refereeing-and therefore no monkey business. Any attempts to shut it down or seriously regulate it should be met with savage resistance. Such attempts at regulation has already begun in the form of the vote in 2004 by the Archive board to allow only preprints-presumably because copyright issues could arise that could jeapordize corporate profits. The bottom line is that it's begun-the attempts to control it.
(The copyrighting of concepts as property is a terrifying phenomenon that I hope to tackle in depth another time.Suffice to say this is a Pandora's Box that threatens all original thinking if it's not strictly controlled.)
It's a wonderful reality we live in currently in this regard- free publishing. Let's not let the B.A.D. crew and their wealthy allies wreck it like they've wreaked whatever doesn't serve their purposes.
In other words,business as usual on Planet Earth.
Saturday, July 11, 2009
Monday, July 6, 2009
A Brief (Partial) Apology For Speaking out of Turn: Calculus, CirriculaAnd Constudents...............
I'm not usually one to apologize when I feel someone is being a dick.
Anyone that knows me knows that.
But my guilt has gotten the better of me and I think I need to make amends for my last post.
Namely,my swipe at McMasters' University professor James Stewart.
I think I was angry at the decaying civilization around me and I took it out on him.
I'm really apologizing just for one small part of the rant that I felt was beneath me. It was simply untrue and would be very unfair for me to say about someone I've never even heard lecture or speak once.I referred to Stewart as a "grotesquely overpaid hack without an ounce of mathematical talent".
Well,that was completely untrue and unfair:Professor Stewart is actually a very fine teacher and mathematician from what I know of him. (It turns out he's the mathematical grandson of the famous Oxford mathematician E.C.Titchmarsh. I didn't know that and found that kind of interesting in and of itself.)
I dug out my copy of the third edition of his textbook to act physical evidence in this trial of my conscience.I also borrowed a copy of the 6th edition.
He's made a lot of improvements in the text since I used it-a lot more pictures, the exponential and logarithmic functions are introduced and discussed MUCH earlier (in the first chapter,in fact),and in general a lot more explicit focus on the overall process of problem solving,which was only indirectly stated in the edition I'm familiar with. Stewart was a graduate student of George Polya before moving on to get his PhD-the influence of the Stanford problem solving master is all over this textbook in both editions. Stewart approaches calculus as a problem solving enterprise first and foremost-such an approach is bound to be pragmatic and will intentionally sacrifice rigor where it obscures understanding.
In short, Stewart is trying to teach his students how to become intelligent problem solvers above all else. As teachers (and speaking for myself as an aspiring teacher at the college level), discouraging the good intentions behind such an approach is the last thing we should want to do. It's easy to forget how confusing calculus and physics is when one first seriously tackles it as a college undergraduate-or for the more fortunate and/or talented, high school. As a result, it's easy to get on your high horse and badmouth a text like this from the viewpoint of someone who's mastered a good portion of rigorous mathematics. Stewart offers to take the student by the hand and walk him or her step by step through the fog-showing them tricks of the trade along the way and tried-and-true methods of attacking problems in ways that not only obtain solutions,but a complete understanding of the MEANING of what's being asked of them. "What do they want from you?What will satisfy the question?" This is what Stewart is trying to teach with his book.
It's really informative in this regard to read Stewart's own comments on the text from an interview done by the MAA on July 6th,comparing it to the texts he used as a student at Stanford University and The University Of Toronto in the 1960's:
IP: How have mathematics textbooks changed over the years?
JS: Compared with the textbooks that I had as a student, textbooks are so much better now. I don’t know how kids learned from these old books. There was no motivation. It was very austere. You can go too far in the other direction, but the state of the exposition of mathematics is just so much better than it was three decades ago.
As an author of the high school textbooks in the 70s, I kept my eye on trends in education. The new math had been well ensconced by then. But what I observed and decried was the waves, the extremes, the pendulum going back and forth from the new math back to basics. You still see this, especially in the U.S., especially at the high school level, where it is much more virulent. At that time, I longed to get hold of that pendulum and stop it somewhere in a sensible middle. People get too dogmatic.
Even more insightful into Stewart's thinking is his comments on teaching and what he's doing lately:
IP: Are you still teaching?
JS: Although I am Professor Emeritus at McMaster, a year ago I was appointed professor of mathematics at the University of Toronto, and I have twice taught first-year calculus. Although I don’t teach fulltime anymore, I love teaching. Being an author is a pretty solitary, sedentary occupation, so I miss the social aspect—which is teaching. I do it partly to keep in touch with kids, because it brings out the best in me, and to give me new ideas for new editions of my books. This fall I am introducing a new course at the University of Toronto on problem solving. I introduced such a course at McMaster quite some time ago. When I was a graduate student at Stanford I fell under the spell of George Polya, who was retired but used to come in and give these problem-solving talks. He had all of us—teachers and students alike—literally sitting on the edges of our seats with mathematical excitement, presenting data, asking us to make conjectures. The idea is: Suppose you’re faced with a problem that you have never seen before. How do you get started? The first few lectures introduce some basic principles of problem solving. The remaining lectures start with a “problem of the day.” How would you solve it? What strategy would you use? What about trying a special case or solving a simpler problem first? It’s my favorite course to teach.I’m doing that this fall, working with some of the faculty at the University of Toronto so that they can carry on after me. It will be a kind of capstone course. You’re drawing on everything that you’ve learned up to that point, putting it together. There’s no new content whatsoever. But once you take a problem out of the context of a specific course, it becomes harder.
Now that's a book I'd love to read-a problem solving textbook by Stewart that emerges from that course!
But sadly,Stewart seems to miss that the problem with this approach to calculus which has made his book so successful is also why it's damaging to students used by itself.The result of the "practical" nature of the text is that the fact that it's a book on calculus becomes completely incidental.
He never asks the all important "why" questions that brought the real number system and the structure of real analysis into focus for mathematicians in the 19th century.Everything's given a name-Sum Rule,Product Rule,Method of Secants,etc.-which makes them tailor-made for memorization rather then learning.He gives quite good "geometric" explanations-such as a good discussion of motivating the definition of the derivative as the limit of a sequence of secant lines to a point on a curve.But such a discussion is completely independent of the definition of a derivative as a limit.It might as well appear in a book on physics or geometry. As a result, it's all completely mechanical-the fact that it's a book on calculus almost become irrelevant!
And this is the problem he fails to see:To most of today's students,it is irrelevant.You may as well be teaching them how to play checkers and they memorize the rules.
Sure, a few students will really look at the very nice geometrical arguments and walk away really learning something.
But most students-who make up most of today's colleges and whom the university administrators are aiming to sell calculus to-couldn't care less.
I call such students constudents-a hybrid of conmen and students. They aren't interested in learning-in fact, like thieves excited about stealing and not getting caught or cheating husbands who call their wives to tell them they're going to be late while getting oral sex from their mistress-getting an A while never learning a damn thing is exciting to them.
I know what some of you are thinking: "Come on-that's human nature,there's always going to be students like that!"
Sure,of course.
But the big advantage of the rigorous calculus texts of the past was that it was almost impossible for such students to con their way to a good grade-the fact that rigorous mathematics was an essential part of the structure of the course ensured they actually had to learn something to do reasonably well. And the course acted to ensure that students with impure motives who didn't even try didn't get good grades.
Books like Stewart's have eliminated this fail-safe altogether.
I remember as a premed sitting around with a number of students taking calculus using Stewart and the discussion of the exam was like they were talking about a football game and how they were going to "beat" the exam. They came up with codes,mnemonics,word games-not a single theorem or concept or proof. I made the idiotic mistake of asking if anyone actually learned the material and the whole table erupted with laughter. The President of the Student Medical Association smiled at me like The Grinch.
"Winning is about APPEARING to know what you're doing,not actually doing it.Don't worry-you can always work taking out the trash in my office on 5th avenue."
Our society rewards this kind of behavior.Why?Because letting these monsters use Stewart and get thier A's without learning anything is good for business,that's why. The university gets to pack the classes with 200 paying students by making this a required course,the students get thier A's which the college can use to improve it's ranking standing so that administrators get promoted for making so much money and helping public relations and off they go to Ivy League medical schools thinking urea is made in the kidney-and worse,not giving a shit.
And 5 years later they're killing and crippling patients left and right and being acquitted at malpractice trials because the only one in the room who's a better liar then they are is the son of a bitch defending them.
A book like Stewart's ENABLES this kind of system.
I have no problem with Stewart wanting to make the book of a problem solving nature-as I've said,this leads to the book having many positive qualities.My problem is that including mathematical rigor need not be contrarian to this intention and for someone claiming to be so devout to teaching, Stewart refuses to acknowledge this.
Sadly, I think he's too smart not to see this. I think his position is one of willful ignorance in a corrupt academic culture that's made him not only very wealthy for his occupation, but very famous. I doubt anyone outside of McMaster would have ever heard of him without this text.
And I stand by my earlier criticism of Stewart of his ridiculous excess with his own concert hall.
He loves music, fine. Bless him. But spending more on his hobby then 5 families spend on their homes is nauseating and he should be ashamed of himself.
Of course,he's hardly alone in that in this day and age.
But he's an academic. He should know better.
Frankly,I think he does and his own words betray this:
When I started writing my first book, I had no idea you could make any money writing books. That was not a motivation at all. It was a surprise, but it enabled me to build this house. And I’ve got to continue to work to pay for the house. The house’s cost [$24 million] is double the original estimates.
It sounds like he has a very strong motivation for continuing to enable the sharks. Amazing what people are able to justify to themselves.
I hope Dr.Stewart keeps making money and succeeds in paying for his house so his heirs have the proceeds from using it as a tourist trap when he passes away. I hope all his kids and grandkids go to Harvard from it and maybe follow in his footsteps as a teacher instead of becoming criminal defense attorneys and bankers as the later generations usually do when the first generation creates a fortune for them. And I hope a lot of teachers of calculus use it as a supplementary reference or secondary source for their calculus courses and as the main text in high school courses.
I just hope one day someone has the balls to challenge the American way someday and writes the text that replaces Stewart by combining mathematical rigor with his teaching skills to give us a calculus text for students and not con-students.
And I hope I and my loved ones are never at the mercy of the enabled MDs in a hospital with their lawyers' number constantly in their back pocket.
Welcome To The Twilight Jungle.
Abandon All Honesty And Integrity Ye That Enter Here.................
Anyone that knows me knows that.
But my guilt has gotten the better of me and I think I need to make amends for my last post.
Namely,my swipe at McMasters' University professor James Stewart.
I think I was angry at the decaying civilization around me and I took it out on him.
I'm really apologizing just for one small part of the rant that I felt was beneath me. It was simply untrue and would be very unfair for me to say about someone I've never even heard lecture or speak once.I referred to Stewart as a "grotesquely overpaid hack without an ounce of mathematical talent".
Well,that was completely untrue and unfair:Professor Stewart is actually a very fine teacher and mathematician from what I know of him. (It turns out he's the mathematical grandson of the famous Oxford mathematician E.C.Titchmarsh. I didn't know that and found that kind of interesting in and of itself.)
I dug out my copy of the third edition of his textbook to act physical evidence in this trial of my conscience.I also borrowed a copy of the 6th edition.
He's made a lot of improvements in the text since I used it-a lot more pictures, the exponential and logarithmic functions are introduced and discussed MUCH earlier (in the first chapter,in fact),and in general a lot more explicit focus on the overall process of problem solving,which was only indirectly stated in the edition I'm familiar with. Stewart was a graduate student of George Polya before moving on to get his PhD-the influence of the Stanford problem solving master is all over this textbook in both editions. Stewart approaches calculus as a problem solving enterprise first and foremost-such an approach is bound to be pragmatic and will intentionally sacrifice rigor where it obscures understanding.
In short, Stewart is trying to teach his students how to become intelligent problem solvers above all else. As teachers (and speaking for myself as an aspiring teacher at the college level), discouraging the good intentions behind such an approach is the last thing we should want to do. It's easy to forget how confusing calculus and physics is when one first seriously tackles it as a college undergraduate-or for the more fortunate and/or talented, high school. As a result, it's easy to get on your high horse and badmouth a text like this from the viewpoint of someone who's mastered a good portion of rigorous mathematics. Stewart offers to take the student by the hand and walk him or her step by step through the fog-showing them tricks of the trade along the way and tried-and-true methods of attacking problems in ways that not only obtain solutions,but a complete understanding of the MEANING of what's being asked of them. "What do they want from you?What will satisfy the question?" This is what Stewart is trying to teach with his book.
It's really informative in this regard to read Stewart's own comments on the text from an interview done by the MAA on July 6th,comparing it to the texts he used as a student at Stanford University and The University Of Toronto in the 1960's:
IP: How have mathematics textbooks changed over the years?
JS: Compared with the textbooks that I had as a student, textbooks are so much better now. I don’t know how kids learned from these old books. There was no motivation. It was very austere. You can go too far in the other direction, but the state of the exposition of mathematics is just so much better than it was three decades ago.
As an author of the high school textbooks in the 70s, I kept my eye on trends in education. The new math had been well ensconced by then. But what I observed and decried was the waves, the extremes, the pendulum going back and forth from the new math back to basics. You still see this, especially in the U.S., especially at the high school level, where it is much more virulent. At that time, I longed to get hold of that pendulum and stop it somewhere in a sensible middle. People get too dogmatic.
Even more insightful into Stewart's thinking is his comments on teaching and what he's doing lately:
IP: Are you still teaching?
JS: Although I am Professor Emeritus at McMaster, a year ago I was appointed professor of mathematics at the University of Toronto, and I have twice taught first-year calculus. Although I don’t teach fulltime anymore, I love teaching. Being an author is a pretty solitary, sedentary occupation, so I miss the social aspect—which is teaching. I do it partly to keep in touch with kids, because it brings out the best in me, and to give me new ideas for new editions of my books. This fall I am introducing a new course at the University of Toronto on problem solving. I introduced such a course at McMaster quite some time ago. When I was a graduate student at Stanford I fell under the spell of George Polya, who was retired but used to come in and give these problem-solving talks. He had all of us—teachers and students alike—literally sitting on the edges of our seats with mathematical excitement, presenting data, asking us to make conjectures. The idea is: Suppose you’re faced with a problem that you have never seen before. How do you get started? The first few lectures introduce some basic principles of problem solving. The remaining lectures start with a “problem of the day.” How would you solve it? What strategy would you use? What about trying a special case or solving a simpler problem first? It’s my favorite course to teach.I’m doing that this fall, working with some of the faculty at the University of Toronto so that they can carry on after me. It will be a kind of capstone course. You’re drawing on everything that you’ve learned up to that point, putting it together. There’s no new content whatsoever. But once you take a problem out of the context of a specific course, it becomes harder.
Now that's a book I'd love to read-a problem solving textbook by Stewart that emerges from that course!
But sadly,Stewart seems to miss that the problem with this approach to calculus which has made his book so successful is also why it's damaging to students used by itself.The result of the "practical" nature of the text is that the fact that it's a book on calculus becomes completely incidental.
He never asks the all important "why" questions that brought the real number system and the structure of real analysis into focus for mathematicians in the 19th century.Everything's given a name-Sum Rule,Product Rule,Method of Secants,etc.-which makes them tailor-made for memorization rather then learning.He gives quite good "geometric" explanations-such as a good discussion of motivating the definition of the derivative as the limit of a sequence of secant lines to a point on a curve.But such a discussion is completely independent of the definition of a derivative as a limit.It might as well appear in a book on physics or geometry. As a result, it's all completely mechanical-the fact that it's a book on calculus almost become irrelevant!
And this is the problem he fails to see:To most of today's students,it is irrelevant.You may as well be teaching them how to play checkers and they memorize the rules.
Sure, a few students will really look at the very nice geometrical arguments and walk away really learning something.
But most students-who make up most of today's colleges and whom the university administrators are aiming to sell calculus to-couldn't care less.
I call such students constudents-a hybrid of conmen and students. They aren't interested in learning-in fact, like thieves excited about stealing and not getting caught or cheating husbands who call their wives to tell them they're going to be late while getting oral sex from their mistress-getting an A while never learning a damn thing is exciting to them.
I know what some of you are thinking: "Come on-that's human nature,there's always going to be students like that!"
Sure,of course.
But the big advantage of the rigorous calculus texts of the past was that it was almost impossible for such students to con their way to a good grade-the fact that rigorous mathematics was an essential part of the structure of the course ensured they actually had to learn something to do reasonably well. And the course acted to ensure that students with impure motives who didn't even try didn't get good grades.
Books like Stewart's have eliminated this fail-safe altogether.
I remember as a premed sitting around with a number of students taking calculus using Stewart and the discussion of the exam was like they were talking about a football game and how they were going to "beat" the exam. They came up with codes,mnemonics,word games-not a single theorem or concept or proof. I made the idiotic mistake of asking if anyone actually learned the material and the whole table erupted with laughter. The President of the Student Medical Association smiled at me like The Grinch.
"Winning is about APPEARING to know what you're doing,not actually doing it.Don't worry-you can always work taking out the trash in my office on 5th avenue."
Our society rewards this kind of behavior.Why?Because letting these monsters use Stewart and get thier A's without learning anything is good for business,that's why. The university gets to pack the classes with 200 paying students by making this a required course,the students get thier A's which the college can use to improve it's ranking standing so that administrators get promoted for making so much money and helping public relations and off they go to Ivy League medical schools thinking urea is made in the kidney-and worse,not giving a shit.
And 5 years later they're killing and crippling patients left and right and being acquitted at malpractice trials because the only one in the room who's a better liar then they are is the son of a bitch defending them.
A book like Stewart's ENABLES this kind of system.
I have no problem with Stewart wanting to make the book of a problem solving nature-as I've said,this leads to the book having many positive qualities.My problem is that including mathematical rigor need not be contrarian to this intention and for someone claiming to be so devout to teaching, Stewart refuses to acknowledge this.
Sadly, I think he's too smart not to see this. I think his position is one of willful ignorance in a corrupt academic culture that's made him not only very wealthy for his occupation, but very famous. I doubt anyone outside of McMaster would have ever heard of him without this text.
And I stand by my earlier criticism of Stewart of his ridiculous excess with his own concert hall.
He loves music, fine. Bless him. But spending more on his hobby then 5 families spend on their homes is nauseating and he should be ashamed of himself.
Of course,he's hardly alone in that in this day and age.
But he's an academic. He should know better.
Frankly,I think he does and his own words betray this:
When I started writing my first book, I had no idea you could make any money writing books. That was not a motivation at all. It was a surprise, but it enabled me to build this house. And I’ve got to continue to work to pay for the house. The house’s cost [$24 million] is double the original estimates.
It sounds like he has a very strong motivation for continuing to enable the sharks. Amazing what people are able to justify to themselves.
I hope Dr.Stewart keeps making money and succeeds in paying for his house so his heirs have the proceeds from using it as a tourist trap when he passes away. I hope all his kids and grandkids go to Harvard from it and maybe follow in his footsteps as a teacher instead of becoming criminal defense attorneys and bankers as the later generations usually do when the first generation creates a fortune for them. And I hope a lot of teachers of calculus use it as a supplementary reference or secondary source for their calculus courses and as the main text in high school courses.
I just hope one day someone has the balls to challenge the American way someday and writes the text that replaces Stewart by combining mathematical rigor with his teaching skills to give us a calculus text for students and not con-students.
And I hope I and my loved ones are never at the mercy of the enabled MDs in a hospital with their lawyers' number constantly in their back pocket.
Welcome To The Twilight Jungle.
Abandon All Honesty And Integrity Ye That Enter Here.................
Sunday, July 5, 2009
A Brief Ode To Stewart's Calculus-NOT.........................
I just read a really funny post at Ars Mathematica and had to share it with all of you with commentary. Apparently, the question's come up with what McMaster's University's self-made gazillionare James Stewart did with all his royalites from the famous/infamous-depending on how much you care about mathematics-calculus book every other university's department uses.
Apparently, he built a gigantic house with his own personal concert hall in the middle of it. You don't believe me? See for yourself :
http://online.wsj.com/article_email/SB123872378357585295-lMyQjAxMDI5MzA4NDcwMjQzWj.html
This was so he-a trained violinist-could perform with his friends in the privacy and comfort of his own mansion. Talk about hubris worthy of being struck down by the Gods with the Ceres asteroid.
Apparently the only way Stewart's fragile self esteem could make it as a violinist in a concert hall was to have one built for himself where he'd be the star of the show every single night.
Sigh.
Only in America would that seem like a logical action and not a gigantic excess of self-centered indulgence. I recently passed-on my way to the bus-a recently homeless family of 4 living in their car with their 4 year old daughter crying to the mother, "Mommy,what happened to my bed?"
Meanwhile, this grotesquely overpaid hack without an ounce of mathematical talent is spending 3 times what these poor people's former house was worth because he doesn't want to embarrass himself in public with his violin playing.................
But be that as it may-I was honestly asked:How bad is Stewart's book and what are some of your favorite texts?What would you use to teach calculus given the chance?
Well,sadly,since I was a complete imbecile in high school and didn't know grades mattered in life-and my parents being laborers,well,they didn't know either-I ended up at The City University Of New York instead of a real college.(I made many friends there and learned a lot-but let no one be deceived my lack of pedigree and relatively advanced age will give me a huge battle ahead for any degree of success.)
So my first exposure to calculus was Stewart.
In all fairness,I was being a bit disingenuous up to this point. It's not as bad as many people make it out to be. The real positive about the book is the immense number of exercises with complete solutions.
Unfortunately,that's a double edged sword and it's the main reason it's completely unpalatable for mathematicians:It reduces calculus to a step-by-step, plug-and-chug bag of techniques without any real mathematical insight or thinking. Anything that requires more thought then a baboon is either completely omitted,put in rushed optional sections or shunted to a mythical "advanced calculus" course.
The students don't have to do any real thinking at all. Which is why most students-particularly the non-science majors-love it,of course.
Let's face it-that's why the bottom feeding universities buy it every year-so the premeds,accounting students,actuaries,pre-law and all the rest of the master cheaters that form the vast majority of bodies filling the enormous lecture halls of the average 200 student calculus course can program the solutions of all their exams into their programmable calculators.
"This is Anerica. Let the Japanese waste their time thinking and just give me my f***ing A so I can go out and screw people over for 6 figures a year working for Goldman-Sachs,geek."
It's also why Stewart would never have become so absurdly wealthy writing a book that is the very pinnacle of mediocrity in any other academic system but America's.
It's why a piece of crap like Charmed was on for 7 years while great shows like Farscape vanish, why Transformers:The Revenge Of The Fallen-with a mindless plot and racist "black" Autobots-is the #1 film in America at this writing.
It's why we sold our blood won freedoms to a stupid evil Texan from a rich family we elected king for the illusion of safety while Americans lost the entire Bill of Rights for 8 years.
"Americans aren't stupid!"
Really? You must be living in a different USA then I am.
So it goes.
My favorites? Well,when anyone tells you Micheal Spivak's Calculus
is the best calculus book ever-ever-it's really hard to argue. It's incredibly beautiful and a model of clarity. But much more then that,with every word,picture and exercise, Spivak asks the reader to think about the concepts before him or her before setting the task of doing it. Really THINK about it.
Is it too hard for the average student?
Well,depends on what you mean by the average student.
The average student cheating their way through every homework and test and sleeping with TAs to get a 4.0 to get into Harvard medical school,sure.
But if you're talking about a typical smart and curious undergraduate student-not necessarily a mathematics or natural science student-who reads everything with a real effort,wonders and asks real questions even if they don't understand or particularly like it because they're there to learn something?
I think they can.
Yes,it would be a struggle-particularly in learning how to calculate with epsilonic limit arguments for the first time. But with a good,patient teacher by their side, they could definitely get through it.In the process,they'd learn an enormous amount-not only about calculus, but logical thinking and problem solving. Which is useful in all walks of life, not just the sciences-and they'd be all the better for it.
For the mathematically talented, the book will become a treasured keepsake for a lifetime.The chapter on infinite series alone is worth photocopying and keeping.
I refuse to recommend soft,"applied" books.To me,the pure/applied mathematics distinction is a symptom of the problem above. There is no pure math or applied math-there is only mathematics. If you don't realize that,you're not part of the solution,you're part of the problem. That being said-the main problem with using Spivak is that he has virtually no applications-just one lame application of vector algebra to celestial mechanics late in the book. The main point of calculus is to calcul-ATE. Theory is important and all well and good, but teaching calculus as real analysis completely devoid of application is a little like teaching music students the complete mechanics of writing scores and symphonies,but never teaching them how to play!!!!
A book that fascinates me and I'd love to try using for a basic calculus course one day is Donald Estep's Practical Analysis In One Variable
. Estep,a numerical analyst, teaches a basic real analysis course combined with a basic calculus course, using numerical methods to motivate the rigorous development of the real numbers via Cauchy sequences of rationals and epsilon-delta arguments-with dozens of actual real-world examples from chemistry and physics!!! It's precisely the kind of book I wish there were more of-a book that combines application and fully careful mathematical development. I'd be a little scared to use the book,though-Estep makes a couple of really strange choices. The biggest one is deciding not to discuss infinite series. To Estep, infinite series is best done with complex variables,so he decides to omit them altogether. Huh?!? I hope there's a second edition where he adds a chapter on infinite series. Still, it's a relatively minor flaw in an altogether marvelous book that should be in everyone's library who loves calculus.
My favorite all around calculus book is a nearly forgotten one by a legendary teacher-Calculus by Edwin E.Moise
.
It's based on both the regular and honors versions of the course in calculus that Moise taught for many years at Harvard and won several awards for. It's completely rigorous, yet beautifully intuitive with many,many pictures and geometric insight motivated using Euclidean geometry such as lines,planes and conic sections, as well as many physical applications. It's not quite as rigorous as Estep or Spivak,but it is considerably more careful then the average calculus book.
It breaks my heart this book is out of print and I'd love to republish it myself one day. This is the book I would use to teach my children calculus.Go to the library and check it out for yourself if you're disappointed with the ton of fluff the departments are trying to push on you to teach calculus with.
You'll thank me later,I promise.
Stewart and his private concert hall.Yet another example we are living in the era of the barbarians at the gate. It's so frustrating-with no address,you can't even drive by and throw a firebomb through his window to burn it down...........LOL
Apparently, he built a gigantic house with his own personal concert hall in the middle of it. You don't believe me? See for yourself :
http://online.wsj.com/article_email/SB123872378357585295-lMyQjAxMDI5MzA4NDcwMjQzWj.html
This was so he-a trained violinist-could perform with his friends in the privacy and comfort of his own mansion. Talk about hubris worthy of being struck down by the Gods with the Ceres asteroid.
Apparently the only way Stewart's fragile self esteem could make it as a violinist in a concert hall was to have one built for himself where he'd be the star of the show every single night.
Sigh.
Only in America would that seem like a logical action and not a gigantic excess of self-centered indulgence. I recently passed-on my way to the bus-a recently homeless family of 4 living in their car with their 4 year old daughter crying to the mother, "Mommy,what happened to my bed?"
Meanwhile, this grotesquely overpaid hack without an ounce of mathematical talent is spending 3 times what these poor people's former house was worth because he doesn't want to embarrass himself in public with his violin playing.................
But be that as it may-I was honestly asked:How bad is Stewart's book and what are some of your favorite texts?What would you use to teach calculus given the chance?
Well,sadly,since I was a complete imbecile in high school and didn't know grades mattered in life-and my parents being laborers,well,they didn't know either-I ended up at The City University Of New York instead of a real college.(I made many friends there and learned a lot-but let no one be deceived my lack of pedigree and relatively advanced age will give me a huge battle ahead for any degree of success.)
So my first exposure to calculus was Stewart.
In all fairness,I was being a bit disingenuous up to this point. It's not as bad as many people make it out to be. The real positive about the book is the immense number of exercises with complete solutions.
Unfortunately,that's a double edged sword and it's the main reason it's completely unpalatable for mathematicians:It reduces calculus to a step-by-step, plug-and-chug bag of techniques without any real mathematical insight or thinking. Anything that requires more thought then a baboon is either completely omitted,put in rushed optional sections or shunted to a mythical "advanced calculus" course.
The students don't have to do any real thinking at all. Which is why most students-particularly the non-science majors-love it,of course.
Let's face it-that's why the bottom feeding universities buy it every year-so the premeds,accounting students,actuaries,pre-law and all the rest of the master cheaters that form the vast majority of bodies filling the enormous lecture halls of the average 200 student calculus course can program the solutions of all their exams into their programmable calculators.
"This is Anerica. Let the Japanese waste their time thinking and just give me my f***ing A so I can go out and screw people over for 6 figures a year working for Goldman-Sachs,geek."
It's also why Stewart would never have become so absurdly wealthy writing a book that is the very pinnacle of mediocrity in any other academic system but America's.
It's why a piece of crap like Charmed was on for 7 years while great shows like Farscape vanish, why Transformers:The Revenge Of The Fallen-with a mindless plot and racist "black" Autobots-is the #1 film in America at this writing.
It's why we sold our blood won freedoms to a stupid evil Texan from a rich family we elected king for the illusion of safety while Americans lost the entire Bill of Rights for 8 years.
"Americans aren't stupid!"
Really? You must be living in a different USA then I am.
So it goes.
My favorites? Well,when anyone tells you Micheal Spivak's Calculus
is the best calculus book ever-ever-it's really hard to argue. It's incredibly beautiful and a model of clarity. But much more then that,with every word,picture and exercise, Spivak asks the reader to think about the concepts before him or her before setting the task of doing it. Really THINK about it.
Is it too hard for the average student?
Well,depends on what you mean by the average student.
The average student cheating their way through every homework and test and sleeping with TAs to get a 4.0 to get into Harvard medical school,sure.
But if you're talking about a typical smart and curious undergraduate student-not necessarily a mathematics or natural science student-who reads everything with a real effort,wonders and asks real questions even if they don't understand or particularly like it because they're there to learn something?
I think they can.
Yes,it would be a struggle-particularly in learning how to calculate with epsilonic limit arguments for the first time. But with a good,patient teacher by their side, they could definitely get through it.In the process,they'd learn an enormous amount-not only about calculus, but logical thinking and problem solving. Which is useful in all walks of life, not just the sciences-and they'd be all the better for it.
For the mathematically talented, the book will become a treasured keepsake for a lifetime.The chapter on infinite series alone is worth photocopying and keeping.
I refuse to recommend soft,"applied" books.To me,the pure/applied mathematics distinction is a symptom of the problem above. There is no pure math or applied math-there is only mathematics. If you don't realize that,you're not part of the solution,you're part of the problem. That being said-the main problem with using Spivak is that he has virtually no applications-just one lame application of vector algebra to celestial mechanics late in the book. The main point of calculus is to calcul-ATE. Theory is important and all well and good, but teaching calculus as real analysis completely devoid of application is a little like teaching music students the complete mechanics of writing scores and symphonies,but never teaching them how to play!!!!
A book that fascinates me and I'd love to try using for a basic calculus course one day is Donald Estep's Practical Analysis In One Variable
. Estep,a numerical analyst, teaches a basic real analysis course combined with a basic calculus course, using numerical methods to motivate the rigorous development of the real numbers via Cauchy sequences of rationals and epsilon-delta arguments-with dozens of actual real-world examples from chemistry and physics!!! It's precisely the kind of book I wish there were more of-a book that combines application and fully careful mathematical development. I'd be a little scared to use the book,though-Estep makes a couple of really strange choices. The biggest one is deciding not to discuss infinite series. To Estep, infinite series is best done with complex variables,so he decides to omit them altogether. Huh?!? I hope there's a second edition where he adds a chapter on infinite series. Still, it's a relatively minor flaw in an altogether marvelous book that should be in everyone's library who loves calculus.
My favorite all around calculus book is a nearly forgotten one by a legendary teacher-Calculus by Edwin E.Moise
.It's based on both the regular and honors versions of the course in calculus that Moise taught for many years at Harvard and won several awards for. It's completely rigorous, yet beautifully intuitive with many,many pictures and geometric insight motivated using Euclidean geometry such as lines,planes and conic sections, as well as many physical applications. It's not quite as rigorous as Estep or Spivak,but it is considerably more careful then the average calculus book.
It breaks my heart this book is out of print and I'd love to republish it myself one day. This is the book I would use to teach my children calculus.Go to the library and check it out for yourself if you're disappointed with the ton of fluff the departments are trying to push on you to teach calculus with.
You'll thank me later,I promise.
Stewart and his private concert hall.Yet another example we are living in the era of the barbarians at the gate. It's so frustrating-with no address,you can't even drive by and throw a firebomb through his window to burn it down...........LOL
Monday, January 26, 2009
Re:The Vampires Of American Medicine And WTF Does "Well Defined" Mean?!?
So much for posting regularly at this blog.I may as well just shut it down and start again.
But I won't. I WILL keep trying to post on a regular basis for the rest of the summer until the blog catches on. Or it doesn't. A blog is for the author,no one else.Anyone else reads it,that's a plus.
I AM hoping it does catch on,though. I have a lot of thoughts on many things ongoing-but now's not the time. If anything, small posts will begin appearing regularly.
This summer-my last one before applying to PHD programs has not gone well. Sleep has eluded me for the better part of a month-stolen by gut pain combined with frequent urination. And the wonderful health care system of America has assured my internest can't see me.
I don't have a right to live according to the AMA, you see-not enough money to buy good health.
That's why they let my father die of agonizing prostate cancer at the end-they crunched the numbers and thier profits simply outwieghed my dad's treatment. So they gave us the bullshit story that "There's nothing more we can do." The cancer metastisized througout his bones over his last few weeks, giving him a death you wouldn't wish on Bernie Madoff.
Meanwhile, if he was a drug kingpin who dropped off 5 million in CASH,I wonder if a miraculous treatment they suddenly remembered about would have appeared and extended his life by 5-10 years. Since corporations now control the publication of most medical research as well as the mass media, we'd never know if one existed no matter how much you researched.
I can get fully into this here, but I WILL say this: The fact that Yale Medical School considered seriously adding ACTING CLASSES to it's required cirriculia for the M.D. for all students entering after 2011 to "improve maximally productive patient-practitioner interaction"(translation:to make the doctors the best con-artists possible) speaks volumes of the age of medicine we live in-and why I turned my back on that world years ago. I consider myself VERY lucky to have good and trustworthy doctors-but I can't tell you how hard my family searched to find them.
100 monsters for every one like them.
"We're coming for your money and we'll GET it all. We're the only real winners.The players don't stand a chance." -from the screenplay of Martin Scorsese's CASINO
Changing the subject to something mathematical, something on the web caught my eye yesterday and I just need to share it with the house. Ever wonder what well-defined means? It's amazing how many graduate students-particularly those working in category theory and the higher altitudes of algebra,where the phrase probably comes up most-never ask what that means. It's kind of accepted everyone "sorta" knows what it means. And for most people,that's good enough.
I remember the first time I ever wondered about it-it was in Kenneth Kramer's honors abstract algebra course a few years ago at Queens College. He was sketching the proof of Cayley's theorum on the fact that every group is the same as some group of permutations on a set (i.e. they're isomorphic). ( Actually, he wasn't proving it,he just wanted to sketch the proof because he'd rather spend the classes' time developing the theory of group actions on a set, of which Cayley's theorum is a special case-i.e. a group acting on itself. But I digress.............)
He was constructing the composition map which is the isomorphism of a group G onto it's corresponding permutation group acting on it's underlying set S -I forget what he denoted it as,call it P(S). He commented the map was clearly well-defined. I raised my hand in frustration since I'd asked the question before and never gotten a straight answer from any professor (some of them actually got annoyed with it and made unkind remarks about my age as a student)
What followed was one of the most impressionable moments of my student career as Dr.Kramer and I exchanged comments on what exactly it meant to be well-defined. "It means it's not ambiguous what the value assigned is, Andrew-that we don't get 2 values for the same arguement." "Oh, you mean the relation actually specifies a function?" "Well, not exactly-if the formula IS a function, you're absolutely right. But this may not be a function and still be a well defined mathematical object." I didn't get it. After a few minutes of him giving a few examples, no progress was made. He ultimately asked me to table the question so we don't waste any more of the classes' time.
I did so,but ultimately,it disturbed me. Dr.Kramer is a gifted teacher on all matters mathematical-an early student of John Tate's at Harvard-and usually the most pleasant and patient of people with even the stupidest of students' questions. In fact,I'll be taking a course on elliptic functions with him at the City University Of New York Graduate Center this fall. The sheer wieght of the subsequent coursework-the first 6 chapters of Herstein's classic Topics In Algebra in a VERY intensive 2 semesters,plus his own notes-prevented us from broaching the subject further. All that really got settled was that it was pretty clear what "well-defined" meant if the object under consideration was a function-in fact, it's almost redundant. But how would you describle a general mathematical object as being "well defined"?
Leave it to Tom Gowers to make everybody happy.
There are several blogs online I try so hard not to miss. Peter Woit's Not Even Wrong, Terrance Tao's, John Baez's The N-Category Cafe' , The Secret Blogging Seminar and a few others. But nothing matches Gower's blog for sheer beauty of writing and thinking about mathematics. A lot of people can do mathematics, a lot more people can teach mathematics, and even more people can talk about mathematics .(Sadly, this is whether or not they know what the fuck they're talking about or not...........)
There are so few who can do all of the above.
Elias Stien can do it (sometimes).
Melvyn Nathanson can do it.
James Stasheff can do it. Better then anyone I've ever heard.
William Thurston can do it.
But for my money, no one does it better currently and consistently then Tom Gowers. His blog should be required reading for all mathematicans and serious math students. (By the way-his old teacher at Cambridge, Bela Bollabos-is also great at all of the above. I doubt that's an accident. )
Anywho, I was reading Gowers' blog and low and behold, Gowers also wanted to know, after grading the exams for the year at Cambridge and discovering NONE of his students understood it,either-what's it mean for something to be well defined?
People who know me know I'm Socratic to a fault, to the point of making people violent. I almost NEVER agree with EVERYTHING someone says.
But this is rare occasion when I'm speechless with complete conviction and agreement with someone else's analysis. As I said, leave it to Gowers to give the perfect answer to a great question.
I'll simply let the beauty,depth and simplicity of Gowers' blogpost speak for itself-I simply have nothing to add to it. Nothing at all. Anyone asks me this question in the future, I'll simply give them a copy of Gowers' post. For all basic mathematical discussions that may come up in the future, I seriously doubt anyone can debunk this discussion.
It's THAT good.
Oh,screw the self-engratiating pontification,here's Gowers. And if you don't bookmark his blog, shame on you.
Good night to all,fellow travelers. Until next time.
http://gowers.wordpress.com/2009/06/08/why-arent-all-functions-well-defined/#more-605
But I won't. I WILL keep trying to post on a regular basis for the rest of the summer until the blog catches on. Or it doesn't. A blog is for the author,no one else.Anyone else reads it,that's a plus.
I AM hoping it does catch on,though. I have a lot of thoughts on many things ongoing-but now's not the time. If anything, small posts will begin appearing regularly.
This summer-my last one before applying to PHD programs has not gone well. Sleep has eluded me for the better part of a month-stolen by gut pain combined with frequent urination. And the wonderful health care system of America has assured my internest can't see me.
I don't have a right to live according to the AMA, you see-not enough money to buy good health.
That's why they let my father die of agonizing prostate cancer at the end-they crunched the numbers and thier profits simply outwieghed my dad's treatment. So they gave us the bullshit story that "There's nothing more we can do." The cancer metastisized througout his bones over his last few weeks, giving him a death you wouldn't wish on Bernie Madoff.
Meanwhile, if he was a drug kingpin who dropped off 5 million in CASH,I wonder if a miraculous treatment they suddenly remembered about would have appeared and extended his life by 5-10 years. Since corporations now control the publication of most medical research as well as the mass media, we'd never know if one existed no matter how much you researched.
I can get fully into this here, but I WILL say this: The fact that Yale Medical School considered seriously adding ACTING CLASSES to it's required cirriculia for the M.D. for all students entering after 2011 to "improve maximally productive patient-practitioner interaction"(translation:to make the doctors the best con-artists possible) speaks volumes of the age of medicine we live in-and why I turned my back on that world years ago. I consider myself VERY lucky to have good and trustworthy doctors-but I can't tell you how hard my family searched to find them.
100 monsters for every one like them.
"We're coming for your money and we'll GET it all. We're the only real winners.The players don't stand a chance." -from the screenplay of Martin Scorsese's CASINO
Changing the subject to something mathematical, something on the web caught my eye yesterday and I just need to share it with the house. Ever wonder what well-defined means? It's amazing how many graduate students-particularly those working in category theory and the higher altitudes of algebra,where the phrase probably comes up most-never ask what that means. It's kind of accepted everyone "sorta" knows what it means. And for most people,that's good enough.
I remember the first time I ever wondered about it-it was in Kenneth Kramer's honors abstract algebra course a few years ago at Queens College. He was sketching the proof of Cayley's theorum on the fact that every group is the same as some group of permutations on a set (i.e. they're isomorphic). ( Actually, he wasn't proving it,he just wanted to sketch the proof because he'd rather spend the classes' time developing the theory of group actions on a set, of which Cayley's theorum is a special case-i.e. a group acting on itself. But I digress.............)
He was constructing the composition map which is the isomorphism of a group G onto it's corresponding permutation group acting on it's underlying set S -I forget what he denoted it as,call it P(S). He commented the map was clearly well-defined. I raised my hand in frustration since I'd asked the question before and never gotten a straight answer from any professor (some of them actually got annoyed with it and made unkind remarks about my age as a student)
What followed was one of the most impressionable moments of my student career as Dr.Kramer and I exchanged comments on what exactly it meant to be well-defined. "It means it's not ambiguous what the value assigned is, Andrew-that we don't get 2 values for the same arguement." "Oh, you mean the relation actually specifies a function?" "Well, not exactly-if the formula IS a function, you're absolutely right. But this may not be a function and still be a well defined mathematical object." I didn't get it. After a few minutes of him giving a few examples, no progress was made. He ultimately asked me to table the question so we don't waste any more of the classes' time.
I did so,but ultimately,it disturbed me. Dr.Kramer is a gifted teacher on all matters mathematical-an early student of John Tate's at Harvard-and usually the most pleasant and patient of people with even the stupidest of students' questions. In fact,I'll be taking a course on elliptic functions with him at the City University Of New York Graduate Center this fall. The sheer wieght of the subsequent coursework-the first 6 chapters of Herstein's classic Topics In Algebra in a VERY intensive 2 semesters,plus his own notes-prevented us from broaching the subject further. All that really got settled was that it was pretty clear what "well-defined" meant if the object under consideration was a function-in fact, it's almost redundant. But how would you describle a general mathematical object as being "well defined"?
Leave it to Tom Gowers to make everybody happy.
There are several blogs online I try so hard not to miss. Peter Woit's Not Even Wrong, Terrance Tao's, John Baez's The N-Category Cafe' , The Secret Blogging Seminar and a few others. But nothing matches Gower's blog for sheer beauty of writing and thinking about mathematics. A lot of people can do mathematics, a lot more people can teach mathematics, and even more people can talk about mathematics .(Sadly, this is whether or not they know what the fuck they're talking about or not...........)
There are so few who can do all of the above.
Elias Stien can do it (sometimes).
Melvyn Nathanson can do it.
James Stasheff can do it. Better then anyone I've ever heard.
William Thurston can do it.
But for my money, no one does it better currently and consistently then Tom Gowers. His blog should be required reading for all mathematicans and serious math students. (By the way-his old teacher at Cambridge, Bela Bollabos-is also great at all of the above. I doubt that's an accident. )
Anywho, I was reading Gowers' blog and low and behold, Gowers also wanted to know, after grading the exams for the year at Cambridge and discovering NONE of his students understood it,either-what's it mean for something to be well defined?
People who know me know I'm Socratic to a fault, to the point of making people violent. I almost NEVER agree with EVERYTHING someone says.
But this is rare occasion when I'm speechless with complete conviction and agreement with someone else's analysis. As I said, leave it to Gowers to give the perfect answer to a great question.
I'll simply let the beauty,depth and simplicity of Gowers' blogpost speak for itself-I simply have nothing to add to it. Nothing at all. Anyone asks me this question in the future, I'll simply give them a copy of Gowers' post. For all basic mathematical discussions that may come up in the future, I seriously doubt anyone can debunk this discussion.
It's THAT good.
Oh,screw the self-engratiating pontification,here's Gowers. And if you don't bookmark his blog, shame on you.
Good night to all,fellow travelers. Until next time.
http://gowers.wordpress.com/2009/06/08/why-arent-all-functions-well-defined/#more-605
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