6
Jul 09

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.................

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