In my opinion Knuth doesn't go far enough. He still sticks with pointwise notions of differentiability and continuity that still require some heavy tools from classical analysis, such as completeness and compactness, to get any practical results. If his estimates had been uniform in x, he would have ended up with a much simpler theory, based on uniform notions and not requiring these heavy tools. See Calculus without limits in the previous reincarnation of Mathematics Under the Microscope blog.
It's nice to know that Knuth is still around. Almost all of my professors from my youth are dead now. I didn't have Knuth as a professor but he was already famous from writing The Art of Computer Programming back when I was a kid (in 1976).
As far as using the order arithmetic in teaching calculus, I think it's a bad idea.
The better students will understand calculus intuitively. The less better students will survive by memorizing formulas. All that it is possible to do is to make it possible for the less better students to learn how to make a few calculations. Teaching them the theory is hopeless; and they don't need it.
When I taught calculus, I didn't prove anything to the students. If they wanted to learn the proof they could read their book. With the product rule, I wrote out an example of something complicated for f and g, and showed that d(fg) = f dg + g df. And then I said "this is generally true and it is proved in your book; if you're curious, read it." Then I worked unassigned homework problems.
The students learned exactly enough to do their homework quickly and efficiently.
The result of all this was that my class had a high graduation rate, my students rated me very effective, and they did considerably better than average in their later classes (taught by others).
It seems to me that a great deal of mathematics and physics is taught in a way that makes it much more difficult to understand than it really needs to be. Simplify, simplify, simplify. It's supposed to be a freshman class, not an introductory grad student class.
To get a taste of how mathematics used to be taught, my father got a BS in Biology back around 1950. The only college class he had in math was algebra. He studied math by mail; a home study class advertised on the back of a book of matches. From there he taught high school math. And then he went to a teacher's college for a year. From that he transferred to U. T. Austin and got a PhD in math including some classes from R. L. Moore.
You can't do that anymore and I think that's sad. The basic problem is that mathematics has become much larger than it once was, and students are required to be knowledgeable about far too much of it. The PhD should be awarded for demonstrating the ability to extend mathematics knowledge, not for (also) being a walking encyclopaedia.
Carl, you said: "When I taught calculus, I didn't prove anything to the students. If they wanted to learn the proof they could read their book." Did you really mean it? The proofs presentad in most of the calculus books are watered-down versions of the proofs from classical analysis, they are too hard for most of the students. You also say: "Simplify, simplify, simplify. It's supposed to be a freshman class, not an introductory grad student class." Yes, introductory calculus can be simplified, take a look at my talk slides for an example.
The better students will understand calculus intuitively.
And thereby not have the tools to figure out whether or not using the calculus is a viable way to model something. It's a skill, not a talent. Moreover, it's a means to an end, not an end in and of itself.
Re "The proofs presented in most of the calculus books are watered-down versions of the proofs from classical analysis, they are too hard for most of the students."
This is exactly my point, but more; I'm saying that understanding proofs is too hard for them whether I lecture on it or not. For some reason, a lot more students can solve problems than can find proofs. If I want to flunk out a lot of students it's easy, just assign proofs.
I suspect that this was once not so hard, back when high school students were expected to master proofs in geometry.
And maybe students are better now than in the 70s. I knew a math student who was unable to prove anything. Despite this, he somehow managed to get a BS degree and showed up in grad school. The faculty nickname for him was something like "the pathological prover". What's worse, they felt pity on him and gave him the MS degree.
But the world takes all sorts, and it finds uses for people who can't prove things. And they may still find uses for calculus. This is not about honors calculus at a big name college where the average students are in the running for a Fields Medal.
Carl, you said: "I'm saying that understanding proofs is too hard for them whether I lecture on it or not. For some reason, a lot more students can solve problems than can find proofs." But look, the classical proofs are hard because the classical definitions are weak. If we start with the stronger definitions, (as I do) the proofs become much easier, and in fact become rather simple exercises that use only elementary tools familiar to the students. It brings the theory on the same level with the problems solved in the class. The streamlined theory is adequate for most of the applications, and even more relavant for numerical analysis, for example, than the classical stuff. Besides, the simplified treatment provides a stepping stone to the classical analysis for the students who want to study it. Doesn't it make sense?
Carl: "The PhD should be awarded for demonstrating the ability to extend mathematics knowledge, not for (also) being a walking encyclopaedia." Thumbs up for that!
My friend and I were recently discussing about how modern society has evolved to become so integrated with technology. Reading this post makes me think back to that discussion we had, and just how inseparable from electronics we have all become.
I don't mean this in a bad way, of course! Ethical concerns aside... I just hope that as technology further advances, the possibility of copying our brains onto a digital medium becomes a true reality. It's a fantasy that I dream about all the time.
(Posted on Nintendo DS running [url=http://kwstar88.zoomshare.com/2.shtml]R4i SDHC[/url] DS NePof)
10 comments:
Your blog is so interesting, including quite amazing probability analyses!
In my opinion Knuth doesn't go far enough.
He still sticks with pointwise notions of differentiability and continuity that still require some heavy tools from classical analysis, such as completeness and compactness, to get any practical results. If his estimates had been uniform in x, he would have ended up with a much simpler theory, based on uniform notions and not requiring these heavy tools. See Calculus without limits in the previous reincarnation of Mathematics Under the Microscope blog.
It's nice to know that Knuth is still around. Almost all of my professors from my youth are dead now. I didn't have Knuth as a professor but he was already famous from writing The Art of Computer Programming back when I was a kid (in 1976).
As far as using the order arithmetic in teaching calculus, I think it's a bad idea.
The better students will understand calculus intuitively. The less better students will survive by memorizing formulas. All that it is possible to do is to make it possible for the less better students to learn how to make a few calculations. Teaching them the theory is hopeless; and they don't need it.
When I taught calculus, I didn't prove anything to the students. If they wanted to learn the proof they could read their book. With the product rule, I wrote out an example of something complicated for f and g, and showed that d(fg) = f dg + g df. And then I said "this is generally true and it is proved in your book; if you're curious, read it." Then I worked unassigned homework problems.
The students learned exactly enough to do their homework quickly and efficiently.
The result of all this was that my class had a high graduation rate, my students rated me very effective, and they did considerably better than average in their later classes (taught by others).
It seems to me that a great deal of mathematics and physics is taught in a way that makes it much more difficult to understand than it really needs to be. Simplify, simplify, simplify. It's supposed to be a freshman class, not an introductory grad student class.
To get a taste of how mathematics used to be taught, my father got a BS in Biology back around 1950. The only college class he had in math was algebra. He studied math by mail; a home study class advertised on the back of a book of matches. From there he taught high school math. And then he went to a teacher's college for a year. From that he transferred to U. T. Austin and got a PhD in math including some classes from R. L. Moore.
You can't do that anymore and I think that's sad. The basic problem is that mathematics has become much larger than it once was, and students are required to be knowledgeable about far too much of it. The PhD should be awarded for demonstrating the ability to extend mathematics knowledge, not for (also) being a walking encyclopaedia.
Carl, you said: "When I taught calculus, I didn't prove anything to the students. If they wanted to learn the proof they could read their book." Did you really mean it? The proofs presentad in most of the calculus books are watered-down versions of the proofs from classical analysis, they are too hard for most of the students. You also say: "Simplify, simplify, simplify. It's supposed to be a freshman class, not an introductory grad student class." Yes, introductory calculus can be simplified, take a look at my talk slides for an example.
The better students will understand calculus intuitively.
And thereby not have the tools to figure out whether or not using the calculus is a viable way to model something. It's a skill, not a talent. Moreover, it's a means to an end, not an end in and of itself.
misha; I liked your slides.
Re "The proofs presented in most of the calculus books are watered-down versions of the proofs from classical analysis, they are too hard for most of the students."
This is exactly my point, but more; I'm saying that understanding proofs is too hard for them whether I lecture on it or not. For some reason, a lot more students can solve problems than can find proofs. If I want to flunk out a lot of students it's easy, just assign proofs.
I suspect that this was once not so hard, back when high school students were expected to master proofs in geometry.
And maybe students are better now than in the 70s. I knew a math student who was unable to prove anything. Despite this, he somehow managed to get a BS degree and showed up in grad school. The faculty nickname for him was something like "the pathological prover". What's worse, they felt pity on him and gave him the MS degree.
But the world takes all sorts, and it finds uses for people who can't prove things. And they may still find uses for calculus. This is not about honors calculus at a big name college where the average students are in the running for a Fields Medal.
Carl, you said: "I'm saying that understanding proofs is too hard for them whether I lecture on it or not. For some reason, a lot more students can solve problems than can find proofs." But look, the classical proofs are hard because the classical definitions are weak. If we start with the stronger definitions, (as I do) the proofs become much easier, and in fact become rather simple exercises that use only elementary tools familiar to the students. It brings the theory on the same level with the problems solved in the class. The streamlined theory is adequate for most of the applications, and even more relavant for numerical analysis, for example, than the classical stuff. Besides, the simplified treatment provides a stepping stone to the classical analysis for the students who want to study it. Doesn't it make sense?
Carl: "The PhD should be awarded for demonstrating the ability to extend mathematics knowledge, not for (also) being a walking encyclopaedia."
Thumbs up for that!
My friend and I were recently discussing about how modern society has evolved to become so integrated with technology. Reading this post makes me think back to that discussion we had, and just how inseparable from electronics we have all become.
I don't mean this in a bad way, of course! Ethical concerns aside... I just hope that as technology further advances, the possibility of copying our brains onto a digital medium becomes a true reality. It's a fantasy that I dream about all the time.
(Posted on Nintendo DS running [url=http://kwstar88.zoomshare.com/2.shtml]R4i SDHC[/url] DS NePof)
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