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A Mathematician’s Lament (2002) [pdf]

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Re: A Mathematician’s Lament (2002) [pdf]

#81

Post by nileshtrivedi »

jfarmer wrote: post #62
ctl wrote: post #57

I love this article, but: what can a practicing math teacher take away from it? How can you apply this stuff if you still have to teach a standard curriculum? I'm really asking -- my friend is about to start as a high-school math teacher. I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Loc…

I co-founded Dev Bootcamp and while I was still there one of my not-so-secret missions was to make mathematics less alienating. I only say that because it was incredibly difficult , even in an environment where I had complete autonomy and authority to make whatever curricular and pedagogical decisions I wanted. The problem becomes combinatorially more complex in a public school where teachers have much less autonomy,…

> when most mathematicians look at a set of abstract symbols they don't "see" the symbols per se, they see what those symbols are meant to represent.

Might we benefit from a different set of symbols that actually convey the geometrical meaning behind them? If instead of π, we used a glyph that shows a circle over a diameter, instead of x for a variable, we show an empty rectangle that shows that it's a placeholder for a value?

> Students who said the answer was 44, how did the students who said 26 arrive at their answer?

I came across a great example of this approach in Chess: The Complete Self-Tutor by Edward Lasker. Instead of just showing the right answer for a chess puzzle, he tells you what you did right in your answer and what you missed in getting an even better answer. This was a printed book that was completely interactive.

Re: A Mathematician’s Lament (2002) [pdf]

#82

Post by saraid216 »

gizmo686 wrote: post #34
dagw wrote: post #31

Earlier quoted context omitted.

yet struggle to convince "artsy" persons that what I do is creative. As another mathematician I've almost found the exact opposite. As soon as I mention math to an arts person they instantly start babbling about fractals and chaos and Fibonacci and all kinds of other vague pop-culture terms they've heard of but don't really understand. Artsy types almost seem to find math much more artistic than I do.

There is a different between artsy and creative. Artsy (roughly) refers to things that appeal to the senses. Creative is a much more general notion of creating. In math, you defiantly create things. And you arguably create beautiful thing. But the beauty is not in the senses, it is in the mind. The senses are involved only as a form of communication.

It's the same kind of beauty you'd find in a well-stated argument.

Re: A Mathematician’s Lament (2002) [pdf]

#83

Post by saraid216 »

dnautics wrote: post #73
saraid216 wrote: post #55

Earlier quoted context omitted.

Please give one example applicable to a reasonable majority of human beings on the planet at this current time where "understanding sums and compounding processes" are a matter of life and death.

did you not read what I wrote? "or quality of life [or quality of death]". Nor did I proclaim that the life-or-death situation was applicable to a majority of people.

Your point is lost if it's not applicable to a significant majority. If it's only applicable to a minority, then that minority ought to be identified and specially trained, like we do with peanut allergies.

Re: A Mathematician’s Lament (2002) [pdf]

#84

Post by giardini »

stiff wrote: post #37

Please read the article with a critical eye, some of it is complete non-sense, for example: CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-ba…

"... earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" ..."

I don't think so.

99.99% of the mathematics of motion consists of continuous functions with continuous derivatives of all orders. Infinitesimals are just fine for that. I find the epsilon-delta construction (EDC) to be not only inordinately clunky but to provide no insight whatsoever insofar as motion is concerned. From a pedagogical standpoint EDC is roughly the equivalent of tossing a monkey wrench into the smoothly oiled and finely-working innards of the mathematics of motion. The student's mind grinds to a screeching halt as (s)he attempts to come to terms with a new construct that not even a mathematician's mother could love. I consider any praise of the epsilon-delta construction of calculus to be little more than turd-polishing and while it is unfortunately necessary, there is no need to call attention to it more than once. And once pointed out, best forgotten.

One of the most underrated advancements in mathematics is Descarte's introduction of Cartesian coordinates. The formalization of the calculus, in contrast, was a step backwards in creative terms. Although necessary, it is of interest mostly to pure mathematicians.

Re: A Mathematician’s Lament (2002) [pdf]

#85

Post by saraid216 »

jameshart wrote: post #68
saraid216 wrote: post #66

Earlier quoted context omitted.

But (1) The subprime mortgage crisis could not have been averted by a larger percentage of the population understanding sums and compounding processes. (2) The advent of civilization was a contributing factor to everything that's happened in the last several thousand years, including the paper cut I just got.

you understand that interest-bearing loans are a compound process, right - and that an understanding thereof might be of relevance to someone entering into a loan agreement which they probably, if they really understood it, were not going to be able to repay... But let's not only blame the borrowers of subprime loans, let's also ask whether the banks didn't entirely understand the risk models of the derivatives they…

> an understanding thereof might be of relevance to someone entering into a loan agreement which they probably, if they really understood it, were not going to be able to repay...

Oh, it's relevant. It's sort of like how being able to load and fire a gun is relevant to the decision whether or not to commit suicide. Sure, it can modify one of the many, many details, but some people just make a noose and hang themselves.

Keep in mind, this is your argument: if more people in the world understood sums and compounding processes, this would have prevented the subprime mortgage crisis and thus the many deaths that were inspired by the resultant fallout. There is no possibility that anything else caused the crisis, and no possibility that anyone is at fault for these deaths other than parents and teachers.

This claim, if true, actually absolves the lenders that you talk about below, because they can be held responsible only for their own understanding of sums and compounding processes.

Amusingly, if you accept your argument, you can also make an interesting inverse version. The fact that lenders understood sums and compounding processes led to their employment at unscrupulous institutions which then mandated their sign-off on high-risk mortgages, which then caused the deaths of all those people.

Math, apparently, kills.

Re: A Mathematician’s Lament (2002) [pdf]

#86

Post by stiff »

abecedarius wrote: post #51
stiff wrote: post #37

Please read the article with a critical eye, some of it is complete non-sense, for example: CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-ba…

This kind of approach simply doesn't make sense It was poorly explained there, but essentially that style of reasoning does work: http://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bel... And I at least find that approach easier and more useful. (I learned it from the Feynman lectures on physics, where he didn't axiomatize it; the above link does.)

I find the logically sound version of the infinitesimals approach to be much more difficult to understand than the approach using limits. For example, you have to introduce hyperreals to make it work (most common approach):

http://en.wikipedia.org/wiki/Hyperreal_number

Re: A Mathematician’s Lament (2002) [pdf]

#87

Post by abecedarius »

stiff wrote: post #86
abecedarius wrote: post #51

Earlier quoted context omitted.

This kind of approach simply doesn't make sense It was poorly explained there, but essentially that style of reasoning does work: http://www.amazon.com/Primer-Infinitesimal-Analysis-John-Bel... And I at least find that approach easier and more useful. (I learned it from the Feynman lectures on physics, where he didn't axiomatize it; the above link does.)

I find the logically sound version of the infinitesimals approach to be much more difficult to understand than the approach using limits. For example, you have to introduce hyperreals to make it work (most common approach): http://en.wikipedia.org/wiki/Hyperreal_number

The book I linked to uses http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis (a quite different approach). I haven't studied nonstandard analysis.

Re: A Mathematician’s Lament (2002) [pdf]

#88

Post by stiff »

AsymetricCom wrote: post #67
stiff wrote: post #37

Please read the article with a critical eye, some of it is complete non-sense, for example: CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-ba…

are we upvoting for length of comments now?

Do you have any particular criticism to make?

Re: A Mathematician’s Lament (2002) [pdf]

#89

Post by stiff »

giardini wrote: post #84
stiff wrote: post #37

Please read the article with a critical eye, some of it is complete non-sense, for example: CALCULUS: This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-ba…

"... earlier and more general lack of any decent understanding of how calculus works at all that was solved by Cauchy, Weierstrass and others. It is their introduction of what the author considers "unnecessary formalism" that made us finally really understand "mathematics of motion" ..." I don't think so. 99.99% of the mathematics of motion consists of continuous functions with continuous derivatives of all orders. I…

99.99% of the mathematics of motion consists of continuous functions with continuous derivatives of all orders. Infinitesimals are just fine for that.

There were logical contradictions even in Newtons and Leibniz works, far before anyone considered continuous functions without derivatives etc., they were basically making decisions about when a given operation or transformation can be applied based on intuition alone and not any logical deductions, and it wasn't rare they arrived at incorrect conclusions. Also, Newton himself did epsilon-delta reasonings, he just did not notice their generality:

http://www.sciencedirect.com/science/article/pii/S0315086000...

Again, you are confusing the work of Cauchy and Weierstrass and the epsilon-delta stuff with all the latter even more formal approaches to treat more complicated functions, which by the way were developed in response to Fourier examining heat transfer and trying to describe it mathematically (so it's actually rooted in physics).

People were saying the same things you are just saying about limits about the geometry of Euclid, in fact Newton at one time was of the opinion all the formal development of geometry is useless. He reconsidered after obtaining nonsense geometrical results a few times...

The formalization of the calculus, in contrast, was a step backwards in creative terms. Although necessary, it is of interest mostly to pure mathematicians.

I wish you luck doing quantum mechanics with Newton-style calculus.

Re: A Mathematician’s Lament (2002) [pdf]

#90

Post by calibraxis »

jfarmer wrote: post #62
ctl wrote: post #57

I love this article, but: what can a practicing math teacher take away from it? How can you apply this stuff if you still have to teach a standard curriculum? I'm really asking -- my friend is about to start as a high-school math teacher. I guess the first recommendation would be: motivate every new technique by starting with one or more problems that the technique helps to solve. (Here "problems" is meant in the Loc…

I co-founded Dev Bootcamp and while I was still there one of my not-so-secret missions was to make mathematics less alienating. I only say that because it was incredibly difficult , even in an environment where I had complete autonomy and authority to make whatever curricular and pedagogical decisions I wanted. The problem becomes combinatorially more complex in a public school where teachers have much less autonomy,…

People should note that * are turned into italics, when text is between them. So, the original equation is 4 * 5 + 6.

(Here, I use spaces to "escape" the * . '\' doesn't work as an escape character.)

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