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Manifesto on the Teaching of Mathematics (2015)

intellectualmathematics.com

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Re: Manifesto on the Teaching of Mathematics (2015)

#31

Post by dan-robertson »

If you want to read a long text about one way of teaching mathematics, I recommend https://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf.

I have a weak interest in the best way to teach things. There are two interesting things in the U.K. around it. One is that methods for teaching children to read seem to have improved recently (that is, the results in tests used to measure reading ability have improved) and there has been a claim that this was mainly due to doing and then applying the results of some research into how to best teach it. The research was cheap and one wonders if it could be applied to more situations. Meanwhile in mathematics there is a regular desire from governments to improve the mathematics that is learned but it feels like the efforts don’t go so well. Usually any time mathematics education is in the news, Simon Jenkins will trot out the same ridiculous tired old article against mathematics education. The disconnect between the way that mathematicians and non-mathematicians think about mathematics education seems pretty bad to me.

Re: Manifesto on the Teaching of Mathematics (2015)

#32

Post by KineticLensman »

dizhn wrote: post #22
throw_pm23 wrote: post #15

Earlier quoted context omitted.

Interesting perspective. Why someone would downvote it is beyond me.

I just got voting powers and did not downvote parent but the icons are tiny on mobile and they give no indication of which you click after the fact. I have no idea which took when I click the icon.

I sometimes zoom in on mobile so that I don't mis-tap the wrong button.

Re: Manifesto on the Teaching of Mathematics (2015)

#33

Post by Tainnor »

zmgsabst wrote: post #27
Tainnor wrote: post #17

I do definitely agree that something is lacking in current mathematical education, but calling your own counter-program "intellectual mathematics" is a bit ... on the nose. Judging from the rest of the website, the author appears to have some rather idiosyncratic opinions. For example, he seems to be unconvinced that rigour is an essential component of mathematics (even going as far as claiming not to understand what…

Talking about not understanding rigor in the context of calculus is silly. Modern formalism is because of calculus!… and the fact that contrary results were obtained regarding continuity and derivatives due to subtly different conceptions of the terms. Enter the Weierstrass function: > Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning severa…

I mean I'm open to the idea that pedagogically, full rigour is not always required (depending on the audience)... although there are certainly always certain people who are not going to be satisfied with seemingly "intuitive" explanations such as "infinitesimally" small, people are different after all.

But instead of saying "I'm not proving things rigorously because [reasons]", he's claiming that rigour doesn't really exist or has no importance which is kind of crazy for the reasons you mentioned.

Re: Manifesto on the Teaching of Mathematics (2015)

#34

Post by rramadass »

Tainnor wrote: post #33
zmgsabst wrote: post #27

Earlier quoted context omitted.

Talking about not understanding rigor in the context of calculus is silly. Modern formalism is because of calculus!… and the fact that contrary results were obtained regarding continuity and derivatives due to subtly different conceptions of the terms. Enter the Weierstrass function: > Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning severa…

I mean I'm open to the idea that pedagogically, full rigour is not always required (depending on the audience)... although there are certainly always certain people who are not going to be satisfied with seemingly "intuitive" explanations such as "infinitesimally" small, people are different after all. But instead of saying "I'm not proving things rigorously because [reasons]", he's claiming that rigour doesn't reall…

Rigour is over-emphasized in the teaching of Mathematics to the detriment of its Understanding. Teaching should always be conceptual first before representing them formally and using rigour.

The best example is Faraday vs. Maxwell. In one communication Faraday actually says it "frightened" him to see Maxwell's mathematics but that his "Conclusions" were so clear that he could think and work from them. Faraday thought and worked with concepts and mental models while Maxwell gave them a formal and rigorous representation.

Re: Manifesto on the Teaching of Mathematics (2015)

#35

Post by danielmarkbruce »

murkt wrote: post #13
danielmarkbruce wrote: post #9

Earlier quoted context omitted.

If your kid likes sports, there are endless lessons to teach around it.

True, we’re going ice skating with him regularly, and it’s a great source of lessons. Doesn’t help very much with math, though :)

Baseball? Cricket? Soccer? Basketball? Football? They are all full of numbers - ie the comment was supposed to imply math lessons.

Re: Manifesto on the Teaching of Mathematics (2015)

#36

Post by zmgsabst »

Tainnor wrote: post #33
zmgsabst wrote: post #27

Earlier quoted context omitted.

Talking about not understanding rigor in the context of calculus is silly. Modern formalism is because of calculus!… and the fact that contrary results were obtained regarding continuity and derivatives due to subtly different conceptions of the terms. Enter the Weierstrass function: > Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning severa…

I mean I'm open to the idea that pedagogically, full rigour is not always required (depending on the audience)... although there are certainly always certain people who are not going to be satisfied with seemingly "intuitive" explanations such as "infinitesimally" small, people are different after all. But instead of saying "I'm not proving things rigorously because [reasons]", he's claiming that rigour doesn't reall…

Yep — I’m a big proponent of explaining the intuition; I even go for intuition first, because it’s only by having a notion of what you want to discuss that you can make a meaningful formalism. After all, how do you know what the right formalism would be without an intuitive notion of what you’re trying to model?

But when you want to resolve a question like “what can we say about continuity and differentiation?” suddenly the answer depends intimately on your formalism. And the only way to resolve that we have conflicting ideas about the intuition for “continuous” and “differentiable” is for us both to formalize that intuition in a model — then compare the two to see where we disagree.

- - - - -

As an aside, you can formalize the notion of infinitesimals, but it requires a lot more machinery which introduces its own quirks. And it was only because we formalized calculus (and tried to formalize all of math) that we had sufficiently advanced model theory. I won’t claim to be an expert, but the topic is Nonstandard Analysis. I believe similar constructions show up in game theory and quantum mechanics.

https://en.wikipedia.org/wiki/Nonstandard_analysis

Re: Manifesto on the Teaching of Mathematics (2015)

#37

Post by murkt »

danielmarkbruce wrote: post #35
murkt wrote: post #13

Earlier quoted context omitted.

True, we’re going ice skating with him regularly, and it’s a great source of lessons. Doesn’t help very much with math, though :)

Baseball? Cricket? Soccer? Basketball? Football? They are all full of numbers - ie the comment was supposed to imply math lessons.

I know nothing about baseball, cricket and American football, they’re practically non-existent here. Basketball and soccer are full of numbers? For 3-4 years old they are. Maybe there could be some lessons for older kids, I just can’t think of how to spin it that way.

A month ago I told him an old and tired joke. 90 degrees is a right angle, and 100 degrees is when water boils (celsius FTW). He howled with laughter and sure knows now some things about angles and circles.

Re: Manifesto on the Teaching of Mathematics (2015)

#38

Post by Tainnor »

zmgsabst wrote: post #36
Tainnor wrote: post #33

Earlier quoted context omitted.

I mean I'm open to the idea that pedagogically, full rigour is not always required (depending on the audience)... although there are certainly always certain people who are not going to be satisfied with seemingly "intuitive" explanations such as "infinitesimally" small, people are different after all. But instead of saying "I'm not proving things rigorously because [reasons]", he's claiming that rigour doesn't reall…

Yep — I’m a big proponent of explaining the intuition; I even go for intuition first, because it’s only by having a notion of what you want to discuss that you can make a meaningful formalism. After all, how do you know what the right formalism would be without an intuitive notion of what you’re trying to model? But when you want to resolve a question like “what can we say about continuity and differentiation?” sudde…

You can introduce nonstandard analysis, but it's not necessarily incredibly intuitive either, even if you skip a formal construction (which requires either ultrafilters or model theory). I think that's still a bit removed from just giving ad-hoc reasoning about "infinitely small" quantities without specifying precisely what you can and cannot do with them.

Re: Manifesto on the Teaching of Mathematics (2015)

#39

Post by danielmarkbruce »

murkt wrote: post #37
danielmarkbruce wrote: post #35

Earlier quoted context omitted.

Baseball? Cricket? Soccer? Basketball? Football? They are all full of numbers - ie the comment was supposed to imply math lessons.

I know nothing about baseball, cricket and American football, they’re practically non-existent here. Basketball and soccer are full of numbers? For 3-4 years old they are. Maybe there could be some lessons for older kids, I just can’t think of how to spin it that way. A month ago I told him an old and tired joke. 90 degrees is a right angle, and 100 degrees is when water boils (celsius FTW). He howled with laughter a…

There are a lot of stats in basketball. There are measurements, counting, probabilities, geometry, algebra and calculus. You can do physics too.
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