What is a Riemannian Manifold?

nomadthethird

more issues than Time mag
philosophy being, after all, a really stupid thing to want to do.
Or maybe it's a really decadent thing to want to do. When the philosophers start getting massive recognition and obsequious praise isn't that usually when it's widely acknowledged that the empire is doomed?

Sort of how it is decadent that I can spend all day reading or listening to whatever I can steal online while watching The Anatomy of a Pandemic and eating candy.
 

vimothy

yurp
Obviously I'm not against teachers/lecturers encouraging their students to think about what things mean, as opposed to merely how things are. That said, an exhaustive analysis of Newtonian vs. Leibnitzian conceptions of infinitesimal quantities is probably not a prerequisite for a class of 16-year-olds to get to grips with the basics of calculus.
But they do not get to grips with the basics of calculus; they learn how to perform certain operations, and so the knowledge they acquire is for the most part purely procedural. A major problem with undergraduate maths courses is that the lack of conceptual knowledge doesn't prepare them well for the transition to HE programmes that rely on maths. There is quite a literature on this.
 
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Mr. Tea

Shub-Niggurath, Please
Staff member
But they do not get to grips with the basics of calculus; they learn how to perform certain operations, and so the knowledge they acquire is for the most part purely instrumental. A major problem with undergraduate maths courses is that the lack of conceptual knowledge doesn't prepare them well for the transition to HE programmes that rely on maths. There is quite a literature on this.
Fair enough. Do you think more emphasis on the history and philosophy of the subject would facilitate students' conceptual understanding of it? It certainly seems plausible, I guess.
 

poetix

we murder to dissect
Well personally I couldn't care less if some philosophy professor gets a hard-on for writing about Riemannian manifolds, but it would seem to call into question his credibility if he does so in a sloppy, capricious way.
But it's liberated to be sloppy and capricious. And authoritarian to be otherwise. The great totalitarian monsters of the 20th century were all noted for their keen, analytic intelligence and ruthless dedication to mastering complex technical domains! As they marshalled their theorems, so did they command men! Vast, methodically drilled armies of indoctrinated rationalists, sweeping across Europe, eliminating all traces of empathy, creativity and fun...
 

vimothy

yurp
Fair enough. Do you think more emphasis on the history and philosophy of the subject would facilitate students' conceptual understanding of it? It certainly seems plausible, I guess.

Not necessarily. I'm thinking in much less grand terms -- though contextualisation would surely help students appreciate the value of the subject -- more time understanding stuff (making mistakes, understanding mistakes), less time racing through procedures.
 
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josef k.

Dangerous Mystagogue
Or maybe it's a really decadent thing to want to do. When the philosophers start getting massive recognition and obsequious praise isn't that usually when it's widely acknowledged that the empire is doomed?

Sort of how it is decadent that I can spend all day reading or listening to whatever I can steal online while watching The Anatomy of a Pandemic and eating candy.
Praising famous men is clearly a pretty dangerous idea, and a perpetual danger in all fields of achievement, often exacerbated by certain rhetorical fly-trap moves which people leave in their text (perhaps unconsciously) for precisely this reason... then again, every generation throws a hero up the pop charts, and society is perpetually manufacturing idols. The image of the philosopher as icon is really just a surrogate image of Athene in this sense... then again, there is also the way in which the culture industry encourages this attitude, through outlandish, authority-condensing blurbs and other forms of marketese.
 

nomadthethird

more issues than Time mag
Not necessarily. I'm thinking in much less grand terms -- though contextualisation would surely help students appreciate the value of the subject -- more time understanding stuff (making mistakes, understanding mistakes), less time racing through procedures (our data are full of teachers saying things like "you don't need to understand this, let's move on").
Yeah, people always say this stuff, but in the end time constraints seem to be a huge enemy of this kind of conceptually comprehensive meta-education that everyone seems to agree would be best.
 

josef k.

Dangerous Mystagogue
But it's liberated to be sloppy and capricious. And authoritarian to be otherwise. The great totalitarian monsters of the 20th century were all noted for their keen, analytic intelligence and ruthless dedication to mastering complex technical domains! As they marshalled their theorems, so did they command men! Vast, methodically drilled armies of indoctrinated rationalists, sweeping across Europe, eliminating all traces of empathy, creativity and fun...
You are an idiot.
 

poetix

we murder to dissect
The problem in my experience with maths teaching (and curricula) in the UK is not so much the lack of "philosophical" context as the neglect of "motivation" - it's seldom apparent what the problem, or class of problems, is that a particular technique is supposed to give you some purchase on. It's just: "These are matrices. Learn them". I still don't know what matrices are for, outside of the rather limited context of composing transformations to apply to vector graphics in silly computer games.
 

vimothy

yurp
Yeah, people always say this stuff, but in the end time constraints seem to be a huge enemy of this kind of conceptually comprehensive meta-education that everyone seems to agree would be best.
The time constraints are a function of a different phenomenon -- the "audit culture" of testing and measurement -- they are not naturally occurring. If we want to change the way teaching and learning occurs, one of the things we need to think about is our system of examinations.
 

josef k.

Dangerous Mystagogue
Yeah, people always say this stuff, but in the end time constraints seem to be a huge enemy of this kind of conceptually comprehensive meta-education that everyone seems to agree would be best.
The outstanding question is: what is education (broadly defined) for? It seems clear that educational institutions are devoted at least in part to the furtherance of the interests of that institution, and its supporting networks of institutions (e.g. university socializes people into the middle class, public schools-Oxbridge (or the Ivy League in the US) provide recruitment facilities for the elites... from a different angle, teachers and special interest groups (faith schools) courted as voters in various ways that do not necessarily coincide with the interests of their students, society-at-large, or humanity, whatever that is...
 

vimothy

yurp
It's just: "These are matrices. Learn them". I still don't know what matrices are for, outside of the rather limited context of composing transformations to apply to vector graphics in silly computer games.
Exactly -- connections are not made, and instead teaching is the transmission of procedures.
 

nomadthethird

more issues than Time mag
I think in some ways, you have to learn (lower level) math by trial and error and then it will slowly become conceptually clear to you. It's hard to work backwards from the concepts down to the rules, isn't it? I remember thinking wtf is trig supposed to apply to, and I had no idea why the rules worked, but they did. And the more I worked them the more trig made sense. Then when I took calculus I understood why they made us take geometry and trig.
 

vimothy

yurp
The outstanding question is: what is education (broadly defined) for?
That question even breaks down into two questions, because there is another question hidden inside it, viz. what should education be for? Or (I prefer), what does education do? And what should education do?
 

josef k.

Dangerous Mystagogue
Exactly -- connections are not made, and instead teaching is the transmission of procedures.
And routines... Which is why the practice of philosophy as a "nomad science" of making unusual and unanticipated connections is valuable... and why philosophy is required to break or mutate disciplinary models (transgressing police boundaries) in order to do this... and why any attempt by a philosopher is to set-itself as a uber-policeman equipped with a masterful method and meta-language is threatening.
 

josef k.

Dangerous Mystagogue
That question even breaks down into two questions, because there is another question hidden inside it, viz. what should education be for? Or (I prefer), what does education do? And what should education do?
There are at least two different sides to education... one being the disciplinary machining of bodies into particular roles and identities, the other being a kind of savage education which transcends roles and identities...
 
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