Mathematics

Started by Florestan, June 21, 2022, 11:51:03 AM

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Florestan

Quote from: DavidW on July 04, 2022, 01:04:56 PM
Don't let the flat earthers know!  They would be so upset if they could read...

The best way to reason the flat earthers out of their belief is the Bible --- which is probably the most quoted and the least read book in the world.  :D
Every kind of music is good, except the boring kind. — Rossini

KevinP

1. Did you like it / were you good at it during elementary / secondary / high school / college years?
Not particularly good at it. Had a horrible teacher in third grade who made it all the worse. I'm pretty sure I had/have dyscalculia--not to the point where I couldn't grasp concepts but to the degree that I was slower in processing numbers, felt left behind, and got no joy.


2. Are you a graduate of a college where mathematics was among the core courses?
Undergrad was music composition. People who like math love to talk about how mathematical music is. People who don't are like, 'Whatever, dude.' (Yes, I see the maths there, but I don't get any jollies from it.)
MA and PhD were in applied linguistics and so statistics is usually around.


3. If yes, do you use it in your current / former job?
Despite my previous answers, I actually teach a course or two in statistics. I've come to appreciate mathematics for its own sake in my middle-aged years and go through spurts where I read books about it.

4. If no, do you think, in retrospective, that learning it during elementary / secondary / high school was of any use?
I fully believe maths should be taught (and not because of the usual balance-your-chequebook BS), though I don't agree with the way it's usually taught where there's no sense of discovery or figuring things out. And where it's all for the test.

Mapman

Quote from: Florestan on July 04, 2022, 11:31:37 AM
If I'm not mistaken the sum of the angles of a triangle drew on the terrestrial globe is larger than 180.  ;)

Yes, one of the simplest examples is the triangle formed by the prime meridian, line of 90 degrees East longitude, and equator. Three right angles!

coffee

Quote from: Mapman on July 04, 2022, 06:56:53 PM
Yes, one of the simplest examples is the triangle formed by the prime meridian, line of 90 degrees East longitude, and equator. Three right angles!

No rectangles (shapes with four right angles), though. What we might think of as a rectangle would have four obtuse angles. 
Liberty for the wolf is death for the lamb.

staxomega

#84
1. I started to get a better feel for it by the time I was in 10th grade with Algebra II. I entered the more advanced math class in 9th grade (Advanced Geometry) and struggled with it, but that was because our math prior to that point in Canada was not taught too well. By 12th grade AP Calc it was smooth sailing and I got a 4 (out of 5) on the AP exam. That was my favorite math class, derivatives and integrals felt quite natural.

2. No

4. No. A statistics course would have been helpful, they added AP Stats a few years after I graduated. I ended up learning/relearning all the stats I needed for our early medical licensing exams. None of that is of much use to me in private practice.

If I was independently wealthy I would have went into astrophysics which is heavy on the math, so I did like it enough to consider that much.

71 dB

By the way, math(s) is taught differently in school in different countries. The emphasis differs.

It seems for example that in the US math is "mechanical execution of rules". For example rules of how to use trigonometric substitutions in integration. Americans seem to be very good at being able to calculate almost any antiderivative using whatever substitution trick. In Finland trigonometric substitution in integration is super-advanced stuff at high-school level The emphasis is in solving problems using math. The idea is that if you know what you are doing you can always check somewhere how to calculate difficult integrals.

Also, in Finland trigonometric functions sin, cos and tan are almost only used while csc, sec and cot are hardly even mentioned. In the US it seems vectors are not a thing in high-school math, while in Finland there's a lot of geometric problem solving using vector math. I still have my old high-school math textbooks from late 80's and the content is:

Year 1

Real Numbers
Introduction to statistics
Equations and in-equations
Vectors
Geometry
Analytic geometry

Year 2

Functions
Trigonometry
Limits & continuity
Derivative
Differential calculus
Series and number sequencies
Probability and statistics

Year 3

Integral function
Definite integral
Three-dimensional geometry
Three-dimensional vectors
Complex numbers


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71 dB

Quote from: Florestan on July 04, 2022, 01:20:33 PM
The best way to reason the flat earthers out of their belief is the Bible --- which is probably the most quoted and the least read book in the world.  :D

Has any flat earthers ever been reasoned out of their beliefs? Aren't they too far gones who want to feel special for not being brainwashed by NASA?
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71 dB

Quote from: DavidW on July 04, 2022, 11:27:16 AM
Matter curves spacetime which makes it non-Euclidean. 

Einstein had as early as 1907 ideas about matter and energy slowing down the speed of light and apparently you can think about it in two ways:

- Speed of light is constant, spacetime curves
- Speed of light isn't constant, spacetime is flat.

Why does matter and energy curve spacetime? I think it has something to do with entropy, entanglement and the relations of the surface area and the amount of entropy inside it. Entanglement happens when particles interact with each other (short range action) and that means that entanglement happen "surface-wise". For some reason spacetime seems to balance out the entropy inside a surface and the surface are so that increase of entropy leads to the need to "eat" space around the surface. This means that space is being sucked in, but it has to fit into the same volume. So, spacetime bends toward past. It is like running out of shelf-space for new CDs and going into the past to the time when you didn't have so many CDs to find room for your new CDs. That's why gravity slows down time: It is making room in the future for the spacetime flowing in.

Because space is being sucked toward the center of the earth, it moves throught us because we can't move with it. The ground stops us moving. The closer the space moves to the center of the earth, the faster it has to move because the volume gets smaller. So, space accelerates through us, but the equaty principles says it doesn't matter if we accelerate though space or if space accelerates through us. So, we feel like accelerating, but our senses tell us we are not accelerating, but still on the ground. This sensory confusion is what we call gravity.

That's the best explanation for gravity I have, but it still has unsolved aspects, especially the entropy & surface area thing. I don't understand that stuff at all.
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DavidW

Quote from: 71 dB on July 09, 2022, 01:41:44 AM
It seems for example that in the US math is "mechanical execution of rules". For example rules of how to use trigonometric substitutions in integration. Americans seem to be very good at being able to calculate almost any antiderivative using whatever substitution trick. In Finland trigonometric substitution in integration is super-advanced stuff at high-school level The emphasis is in solving problems using math. The idea is that if you know what you are doing you can always check somewhere how to calculate difficult integrals.

Actually trig sub is only taught in BC calculus.  Most students here in the US take AB calculus.  Your impressions of US math are no longer valid, but they were 20-30 years ago when I took AP Calculus.  It is much more conceptual and less mechanical these days.  Just take a look at the recent exam:

https://secure-media.collegeboard.org/apc/ap22-frq-calculus-ab.pdf

I've encountered the same thing teaching AP Physics and I've had to change my approach to how I teach since calculation skills are not as valued as they used to be.

QuoteIn the US it seems vectors are not a thing in high-school math,

That is not correct, vectors are taught in precalculus.  I've only encountered three students in my career that didn't come into my AP Physics C class with vectors under their belt.  And I've taught hundreds of students over the years.

71 dB

Quote from: DavidW on July 09, 2022, 07:22:20 AM
Actually trig sub is only taught in BC calculus.  Most students here in the US take AB calculus.
In Finland we have "short" and "long" math options in high-school. Back when I was in high-school, maybe 60 % took short math. I think it contains all the same things, but it isn't as deep as long math. I suppose BC and AB are the same as "long" and "short" here.


Quote from: DavidW on July 09, 2022, 07:22:20 AMYour impressions of US math are no longer valid, but they were 20-30 years ago when I took AP Calculus.  It is much more conceptual and less mechanical these days.  Just take a look at the recent exam:

https://secure-media.collegeboard.org/apc/ap22-frq-calculus-ab.pdf

Okay, but this is college math? Post high-school? I have never understood US school system.

Quote from: DavidW on July 09, 2022, 07:22:20 AMI've encountered the same thing teaching AP Physics and I've had to change my approach to how I teach since calculation skills are not as valued as they used to be.

That is not correct, vectors are taught in precalculus.  I've only encountered three students in my career that didn't come into my AP Physics C class with vectors under their belt.  And I've taught hundreds of students over the years.

Oh.
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Florestan

I am an engineer. In my experience, mathematicians inhabit quite a different planet than engineers do. We engineers use mathematics to solve practical problems  --- mathematicians use practical problems as a mere pretext to perform mathematical tricks and get a Ph. D. for it. I had plenty of opportunities to learn that during my stay in The Netherlands where I did an aborted Ph. D. in Computational Mathematics. My thesis was supposed to deal with shrinkage (https://www.corrosionpedia.com/definition/2482/shrinkage-corrosion) but don't think even for a second that they were interested in understanding the physics and chemistry behind it. No way, guys, not at all! As soon a a set of equations was obtained, their only goal and interest was in applying strictly mathematical tools and tricks in order to solve them, with complete disregard of any practical use. The exact equivalent of composing music for the sake of combining sounds, with complete disregard of any effect the combination might have on the listener. No wonder I quit after two years.

To illustrate this, I vividly remember a brainstorming session in which the conservation of energy, aka the 1st Law of Thermodynamics was discussed. Now, I think we are all familiar with it, namely:



and I also think that none of us would lose much time over it.

Well, those guys spent a whole afternoon in heated discussions about which term should be on what side (left or right) in order to make the whole equation more convenient for a numerical solution. I could really not believe my eyes and ears watching and hearing them, and after a few long hours I was literally about to stand and cry out loud "Gentlemen, for fucking's sake, just write the fucking equation any fucking way you please and let's just fucking go home 'cause I'm fucking hungry and I could fucking use a fucking drink!" --- when the Chair of the Mathematics Department called the meeting off and announced it will continue next afternoon. Needless to say, next afternoon a (fake) headache made me unable to attend.
Every kind of music is good, except the boring kind. — Rossini

staxomega

#91
Quote from: 71 dB on July 09, 2022, 01:41:44 AM
By the way, math(s) is taught differently in school in different countries. The emphasis differs.

It seems for example that in the US math is "mechanical execution of rules". For example rules of how to use trigonometric substitutions in integration. Americans seem to be very good at being able to calculate almost any antiderivative using whatever substitution trick. In Finland trigonometric substitution in integration is super-advanced stuff at high-school level The emphasis is in solving problems using math. The idea is that if you know what you are doing you can always check somewhere how to calculate difficult integrals.

Ahh the tired internet trope of Europeans knowing more than Americans about their own system and how things are  ;D I'm glad a teacher was here to set this straight.

Quote from: 71 dB on July 09, 2022, 07:35:16 AM

Okay, but this is college math? Post high-school? I have never understood US school system.

Oh.

All high school. I think Commonwealth system would call this A levels or O levels.

Ages 15-18 (though you could be as much minus 6-8 months in 12th grade depending on your birthday), is grades 9-12 respectively to correlate with my previous post. Or in other words pre-calc in grade 11, AP Calc in grade 12, all before college/university.

I took my AP Calc exam nearly 20 years ago to the T and the exam was very much application and problem solving based, guaranteed you wouldn't have passed (> 3) with anything less. What you were describing in your previous post was what we called "plug and chug" and that might have got you through pre-calc with a passing grade, but it wouldn't have given you a solid foundation for next year.

My AP Physics and AP Comp Sci exams were also heavily application based. Comp Sci was even pen and paper so you couldn't rely on a compiler for trial and error either.

DavidW

Florestan, I have to say I love your story!  It painfully reminds me of grad school. :laugh:

Florestan

Quote from: DavidW on July 09, 2022, 10:55:43 AM
Florestan, I have to say I love your story!  It painfully reminds me of grad school. :laugh:

Every kind of music is good, except the boring kind. — Rossini

71 dB

Quote from: hvbias on July 09, 2022, 09:08:57 AM
Ahh the tired internet trope of Europeans knowing more than Americans about their own system and how things are  ;D I'm glad a teacher was here to set this straight.

I'm not claiming to know more. Seems like I know nothing. A misunderstanding from my part.
Spatial distortion is a serious problem deteriorating headphone listening.
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