Where is balance? -- Re: Academic priorities
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From: Terje Mathisen on 2 May 2005 03:03 Steve Richfie1d wrote: > In short, the curriculum in our schools is about a century out of date. > Numerical methods now usually work better than analytical methods, yet > our schools spend years teaching analytical methods and usually *NO* > time on the subtleties of numerical methods. I didn't learn Runga Kutta > until COLLEGE! This should be taught just after algebra. Here's a key point: If you're at all interested in this stuff, then you don't wait for somebody to teach you whatever you need, or would like to know: Instead you explore, and in my case, discover RK and Taylor series while reading my father's university textbooks. I can still (30 years later!) remember the feeling of _power_ I got from grokking Taylor and realizing that I could now, modulo not having a computer to do the actual drudge work, calculate any trancendental function. Yeah, I know that spending a lot of physics classes (which were quite boring because they didn't ever teach me anything new) manually calculating 20+ digits of pi using the Taylor series for arctan were sort of worthless, but otoh, I have never regretted learning anything. In the current context of math vs arithmetic and mental calculation, I feel certain that having spent a lot of time on stuff like this do give me a big advantage when trying to spot stupid mistakes like misplaced decimal points, sums that don't add up etc. One example is that I find it impossible to watch a presentation with _any_ kind of graph or numbers on it, without automatically checking that they seem reasonable. It is very sad how often they do not, and how often noone else have ever noticed. :-( Terje -- - <Terje.Mathisen(a)hda.hydro.com> "almost all programming can be viewed as an exercise in caching"
From: glen herrmannsfeldt on 2 May 2005 04:36 Morten Reistad wrote: (snip) > Formulas used are from general relativity. They have to account for > the gravity well, the resultant time dilution, doppler effects, and > bending of space. Without these the GPS receiver would have a best > resolution of a few kilometers. Now it can be as good as a few meters. I just saw an article about it in Scientific American (Sept 2004). It seems that the GPS designers were unsure about the need for the general relativity correction, so they made it optional. It didn't take long to found out that it needed to be on. -- glen
From: Morten Reistad on 2 May 2005 06:31 In article <nsmdnYo0r83he-jfRVn-ug(a)comcast.com>, glen herrmannsfeldt <gah(a)ugcs.caltech.edu> wrote: >Morten Reistad wrote: > >(snip) > >> Formulas used are from general relativity. They have to account for >> the gravity well, the resultant time dilution, doppler effects, and >> bending of space. Without these the GPS receiver would have a best >> resolution of a few kilometers. Now it can be as good as a few meters. > >I just saw an article about it in Scientific American (Sept 2004). >It seems that the GPS designers were unsure about the need for >the general relativity correction, so they made it optional. >It didn't take long to found out that it needed to be on. This is why it can be quantified as good as it is. An unadjusted GPS gives an accuracy worse than the very best astronavigation. An amusing part of it is that astronavigation adjusts for deviation as a function of height over horizon. This table is based on observation, and does in fact take some relativisitic effects into account. -- mrr
From: jmfbahciv on 2 May 2005 04:45 [May I remove the xposting?] In article <d54jch$6id$1(a)osl016lin.hda.hydro.com>, Terje Mathisen <terje.mathisen(a)hda.hydro.com> wrote: >Steve Richfie1d wrote: >> In short, the curriculum in our schools is about a century out of date. >> Numerical methods now usually work better than analytical methods, yet >> our schools spend years teaching analytical methods and usually *NO* >> time on the subtleties of numerical methods. I didn't learn Runga Kutta >> until COLLEGE! This should be taught just after algebra. > >Here's a key point: > >If you're at all interested in this stuff, then you don't wait for >somebody to teach you whatever you need, or would like to know: That is key. I was not allowed to open those books. > >Instead you explore, and in my case, discover RK and Taylor series while >reading my father's university textbooks. You had access to other people's knowledge. I started this objecting when Steve claimed he couldn't find an example when there's oodles of info and examples out there. My thinking went thus: If he couldn't "see" examples, then his kids couldn't be exposed to the knowledge that he couldn't see. Everybody has their blind spots and lacks of knowledge. To limit any student to those blind spots is a disservice. That is all I'm yakking about. > >I can still (30 years later!) remember the feeling of _power_ I got from >grokking Taylor and realizing that I could now, modulo not having a >computer to do the actual drudge work, calculate any trancendental function. I still get goosebumps from the proof of the Fundamental Theorem of Calculus. That is beauty. > >Yeah, I know that spending a lot of physics classes (which were quite >boring because they didn't ever teach me anything new) manually >calculating 20+ digits of pi using the Taylor series for arctan were >sort of worthless, but otoh, I have never regretted learning anything. But the whole point of taking those classes was to get a key to the labs :-))). > >In the current context of math vs arithmetic and mental calculation, I >feel certain that having spent a lot of time on stuff like this do give >me a big advantage when trying to spot stupid mistakes like misplaced >decimal points, sums that don't add up etc. > >One example is that I find it impossible to watch a presentation with >_any_ kind of graph or numbers on it, without automatically checking >that they seem reasonable. It is very sad how often they do not, and how >often noone else have ever noticed. :-( Can you teach that? Some of these techniques seem to be something each individual picks up and is specialized to match his/her thinking styles. This is why mentorship is so valuable. The younger watch the older perform, pick up cues that can't be written down, modify them to match their own style and then later teach their styles to the young. /BAH Subtract a hundred and four for e-mail.
From: Terje Mathisen on 2 May 2005 08:36
Morten Reistad wrote: > Formulas used are from general relativity. They have to account for > the gravity well, the resultant time dilution, doppler effects, and > bending of space. Without these the GPS receiver would have a best > resolution of a few kilometers. Now it can be as good as a few meters. > > They also have to do advanced Maxwell and some resultant signal > analysis to recover the signal. I believe it was Yagi that first > formulated this theory. The GPS signal is well below the ambient noise floor, it is only recoverable at all due to spread spectrum techniques, where a 1023 bit (afair) spreading code is unique to each satellite. Matching the proper spreading code in both time and doppler stretching/contraction is what makes it possible to detect the signal at all. Since the sat orbits aren't totally circular, the GPS receivers also have to (or at least should?) correct for the ~12 hour period by which the onboard clocks speed up and slow down again as the orbit altitude changes. Terje -- - <Terje.Mathisen(a)hda.hydro.com> "almost all programming can be viewed as an exercise in caching" |