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Robert J. Kolker
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 Posted: Sun Jun 08, 2008 6:17 am Post subject: Re: Not Just the US With Education Problems |
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Walter Bushell wrote:
| Quote: |
Cheaper, faster and smaller will win every time. Not to mention quieter.
Also, they quickly became more capable.
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A twelve digit TI handheld can be had for less than $100.00 and it can
do a limited range of programming as well. One can enter quantities with
up to three levels of parens or braces and one can do statistic stuff
like correlations, averaging and least squares fitting in a jiffy. The
old mechanicals simply did not have the capability.
But the old fashioned devices, limited though they wre could be
comprehended as -mechanical- entities who principle of operations were
readily understood. Try explaiing to your grandma what is going on
inside a modern solid state computer.
Bob Kolker
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Robert J. Kolker
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| Posted: Sun Jun 08, 2008 6:19 am Post subject: Re: Not Just the US With Education Problems |
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Paul J Gans wrote:
| Quote: |
Yup. They killed off base 10 logs too.
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Actually the logarithmic algorithm (base e) is alive, well, miniaturized
and transistorized. Can you do a Taylor Series approximation of a
natural logarithm. Sure you can, but your hand held device can do it
better and faster. Base ten stuff is gotten from base e by a scaling
factor.
Bob Kolker |
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Robert J. Kolker
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| Posted: Sun Jun 08, 2008 6:24 am Post subject: Re: Not Just the US With Education Problems |
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Tim Norfolk wrote:
| Quote: |
And, without doing logs with tables or slide rules, and all the fun of
interpolation, the students reaching Calculus II and Differential
Equations now have no clue about size estimates, or how logarithmic
differentiation works, or how the standard exponential growth model
works.
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This can be gotten by proper discipline in classrooms. The rule is no
machinery, no hand held devices until one masters the underlying
principles of calculation (series approximations, error estimates based
on first order infinitesimals etc etc). First one must do it the old
fashioned way until the principles are mastered, then one can use the
machiner safely. Just using the machinery reduces numerical analysis to
mechanical plugging into formulas or pushing button. The brainwork is
removed.
I have four children (in their late 40's), and I would not let them use
devices to calculate until they could do it manually or with pencil and
paper. I absolutely insisted that they be able to do order of magnitude
estimates of outcomes without the detailed arithmetic. That gave them a
"feel" for the numbers.
I hope my children will impose the same discipline on the grandchildren.
Bob Kolker
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Vernon Balbert
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| Posted: Sun Jun 08, 2008 10:26 am Post subject: Re: Not Just the US With Education Problems |
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On 6/7/2008 6:09 PM, Robert J. Kolker went clickity clack on the
keyboard and produced this interesting bit of text:
| Quote: |
Walter Bushell wrote:
Just how old are you. Frieden calculators from the styling my predate
WWII and maybe I.
The Freiden all electromechanic twelve digit calculator goes back to the
middle 30s at least. Watch the thing multiply is a gasser. The carriage
skips one place at a time so it is really doing addition and shifting.
Likewise, division is substraction and shifting. When electronic
calculators (as they were first called) came along the shifting was all
electronic in registers, but the principle was identical.
My pop was a CPA and he had Frieden calculators in the office but he
would not let me use them until I became proficient at good old paper
and pencil addition, substraction, multiplication and division. Once he
was satisfied I could do accurate order of magnitude estimates he let me
use the machinery.
I also had a Kuder calculator, a German made mechanical job. It was
cylindrical and portions rotated. So the register was really a circular
gear track and it was moved by twisting the top of the device. I always
thought of it as a pepper mill that could do arithmetic. I wish I had
kept it. It would be worth over a thousand dollars as a collectors item.
Once battery or photoelectric hand calculators became inexpensive it was
the end of mechanical calculators and slide rules. Slide rules were in
heavy use in the 60's and were mostly gone by the 80's.
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My dad had one of these. I don't know if it was made by the same
company or not, but when I was about 15 or 16 I got to play with it. It
was a lot of fun to watch. I remember being incredibly delighted that
it could multiply 123456789 by 987654321 which my TI-59 (the calculator
I used at the time which had 10 digits with 2 internal) could not do
with the same precision. It took a bit for it to do, but boy, it was
incredible to watch. I wanted to look inside it to see how it worked,
but I hated the idea of breaking the device.
--
"Pinky, are you pondering what I'm pondering?"
"Wuh, I think so, Brain, but isn't Regis Philbin already married?" |
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Mike Dworetsky
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| Posted: Sun Jun 08, 2008 11:01 am Post subject: Re: Not Just the US With Education Problems |
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"Paul J Gans" <gans@panix.com> wrote in message
news:g2f2bi$87q$4@reader2.panix.com...
| Quote: |
Walter Bushell <proto@xxx.com> wrote:
In article <6ask8qF38sc6eU1@mid.individual.net>,
"alwaysaskingquestions" <alwaysaskingquestions@gmail.com> wrote:
Some of it may have to do with personal inclination - I have always been
fascinated with both puzzles in general and with knowing how things
work; in
my first job - just before the introduction of desktop calculators - I
had
to learn how to use a slide rule and was thought it was a fantastic
invention.
Just how old are you. Frieden calculators from the styling my predate
WWII and maybe I.
The Friden calculator was in major use up through 1970. I recall
doing the calculations for a paper on one in that year.
Fridens had 10 digit keyboards and the high end models could
automatically extract square roots of 20 digit numbers.
See <http://www.oldcalculatormuseum.com/fridenstw.html> for
a view of a late model.
|
I was at an advanced summer school for teens in 1961 and we had to do orbit
calculations with calculators (!). During the course of the programme we
took delivery of the latest Friden calculator that could extract square
roots. For amusement one day we tried taking concatenated square roots of a
number several times then squared it back up again. And that's when we
learned a practical lesson about the meaning of precision and significant
digits in computations.
We also used an early digital computer, the Bendix G-15, about the size of a
large freezer.
| Quote: |
The mechanical calculator is arguably the finest example of the
mechanical engineer's art. They were enormously complex
instruments powered by an electric motor. The insides were
a three dimensional mass of gears, levers, cams, and whatnots.
The reference above shows some "inside" shots.
The pocket calculator killed them off fairly quickly.
--
--- Paul J. Gans
|
--
Mike Dworetsky
(Remove pants sp*mbl*ck to reply) |
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Paul J Gans
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| Posted: Mon Jun 09, 2008 1:08 am Post subject: Re: Not Just the US With Education Problems |
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Robert J. Kolker <bobkolker@comcast.net> wrote:
| Quote: |
Paul J Gans wrote:
You are working from memory, which is OK. But the Fridens were
10 digit machines, not 12 and the mechanical hand-held calculator
was a Curta. I *really* wanted one of those...
Yup. That is it. Memory slip. It was such a beautiful machine. I did not
give it away or sell. I LOST it and I still weep tears thinking about
it. It was such an elegant device, it did not matter to me whether it
was a practical calculating device or not. It was perfection of the
mechanical arts.
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Yes they were. The Scientific American had an article on the
Curtas a year or two ago. You could probably find it on line.
It was interesting.
Those things were mechanical marvels.
--
--- Paul J. Gans |
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Tim Norfolk
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| Posted: Mon Jun 09, 2008 4:58 am Post subject: Re: Not Just the US With Education Problems |
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On Jun 8, 6:44 pm, Paul J Gans <g...@panix.com> wrote:
| Quote: |
Robert J. Kolker <bobkol...@comcast.net> wrote:
Paul J Gans wrote:
Yup. They killed off base 10 logs too.
Actually the logarithmic algorithm (base e) is alive, well, miniaturized
and transistorized. Can you do a Taylor Series approximation of a
natural logarithm. Sure you can, but your hand held device can do it
better and faster. Base ten stuff is gotten from base e by a scaling
factor.
Which makes me wrong how?
By the way, base e stuff is gotten from base 10 by
using a scaling factor too.
Neither electronic calculators nor computers use a simple
Taylor series expansion for logs (or most other functions
either). There are more efficient methods.
--
--- Paul J. Gans
|
The Cordic algorithm, I believe, which behaves very like a Pade
approximant, still derived from the Taylor series. |
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Tim Norfolk
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| Posted: Mon Jun 09, 2008 4:59 am Post subject: Re: Not Just the US With Education Problems |
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On Jun 8, 6:44 pm, Bob Casanova <nos...@buzz.off> wrote:
| Quote: |
On Sat, 7 Jun 2008 17:53:48 -0700 (PDT), the following
appeared in talk.origins, posted by Tim Norfolk
timsn...@aol.com>:
On Jun 7, 8:23?pm, Paul J Gans <g...@panix.com> wrote:
Walter Bushell <pr...@xxx.com> wrote:
In article <g2f2bi$87...@reader2.panix.com>,
Paul J Gans <g...@panix.com> wrote:
Walter Bushell <pr...@xxx.com> wrote:
In article <6ask8qF38sc6...@mid.individual.net>,
"alwaysaskingquestions" <alwaysaskingquesti...@gmail.com> wrote:
Some of it may have to do with personal inclination - I have always been
fascinated with both puzzles in general and with knowing how things work;
in
my first job - just before the introduction of desktop calculators - I had
to learn how to use a slide rule and was thought it was a fantastic
invention.
Just how old are you. Frieden calculators from the styling my predate
WWII and maybe I.
The Friden calculator was in major use up through 1970. ?I recall
doing the calculations for a paper on one in that year.
I did a least squares fit in 1967 on one. I think it took a week or two.
Fridens had 10 digit keyboards and the high end models could
automatically extract square roots of 20 digit numbers.
See <http://www.oldcalculatormuseum.com/fridenstw.html> for
a view of a late model.
The mechanical calculator is arguably the finest example of the
mechanical engineer's art. ?They were enormously complex
instruments powered by an electric motor. ?The insides were
a three dimensional mass of gears, levers, cams, and whatnots.
The reference above shows some "inside" shots.
The pocket calculator killed them off fairly quickly.
Cheaper, faster and smaller will win every time. Not to mention quieter.
Also, they quickly became more capable.
Yup. ?They killed off base 10 logs too.
And, without doing logs with tables or slide rules, and all the fun of
interpolation, the students reaching Calculus II and Differential
Equations now have no clue about size estimates, or how logarithmic
differentiation works, or how the standard exponential growth model
works.
Could be the case today, although when I took undergrad
engineering math in the late 70s/early 80s (through diff eq)
all those were covered pretty thoroughly.
--
Bob C.
"Evidence confirming an observation is
evidence that the observation is wrong."
- McNameless- Hide quoted text -
- Show quoted text -
|
They still are covered, but are much harder for the students, for the
reasons mentioned above. |
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Robert J. Kolker
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| Posted: Mon Jun 09, 2008 6:00 am Post subject: Re: Not Just the US With Education Problems |
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Paul J Gans wrote:
| Quote: |
By the way, base e stuff is gotten from base 10 by
using a scaling factor too.
Neither electronic calculators nor computers use a simple
Taylor series expansion for logs (or most other functions
either). There are more efficient methods.
|
What method do they use? A Pade approximation?
Bob Kolker
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Robert J. Kolker
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| Posted: Mon Jun 09, 2008 6:03 am Post subject: Re: Not Just the US With Education Problems |
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Paul J Gans wrote:
| Quote: |
By the way, base e stuff is gotten from base 10 by
using a scaling factor too.
Neither electronic calculators nor computers use a simple
Taylor series expansion for logs (or most other functions
either). There are more efficient methods.
|
Oh yes. Integrating 1/(1-x) will do the trick, but you still end up with
a power series.
Bob Kolker
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Rupert Morrish
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| Posted: Mon Jun 09, 2008 7:32 am Post subject: Re: Not Just the US With Education Problems |
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alwaysaskingquestions wrote:
| Quote: |
"Walter Bushell" <proto@xxx.com> wrote in message
news:proto-059A39.17420907062008@70-1-84-166.area1.spcsdns.net...
In article <6ask8qF38sc6eU1@mid.individual.net>,
"alwaysaskingquestions" <alwaysaskingquestions@gmail.com> wrote:
Some of it may have to do with personal inclination - I have always been
fascinated with both puzzles in general and with knowing how things work;
in
my first job - just before the introduction of desktop calculators - I
had
to learn how to use a slide rule and was thought it was a fantastic
invention.
Just how old are you. Frieden calculators from the styling my predate
WWII and maybe I.
I started my first job as a trainee Industrial Engineer in 1967. The company
I worked with had Frieden machines but only in the wages office - the
company had 1500 employees, most of them on piece rate schemes so wages
calculation involved some serious number crunching. (On a side note, the
company computerised its wages calculations about 1975; this involved
putting the raw data onto punched cards which were couriered to a computer
bureau for overnight processing; I assume computer bureaux are well and
truly a thing of the past now.)
|
Not at all: http://code.google.com/appengine/
[snip]
-----------------
www.Newsgroup-Binaries.com - *Completion*Retention*Speed*
Access your favorite newsgroups from home or on the road
----------------- |
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Walter Bushell
Guest
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| Posted: Mon Jun 09, 2008 8:12 am Post subject: Re: Not Just the US With Education Problems |
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In article <g2hqdq$in7$4@reader2.panix.com>,
Paul J Gans <gans@panix.com> wrote:
| Quote: |
What happens is that many kids, not understanding how
to compute a square root decide that they will never
master math.
The algorithm for the computation isn't important. The
concept of the square root is.
A caculator allows students to play with square roots
without having to know the algorithm -- which is complex
and very difficult to prove to grade or high school
students.
|
x[n+1]=(x[n]+c/x[n])/2 where c is the number we are taking the square
root of.
is hard to prove, given a sloppy guess it converges fast as a little
experience will show. Not only that errors do not accumulate.
--
What is done in the heat of battle is (normatively) judged
by different standards than what is leisurely planned in
comfortable conference rooms. |
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John McKendry
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| Posted: Mon Jun 09, 2008 8:43 am Post subject: Re: Not Just the US With Education Problems |
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On Sun, 08 Jun 2008 23:27:54 +0000, Paul J Gans wrote:
| Quote: |
tgdenning@earthlink.net wrote:
On Jun 8, 2:58Â am, "Mike Dworetsky" <platinum...@pants.btinternet.com
wrote:
"Paul J Gans" <g...@panix.com> wrote in
messagenews:g2f2bi$87q$4@reader2.panix.com...
Walter Bushell <pr...@xxx.com> wrote:
In article <6ask8qF38sc6...@mid.individual.net>,
"alwaysaskingquestions" <alwaysaskingquesti...@gmail.com> wrote:
Some of it may have to do with personal inclination - I have
always been fascinated with both puzzles in general and with
knowing how things work; in
my first job - just before the introduction of desktop calculators
- I had
to learn how to use a slide rule and was thought it was a
fantastic invention.
Just how old are you. Frieden calculators from the styling my
predate WWII and maybe I.
The Friden calculator was in major use up through 1970. Â I recall
doing the calculations for a paper on one in that year.
Fridens had 10 digit keyboards and the high end models could
automatically extract square roots of 20 digit numbers.
See <http://www.oldcalculatormuseum.com/fridenstw.html> for a view
of a late model.
I was at an advanced summer school for teens in 1961 and we had to do
orbit calculations with calculators (!). Â During the course of the
programme we took delivery of the latest Friden calculator that could
extract square roots. Â For amusement one day we tried taking
concatenated square roots of a number several times then squared it
back up again. Â And that's when we learned a practical lesson about
the meaning of precision and significant digits in computations.
An excellent example. By removing the time, tediousness, and human error
associated with doing monkey-work by hand, a far more expansive lesson
can be delivered. And that was with technology from almost 50 years ago.
Carl Friderich Gauss, undoubtedly one of the most brilliant
mathematicians who ever lived, wasted three years of his live manually
calculating the orbit of the moon. Given his productivity, we lost out
on a bunch of "Gauss's Theorems" and tons of insights.
|
First of all, I'm having a hard time finding corroboration of
this story at all, so it may not even be true. One of Gauss' earliest
accomplishments was the calculation of the orbit of the asteroid
Ceres on the basis of three observations, but if that's the story
you're thinking of, it really seems to be more a counterexample
to the claim you seem to be making, because it was the sort of
creative mathematics that absolutely could not have been accomplished
with a calculator alone; it required a deep familiarity with
the geometry of conic sections. And it took three months.
http://www.maa.org/mathland/mathtrek_4_19_99.html
But if he had taken three years to calculate the orbit of the moon,
I still doubt that it would have been lost time. You make it sound
like grunt-work arithmetic, like calculating seven hundred digits
of pi. Calculating the orbit of Ceres meant inventing a new method
of calculating.
The broader argument here seems to be about what counts as monkey
work, and I have to vote with Tim Norfolk overall. I agree that
calculating the sine of 39 degrees is monkey work, but writing out
an expression for the sine of 67.5 degrees is not. Anyone who
wants to do any level of creative work involving math should
recognize the latter as a couple of applications of basic trig
identities to the sine of 45 degrees (or better, 135 degrees).
I don't care if he/she looks in a book for the trig identities;
the important thing is to know they exist. You will never learn
the trig identities from a calculator. If all you care about
is the numeric answer, it doesn't matter, but if all you care
about is the numeric answer, you'll never learn what it is
to do math, either.
John |
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John McKendry
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| Posted: Mon Jun 09, 2008 9:03 am Post subject: Re: Not Just the US With Education Problems |
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On Sun, 08 Jun 2008 21:00:37 -0400, Robert J. Kolker wrote:
| Quote: |
Paul J Gans wrote:
By the way, base e stuff is gotten from base 10 by using a scaling
factor too.
Neither electronic calculators nor computers use a simple Taylor series
expansion for logs (or most other functions either). There are more
efficient methods.
What method do they use? A Pade approximation?
Bob Kolker
Something called the CORDIC - Coordinate Rotation Digital Computer - |
algorithm.
http://www.math.utep.edu/Faculty/helmut/presentations/1997/cordic/wcordic.html
and the first two papers at
http://www.math.ufl.edu/~haven/papers.html .
John |
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Michael Siemon
Guest
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| Posted: Mon Jun 09, 2008 9:31 am Post subject: Re: Not Just the US With Education Problems |
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In article <g2hpoq$in7$2@reader2.panix.com>,
Paul J Gans <gans@panix.com> wrote:
....
| Quote: |
Long ago, in the early days of the republic, I memorized the
logs to the base 10 of both 2 and 3 to five decimal place.
It was amazing how much off the cuff calculation I could
do. Logs of all the single digit integers except 7 were
rapidly available.
I still know them. They sit there taking up space in my
limited memory capacity and won't go away.
|
0.30103, as I recall for log 2 (I never bothered with log 3  ). It
prepared the way for my understanding of the approximate equality of
10^3 and 2^10, which plays out in a _lot_ of digital age stuff that
makes no explicit reference to logarithms.
Thanks for the memory Paul; I hadn't thought about log 2 for decades! |
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