Re: The state of education in the USA.

On Apr 14, 1:34 pm, Robert Grumbine <b...@xxxxxxxxxxxxxxxxxx> wrote:
In article <37f1a42b-2425-4dcc-a977-31b1da4df...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, Tim Norfolk wrote:
On Apr 14, 6:54?am, tgdenn...@xxxxxxxxxxxxx wrote:
On Apr 13, 2:31?pm, Cory Albrecht <coryalbre...@xxxxxxxxxxx> wrote:

Tim Norfolk wrote, on 2008/04/12 23:10:

I must ask, since it is germane to the discussion, if you have ever
had to set up a Differential Equation (or set thereof) to solve a
physical problem? Simply doing that set-up requires a lot of tedious
algebra, which can be extremely enlightening, in terms of noting signs
and sizes of the terms involved. I will still claim, until evidence is
provided to the contrary, that those skills are not acquired by simply
teaching how to use software.

My mother, a retired high school mathematics teacher, has always said
this about calculators: If you don't know how to work it out with a
pencil & paper how are you going to know how to do it with a calculator?

I never learned how to find square roots by hand. Does that mean I
can't find square roots with a calculator?

The same thing undoubtedly holds true with software like MathCad, MatLab
or Mathematica.- Hide quoted text -

No, but if you were like many of the high school students that I am my
friends have tutored, you would reach for your calculator to compute
the square root of 9.

Piggy backing ...

  The students who only know math through calculator, I find, have no
way of sanity checking their calculator results.  If you have elementary
arithmetic in your head, when you try to compute sqrt(9) and get 81
for an answer (as often sqrt and x^2 are the same key, just shifted)
you know you oopsed.  Ditto for looking for sqrt(17) and getting 289 instead.
The problem isn't one of typing, but that they have no way of proofreading,
so to speak.


I will answer your other post and apologies for the one I missed.

Here's the problem. If you look for the sqrt of 9 and get 81, and
don't recognize that this must be wrong, then you haven't learned the
basic rule that the sqrt should be smaller than 9 not larger. That's
not a problem with calculators, but with teachers who haven't drilled
their students enough. Likewise, your statement that there is no sane
way to check the result is completely wrong. What could be easier
than trying 81*81? Again, this kind of discipline (always check your
result) can be taught just as well---more easily if you ask me---with
a calculator. How would you check finding the sqrt of a large number
calculated by hand? You would have to do a long multiplication. How
would you know whether a disagreement was the result of an error in
determining the sqrt or in squaring the number obtained?

  But the square root thing brings us to a different point about
numerate vs. non-.  I did once know an algorithm for extracting square
roots by hand.  That was a long time ago.  Since then (and even since
I learned calculus, which was also a long time ago) I've occasionally
wanted the square root of a number, but didn't have a calculator handy
with a sqrt key and didn't feel like reworking the Newton's method
iterative scheme (while good for the problem, I just didn't feel like
working it out).

   No problem -- if you understand arithmetic.  For example, find the
square root of 17 only by arithmetic.  Well, that'll be some number
that if we multiply it by the same number, we get 17.  So let's
guess a number.  Since we know arithmetic, we know that 4*4 is 16
and 5*5 is 25.  (Right off, an important mathematical concept --
bounding your answer from above and below.  Whatever the square
root of 17 is, even though we don't know it, we _do_ know that it's
between 4 and 5, a lot closer to 4.)  Being a lazy sort, I'll guess 4
to start with, then.  Now when we divide 17 by 4 (aha, if X*X = 17,
then 17/X = X also -- I should get back the same number in doing the
division), we get 4.25.  Not the same number, but pretty close.

  What to do now?  Well, 4 was obviously too small (since we got a larger
number back in the division) and 4.25 is obviously too large.  So,
take the average and try again.  4.125 gives 4.1212 (repeating as it
turns out).  If I only wanted 2 decimal digits, I could already stop.
But let's say I want 4.  Next trial is 4.1231, which gives 4.1231 and
we're done.

  Different benefit here, vs. a square root key, is that you _must_
consider how much precision is warranted for your problem.  

  Such thinking also sets you up for later calculus and numerical
modeling if you go there as it has implicitly involved limits and
iterative processes.  Whether or not you go to calculus and numerical
modeling, those concepts are useful in more mundane things including
navigating to a location.


And nothing about using a calculator *precludes* learning these
things. Again, this is up to the person teaching, who could use the
time saved for that purpose. All part of finishing math by 4th grade.
(see other post)

-tg




--
Robert Grumbinehttp://www.radix.net/~bobg/Science faqs and amateur activities notes and links.
Sagredo (Galileo Galilei) "You present these recondite matters with too much
evidence and ease; this great facility makes them less appreciated than they
would be had they been presented in a more abstruse manner." Two New Sciences


.

Relevant Pages

  • Re: The state of education in the USA.
    ... pencil & paper how are you going to know how to do it with a calculator? ... But the square root thing brings us to a different point about ... Robert Grumbine http://www.radix.net/~bobg/ Science faqs and amateur activities notes and links. ... Sagredo (Galileo Galilei) "You present these recondite matters with too much ...
    (talk.origins)
  • Re: The state of education in the USA.
    ... had to set up a Differential Equation (or set thereof) to solve a ... pencil & paper how are you going to know how to do it with a calculator? ... basic rule that the sqrt should be smaller than 9 not larger. ... wanted the square root of a number, but didn't have a calculator handy ...
    (talk.origins)
  • Re: The state of education in the USA.
    ... pencil & paper how are you going to know how to do it with a calculator? ... basic rule that the sqrt should be smaller than 9 not larger. ... wanted the square root of a number, but didn't have a calculator handy ...
    (talk.origins)
  • Re: The state of education in the USA.
    ... had to set up a Differential Equation (or set thereof) to solve a ... pencil & paper how are you going to know how to do it with a calculator? ... for an answer (as often sqrt and x^2 are the same key, ... wanted the square root of a number, but didn't have a calculator handy ...
    (talk.origins)
  • Re: The state of education in the USA.
    ... algebra becomes hard to follow, no matter what technology is used. ... education that left students better prepared for further study. ...   Unfortunately, the 'win by definition' approach is yours. ... It was also striking to me the number of calculator users (attempted ...
    (talk.origins)
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