Re: The state of education in the USA.

On Apr 14, 9:19 pm, Tim Norfolk <timsn...@xxxxxxx> wrote:
On Apr 14, 5:09�pm, tgdenn...@xxxxxxxxxxxxx wrote:



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.

You mean, like sqrt(0.25) = 0.5, and 0.5 < 0.25?


What's your point? The basic rule is that for numbers >1, the sqrt is
smaller than than the number. Are you making some semantic argument
about what's a 'basic rule', or just being silly?

-tg









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/Sciencefaqsand amateur activities notes and links.

<snip>


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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? ... 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.
    ... can't find square roots with a calculator? ... basic rule that the sqrt should be smaller than 9 not larger. ...
    (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.
    ... pencil & paper how are you going to know how to do it with a calculator? ...   The students who only know math through calculator, I find, have no ... 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.
    ... 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)
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