Re: The state of education in the USA.

In article <e3916a3c-f0e2-42fe-bce0-80f3e1a5e382@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, tgdenning@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.

Except or course when the square root is larger than the number
(per Tim's followup, the point of which you totally missed), or when
it doesn't exist in the first place. But why should your calculator-only
student know any such thing?

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.

Trust the software is no different than trust the calculator.
You advise the former, so you're in no position to now invoke
teaching skepticism of tools.

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?

Not relevant since you trust the software and advocate that
others do so.

In fact, those of us who don't trust the software also don't
trust the software that checks the first software. Same thing
for hand computation. The difference being, that we learn the
fact ourselves (after enough oopses of multiplying 2*3 and getting
5, we understand something of human fallibility). Facts learned
for oneself stay much better than anything somebody else tells
you.

[trim my example of an arithmetical method for finding
a square root]

  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)

What other post of yours is that? I saw nothing that explained
how you were going to teach math.

But the goalpost shift is impressive even by local standards.
You've gone from making the unsupported claim that something was
better, now to requiring proof that it is _impossible_ for your
idea to work at all.

The practical result is that real students taught for years by real
teachers in the method that you advocate come out knowing less math and
are less capable of learning any further math than students taught by
older means. Those of us concerned with reality and math education
find that a fair reason to not use your method.


Very related example I encountered, but in reading. There have
been even more fads on teaching reading the last 50 years than
for math. One of them, 'see and say', was -- as you like for
math -- based on an interpretation of how expert readers read.
The researchers noted that experts have a large sight vocabulary,
words being effectively single entities, not things to be
'sounded out'. So they ran a test on students who'd already
had some years of phonetic teaching and showed an 'impressive'
gain in vocabulary in the test period over students who
continued to be taught by phonetics. So the method gained fad
status and was applied to all students from the start.

One reason I know this story is that I volunteered at a local
literacy council for adult education. They knew when the different
school systems in the area adopted see and say by the age of
their adult students. Per your new goal post, it wasn't _impossible_
for see and say students to learn to read, at least some. But
the odds went way down. And the ability to learn more reading
later was also crippled. (Trying to teach phonics to my
adult student was nearly impossible because he simply had been
taught too firmly that words don't have small parts. Also
impaired his ability to hear parts of words; once you can't see
the parts, nor hear them, you're pretty close to toast in learning
a written language like English. He might have been ok with Chinese.)


--
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
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

.

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