On Jun 28, 2007, at 11:05 PM, Stewart Stremler wrote:
begin quoting Christopher Smith as of Thu, Jun 28, 2007 at
07:49:30PM -0700:
Andrew Lentvorski wrote:
Certain things need some rote. Multiplication tables, grammar
rules,
etc.
There is increasingly questionable need for rote learning as
computers
become increasingly prevalent. I know that sounds all "new math"-ish,
but the reality is that just using a calculator a lot (which
should be
an inherent part of the educational process) can help give
students the
same intuitive arithmetic insights that come from memorizing
multiplication tables.
I have cousins that are much younger than myself. They had a
calculator
on hand all through school, and never had to memorize anything.
They have no "intuitive arithmetic insights".
That is purely anecdotal. I work with a bunch of young math whiz's
who are dripping with insights, and never had to memorize
multiplication tables. I also know plenty of kids, classmates of
mine, who successfully memorized multiplication tables and had the
arithmetic insights of a rock. Oddly enough one of them is actually a
pretty good physicist.
Unfortunately neither approach can guarantee results.
I believe the ACM RISKS folks don't either. In searching for a mention
of the dangers, I ran across this:
http://bioinfo.uib.es/~joemiro/opinion/ParShfDgr.html
Ugh. I hate docs like this, although I do think his fundamental
insight correct: when you change teaching methods, there are likely
unintended consequences (duh, this is why education is an empirical
process). Many of the problems he identifies have been addressed by
the process. Others are just silly and suggest an entirely different
set of problems. More importantly, he fails to even acknowledge the
possibility that there may be benefits to the paradigm shift or that
they might outweigh the disadvantages.
The oddest one was his comment about his own experiences. He
demonstrated from his own experiences that it is possible to be
educated properly in the use of calculators such as to avoid the
pitfalls he recognizes later. The one pitfall he personally
experiences, not being able to "understand graphs" is incredibly sad.
By the time I was learning calculus, the use of calculators in math
class was an established practice. Our calculus curriculum however,
required the teaching of the "understanding" of graphs because it was
necessary to develop an intuition about calculus, and ironically our
curriculum used and *required* graphing calculators in order to teach
said intuition, and frankly I feel it was an asset.
Anyway, going through his points one-by-one:
The limits of calculators: this one is by far the lamest of the lot.
You teach kids to use a tool but you don't teach them the limitations
of the tool or how to get around them!? Come on! Furthermore teaching
how to overcome this limitations is intrinsic to teaching the
mathematical concepts being employed. Calculators or not, you still
need to teach those. I agree calculators make it possible to get a
certain kind of "result" without teaching this, but the same could be
said for various "tricks" or "methods" for doing calculations with
pen and pencil. In my father's day you used slide rules, and many of
his classmates memorized the "tricks" to using those without having
the faintest idea as to why they worked too. Yet another reason to
hate standardized tests and rote learning. ;-)
All operations are equal: I don't know where to begin with this one,
it's so laughable. First of all, as I'm sure others have observed,
with modern processors and programming languages, multiplication *is*
as fast doing the equivalent addition or bit shifting. You might
light up a few more transistors in some cases, but learning *that*
can and should be done on a case by case basis. This is part of what
he misses: not only has rote memorization become unnecessary, but
what is hard and what is easy may very well change from when one is
educated in grade school to when one joins the real world. Teaching
students to appreciate and learn from that is key. Furthermore, there
are tons of cases where things that might seem harder or easier for
pen and pencil work when they aren't (or worse still where the
opposite is true) with silicon. For example, using log tables to do
multiplication might teach you how to design a CPU, but if you apply
said teachings to software you'd end up wasting a lot of said CPU's
resources. I see this kind of broken thinking showing up all the time
when dealing with product managers who have just enough mathematical
understanding to be dangerous, but no knowledge of programming. They
keep coming up with specs and ideas about products that are framed by
their faulty frame of reference.
Loss of insight: I'm sure there is some loss of insight by not doing
things without pen and paper, and there is similarly a loss of
insight by not doing this with an abacus or a calculator. Each tool
also provides its insights. It is particularly sad to consider this
case though. You're telling me that if a five month job is done in a
few minutes you won't take the time to cross check your data? You
won't consider the possibility that your data is wrong or that you
input it incorrectly? Indeed, the lesson you learn with a calculator
is that these are the primary reasons you get things wrong, rather
than arithmetic errors while doing your computations. The stuff I do
these days with mathematics entirely uses the computer for all
calculations, but we spend a lot of time doing exactly the kind of
cross checking this Chinese professor did, and part of the reason we
do it so extensively is that it is comparatively cheap to do.
The great sorcerer: I remember reading an Isaac Asimov story about a
future where someone "invents" arithmetic all over again, with these
same kind of confusions and questions being asked of others
questioning his invention. Who knows, maybe it happens in an
Idiocracy world, but it doesn't happen when you drop calculators.
Indeed, with calculators, teachers are taught to *focus* on teaching
estimation and verification skills (one doesn't have to use pen and
pencil to know that probabilities can't be larger than 1!), because
errors can and do occur all the time. You also focus on teaching
rules about tracking levels of accuracy because most hand calculators
don't currently track that for you. Again, failing to teach the
limitations of a tool is not teaching how to use the tool in the
first place. And of course his problem with "knowing" that a binary
calculator cannot represent 0.1 exactly, fails to recognize the
capabilities of calculators that maintain internal results as
rational numbers... again, one must know the tool and its limitations.
The conclusion is the worst part though. There have been countless
changes in education, society, and the mental development of the
population in twenty years, but he fearless asserts the cause of the
changes he sees.
Simply doing lots of reading and writing
(which
should be an inherent part of an education process) with an
instructor/computer program that provides guidance and/or corrections
will produce a student with grammatical aptitude far above the
norm, or
even above those students who had memorized the rules but hadn't been
taught how to apply them.
Agreed. Memorization is not sufficient. You have to /use/ the
knowledge.
I would argue that learning the rules, but not memorizing them, and
using the knowledge will likely produce better results and a better
love for language and learning in many students. More importantly, it
allows one to focus more on instilling such things in the student
that might otherwise learn to hate a classroom filled with rote
learning.
Of course, I could be wrong about all these things. It is
unfortunately difficult to control variables effectively, but that is
all the more reason not to be too confident about drawing lines of
cause and effect. A good teacher will observe and react, and not by
always going back to the "old way" that didn't appear to have new
problems, but which carried with it a host of others.
--Chris
--
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