[MATHEDCC] factoring

RayM (raypublk@san.rr.com)
Thu, 8 Apr 1999 18:24:50 -0700

Factoring quadratics is not a tricky thing to learn. If the coefficient of
x^2 is 1, the nutshell rule is "I'm looking for two numbers. The sum of
the numbers should be B and the product of these two numbers should be C."
At that level, it can be taught in the second grade. To ward off the
storm of protest about 'not understanding what's going on and they're
really just pushing a lot of symbols around', I'll say that my second
grader can factor something like
x^2 + 5x + 6 in his head in less than 10 seconds, then tell you the roots,
then plot the parabola using either the factored form or the original or
both at once. Most of his class can't do that; a few kids are working on
adding mixed numbers, the bulk are working with multiplication and division
problems, and there are some that are still struggling with addition.
Fortunately his teacher is willing to support that kind of differential
performance. One point that is relevant to the discussion here is that it
less than 1 hour of instruction to teach factoring quadratics and to touch
on factoring higher order polynomials. If there are about 300 hours of
instruction in a particular subject in a K-12 year, that does not seem like
huge investment to me. If the amount of factoring homework is kept to less
than another hour, what have you lost? I have to agree that I rarely see a
factorable quadratic in my work, but learning how to factor is part of
learning how to attack problems in general. I know the first few rows of
Pascal's triangle by heart and even if I'm looking at a combinatorics
problem and I see 1,3,3,1 or 1,4,6,4,1 , I can't help but making a
fleeting association with (a+b)^n; that is I factor a problem that isn't
even explicitly there.

On 3/26/99 Beth Hentges posted an article from the Los Angeles Times,
Monday, March 15, 1999. Many of the people in that article are at CC's and
are probably struggling with algebra no more complex than factoring a
quadratic. That is unfortunate and I don't know if there is a solution for
those people. We can reasonably ask if there is a solution for the next
generation. Part of that solution will be communicating to parents and
daycare providers that the years 0 to 5 are crucial for success in K and
like a row of dominos, crucial for success in 1-12. There are two fairly
new books that discuss both how large some of these performance
differentials are and address the genetic and environmental origins of the
differentials:
The Youngest Minds: Parenting and Genes in the Development of Intellect and
Emotion,
Anne Barnet and Richard Barnet,
352 P., Simon & Schuster, c. 1998 ISBN 0-684-81537-0

The Cradle of Knowledge: Development of Perception in Infancy,
Philip J. Kellman and Martha E. Arterberry,
c. 1998 MIT press, 369 p., ISBN 0-262-11232-9
Both books have extensive bibliographies and references into the peer
reviewed literature.

As for graphics calculators, I rarely see them. When I walk through LAX or
Dulles, I do see a lot of laptops. I'm part owner of a $12M/yr dollar
company and every engineer there strongly prefers his Pentium with 21"
monitor over any calculator. Most of the machinists use either 4 banger
or cheap scientifics. I asked a scientist at SAIC (~$1B/yr) if anyone used
graphics calculators there. The response was, "What's that?" NO ONE that
I know uses graphics calculators. Most employees have a computer on their
desk and everyone has access to at least one computer. I work with a
second company of about the same size and I have seen a graphics calculator
there once: a technician was taking a course at a CC and he had to learn
how to use the stupid thing. Now that the course is over he is irritated
at the waste of money. Laptops are now selling in the $700 range. Given
that laptops can support email, spell checkers, web browsing, GUI
interfaces, CDROM drives, and a very long list of related things that will
be useful in Literature, History, English, Chemistry courses, and life
after graduation, my advice would be to use the money that might go into a
graphics calculator to buy the best laptop you can or to upgrade one that
you already own. There may be a few isolated companies where the graphics
calculator culture is strong and I would advise CC instructors to survey
industry in their area before devoting much time to using these devices.

Ray M
raypublk@san.rr.com
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