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Re: [Phys-l] what shall we do about math?



Thsi is discussed in Gelfand and Shen, "Algebra", p.96. Apparently, when mathematicians write \/x, where x is a positive number, they are specifying that the number is the positiive square root. That there are two roots is discussed at length, as is the fact that the equation
x^{2}=p (p a positive number)
has two, and only two solutions, and that the solution is something called an irrational number. The discussion is part of a section on quadratic equations.
Regards,
Jack

On Thu, 16 Jul 2009, Steve Highland wrote:

I've been visiting a couple of summer algebra classes this past week and I'm
depressed by how impossible a situation the students and teachers find
themselves in.

Several of these students have failed the course up to three times. They
are forced to repeat it over and over. The summer class is supposed to
cover the same amount of material the regular class did, so the teacher told
me they have to cover about a chapter a day. And the one group I met today
has four hour sessions each day. That would fry my brain.

This just strikes me as torture. Are there any better approaches out there?
I would think there should be some other option for students who can't pass
the course than just forcing them to fail it repeatedly.


The class I met today had a dispute going (good!) over what they saw as a
contradiction in the book. It defined the square roots of a number a to be
the solutions of

X^2 = a

-- so that both the positive and negative values qualify.

But then it said the value of {radical sign}a was just the positive value.
So the text defined "square root" written out in words one way and square
root written as a radical a different way. It seemingly made a distinction
between "a square root" and "the square root."

Man, that's confusing. Does anybody have a better way out of this language
dilemma?

Steve Highland



Quoting Paul Lockhart:

A musician wakes from a terrible nightmare. In his
dream he finds
himself in a society where music education has been made
mandatory.
³We are helping our students become more competitive in an

increasingly sound-filled world.² Educators, school systems, and the
state
are put in charge of this vital project. Studies are
commissioned,
committees are formed, and decisions are made‹ all
without the advice or
participation of a single working musician or
composer.

Since musicians
are known to set down their ideas in the form of
sheet music, these curious
black dots and lines must constitute the
³language of music.² It is
imperative that students become fluent in
this language if they are to
attain any degree of musical competence;
indeed, it would be ludicrous to
expect a child to sing a song or
play an instrument without having a
thorough grounding in music
notation and theory. Playing and listening to
music, let alone
composing an original piece, are considered very advanced
topics and
are generally put off until college, and more often graduate
school.

...

Sadly, our present system of mathematics education is
precisely this
kind of nightmare. In fact, if I had to design a mechanism
for the
express purpose of destroying a child¹s natural curiosity and love
of
pattern-making, I couldn¹t possibly do as good a job as is currently

being done‹ I simply wouldn¹t have the imagination to come up with
the kind
of senseless, soulcrushing ideas that constitute
contemporary mathematics
education.

It goes on like this for 25 pages:

http://www.maa.org/devlin/LockhartsLament.pdf

It's well worth reading. You
won't soon forget it.

Other reviewers have called the article "humorous" but
not
everyone will find it so; slapstick stops being funny if/when
you start
to identify with the guy whose kiester is getting
slapped with a stick. And
an insult hurts in direct proportion
to its accuracy.

Background on the
article and the author:
http://www.maa.org/devlin/devlin_03_08.html

I'm
surprised Lockhart hasn't yet been arrested and charged with
corrupting the
youth of Athens.

_______________________________________________
Forum for
Physics
Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu
/mailman/listinfo/phys-l



_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l


--
"Trust me. I have a lot of experience at this."
General Custer's unremembered message to his men,
just before leading them into the Little Big Horn Valley