(warning: rant ahead)
nobody's talking about the kind of teacher who's leading this [ideal
pythagorean] activity. until elementary school teachers are paid as well as
corporate VPs, few of them will have the kind of mental energy and the
combination of interpersonal and mathematical skills that will make robert's
comments relevant to the typical classroom. avoiding the win-lose pitfall is
crucial; if kids get too frustrated (if it's too much like work and not enough
like play) and if they don't know where they're supposed to go or if the answer
is withheld for too long, they'll just learn how to not-care and not how to have
fun with math.
our society generates huge amounts of sanctimony about the importance of
traditional women's work, while expecting to pay little for it because
motherhood is priceless, and because if traditional women were paid as much as
people (particularly conservative men) say they value them nobody could afford
them. there's even more discomfort with paying for mothering than with paying
for sex. meanwhile, professors of education agree that early childhood training
is crucial but the only people willing to take care of them are those who either
care more for kids than for money or who can't get a better job. the
throughness of the prejudice against women and women's work is measured by the
feminist choice to go after jobs traditionally held by men rather than to
require society to pay them [women] what they're really worth.
if taking care of kids really is work, then welfare mothers *are* working and we
should stop forcing them to get fastfood jobs without healthcare that they have
to commute 2 hours to and spend all their money on crummy childcare.
classically, men face death in war and women in childbirth; thus, mothers should
get the same respect as the military, including early pensions.
no wonder kids look at american society and care more about a piece of paper
than about really learning for its own sake. why bother to do the things that
really matter when it doesn't pay?
Robert Dawson wrote:
> Michael Cohen quotes Martin Gardner:
>
> snip
> > Students who never discover the theorem are said to have "lost" the game.
> In this
> > manner,
> ** with no help from teacher, **
> the children are supposed to discover that with [snip] Even worse, the paper
> > game may bore a group of students more than hearing a good teacher explain
> > the theorem on the blackboard.
>
> The point about taking huge amounts of time and boring the students is a
> good one. However, the basic idea of having students (re)discover things for
> themselves is a good one.
> What's the synthesis? Well, too much time is spent doing nothing, for a
> start. With ziplock bags of bristol-board squares, reused from year to year,
> significant time coud be saved. Cutting out squares is not mathematics; one
> suspects that it might be being used here to "dilute" the math with stuff
> that (almost) everybody can do.
>
> True, the squares aren't needed at all - they could use sticks & compute
> the squares themselves; but the diagram produced is a cultural icon, and
> beautiful in its own right, and I would not want to deprive the students of
> that. Also, they need a bigger set - there aren't enough Pythagorean
> triples
> with x,y,z <= 15.
>
> At a deeper level, though, discovery is itself a skill, and skills are
> learned by doing them properly. With a well-constructed outline, the
> students could reach the same conclusions much faster, having been guided
> through a sensible approach (based, perhaps, on a standardized approach to
> problem-solving? I know that approaches such as Polya's are not the
> be-all-and-end-all, but they enable the beginner to work efficiently, and
> like the ban on whittling towards your thumb, when you understand when to
> break the rule you will be ready to break it.)
>
> I also don't like the artificial dichotomy of "winners" and "losers"
> (And I thought these people were the politically-correct self-esteem
> pushers???). Finally, half the class will already know the answer - *and*
> the joke about the squaw on the hippopotamus. There needs to be more
> challenge!
>
> One possibility, if one wants to put some sort of challenge into the
> process, would be a graduated series of hints. The idea is to complete the
> exercise using as few hints as possible. Sort of like playing Myst while in
> possession of a hint book.
>
> Question: If you arrange the edges of squares of edge length 3,4, and 5
> into a triangle (see picture) you will see that one corner appears to be a
> _right_angle_, and the triangle is a _right_triangle_. Find some other
> triples of edge lengths for which this happens (Hint 1).
>
> [Hint 1: 12, 13, ? ]
>
> Try to find a rule that lets you predict, from the edge lengths, whether
> the triangle will be right (Hint 2) (Hint 3)
>
> [Hint 2: Think about the _squares_]
> [Hint 3: For (3,4,5) the squares of the edge lengths are 9,16, and 25.
> For (6,8,10) they are 36,64, and 100.]
>
> And so on....
>
> One could then lead them through the "scissors proof" of Pythagoras'
> theorem; challenge the faster kids to show that there are infinitely many
> Pythagorean triples other than multiples of one; ask if there is a
> Pythagorean triple of odd numbers (find or disprove)...
>
> The basic idea of discovery is a good one. However, the presence of
> absence of a few well-chosen hints makes a huge difference.
--
Any resemblance of any of the above opinions to anybody's official position is
completely coincidental.
Muriel Strand, P.E.
Air Resources Engineer
CA Air Resources Board
2020 L Street
Sacramento, CA 59814
916-324-9661
916-327-8524 (fax)
www.arb.ca.gov
