Michael Cohen quotes Martin Gardner:
> "According to fuzzy math, this is a terrible way to teach the theorem.
Students
> must be allowed to discover it for themselves. As Cheney describes it,
they cut
> from graph paper squares with sides ranging from two to fifteen units.
(Such
> pieces are known as "manipulatives.") Then they play the following "game."
Using
> the edges of the squares, they form triangles of various shapes. The
"winner" is
> the first to discover that if the area of one square exactly equals the
combined
> areas of the other two squares, the triangle must have a right angle with
the
> largest square on its hypotenuse. For example, a triangle of sides 3,4,5.
> 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
> right triangles a^2+b^2=c^2.
>
> ""Constructivism" is the term for this kind of learning. It may take a
group
> several days to "construct" the Pythagorean theorem. 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.
