I've just come across this post by Gary Rubenstein: You-reeka-math.
Here's a question.
For 8th grade students, Rubenstein prefers a visual proof of the Pythagorean Theorem to the algebra-based proof in Eureka Math/engageny math.
I follow the engageny math proof, but I can't make head or tails of the visual proof Rubenstein argues is more appropriate.
Am I missing something?
Is the visual proof easier for the teacher to explain, but harder for me to explain to myself?
Related: I've been meaning to read Rubinstein's Reluctant Disciplinarian forever.
I remember a fabulous passage (which I don't have time to fact-check at the moment, so if I'm remembering some other book, I'll have to come back and revise): something to do with novice teachers (Rubinstein) asking the old hands how they dealt with things like spitballs and talking out of turn and the like.
The old hands' advice: "I don't put up with that."
"I don't put up with that" is a classic example of experts having no idea how they do what they do.
Filed under: cognitive unconscious.
The cognitive unconscious is a sadly neglected concept in K-12.
K-12 education: way too much thinking, way too little knowing.
If you had someone in the same room as you pointing at the parts of the visual proofs and explaining "this thing in this part of the diagram is the SAME as this thing in the other part of the diagram," you'd probably get it.
ReplyDeleteIn my class we printed out the diagrams for the visual proof, cut them up with scissors and rearranged the pieces to show that the areas represented by a^2 and b^2 could be put together to make a square whose area was c^2.
Proof by disection, which is what Gary presents in the second diagram where the squares can be cut up and reassembled doesn't answer basic questions. Does this work for every right triangle? Why? How are the lines drawn to get this result? THere has to be something more than someone pointing and saying "You see? You see? They're the same." Prove it for every right triangle then.
ReplyDeletehow anyone could actually *prefer* that cut-and-pasty
ReplyDeleteversion of proving-PT appears deeply mysterious to me.
okay, second thought. shallowly mysterious. there *are*
people... who may not even consider themselves very mathy...
who *like* doing those dissection problems. as games,
for fun. you can find 'em in GAMES magazine, for heck sake.
"visual learners", i suppose, with better imagine-moving-lines-
-on-paper skills than mine. also better move-pencil-on-paper
skills in certain ways. anyhow.
the very fact that *anyone* can prefer the weird-cut-and-pasty
thing over the proof-from-the-book-in-erdos's-heaven thing...
the very *model*, in my mind of a proof-from-the-book...
well, it's almost enough to make me take "learning styles" seriously.
any class led by me, with a^2 + b^2 = c^2 in it, is likely to get
the "proof from the book" in some form. (sometimes i'll just draw
the icon and wave my hands. sometimes i'll say more and scribble code.)
(also maybe the proof involving two formulas for the radius of an inscribed circle.
i learned that one from a great teacher and have presented it once.
with a great class.)
euclid's proof mystified me for a long time. (how could anyone *prefer*...?)
"pythagoras"
PS
ReplyDeletethe given diagram should, most often, be
supplemented with a geometric diagram for
(a + b)^2 = a^2 + b^2 + 2ab
(the big square "cut up" as
two smaller squares and
two (oblong) rectangles...
the proof from the book
is the *same* as the cut-and-paste
for this theorem as it seems...)
then you don't need so much algebra.
"BEHOLD" was said to be someone's
whole proof. (but his diagram was
inside-out by my lights.)
There is a difference between a demonstration and a proof.
ReplyDeleteAnd there are puzzles designed to fool those who trust every diagram they see.
https://en.wikipedia.org/wiki/Missing_square_puzzle
-- Phil
https://en.wikipedia.org/wiki/Missing_square_puzzle
ReplyDeletea classic. the w'edia entry is badly marred by the animations.
sam loyd might also be usefully mentioned in this context.
https://en.wikipedia.org/wiki/Sam_Loyd
all of stage magic and most political discourse
is "designed to fool" those who "trust" what "they see".
of science and religion?
responsible parties differ.
trust *nobody*.
(anyhow, that works for *me*.)
or, as the epistle has it.
prove everything. hold fast
to that which is true.
i'm not a robot
ReplyDeletesometimes the system works
ReplyDelete"pythagorAs"
ReplyDeletestill not a robot.
ReplyDeletedeath to ggooogl.