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The 4th said she had a 'proof using words', but it was also just a description. I was amazed. They seem to have no conception of the meaning of proof. One student was making a very accurate diagram of a 3-4-5 triangle with the squares attached, and when I called it an example, he said he could change the sides to a, b, and c.
I asked the class, "How do we know this is true?" And they said we could measure it. I asked, "What if the third side is 4.9 inches, instead of 5 inches? Or 4.95?" They had trouble seeing why measurement wasn't enough. I'm going to learn so much from this class.
I showed them a visual proof and an algebraic proof, both starting with this tilted square inside a square. In the visual proof, you swing two of the triangles around, so the 4 triangles make two rectangular areas. What's left is one square area with side length a, and another with side length b. So elegant.
The algebraic proof describes the total area two ways:
(a+b)2= 4*(1/2*a*b)+c2
A few simple algebra steps will do it.
It felt important to discuss the meaning of proof. I have no idea if it stuck, and I'd like to come back to this during this course.
Edit on 9-5: Unknown pointed me to another great Vi Hart video. She does pretty much the same proof I showed above, but using paper that she folds (and rips). It's a great demo.
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I just wanted you to know that I have been enjoying your posts for awhile now, but never commented. I appreciate your honesty and your continual efforts to challenge your students to think deeply about math.
ReplyDeleteHave you thought of using Google Docs Forms to collect reflective information like you asked for in your Pre-Calc class? You can send a bulk email to the entire class with a link to the form, and then when they respond you have all of their thoughts on one spreadsheet rather than a bunch of separate emails to sort through.
I will be attempting to incorporate more communication via technology with my middle school students this year. We shall see how it goes :)
Also: Vi Hart did a visual "proof" of the Pythagorean Theorem recently with paper folding. She is awesome!
This is very similar to what I've been considering lately. I'll be interested in the developments in this class of yours.
ReplyDeleteInteresting. My daughter came home from U of I, where she is finishing up her bachelor's in engineering, with a math minor. She talked about her 400-level math course (don't remember which) and couldn't believe it was the first time many of the students had to deal with proofs. How can one do college math without proofs? I can't even do middle school math without proofs!
ReplyDeleteOn the flip side, the math majors couldn't believe her community college calculus course was swarming with proofs. We knew at the time that her calculus prof was exceptional, but we didn't realize quite how much so.
P.S.: As a biased observer, I think that daughter's younger sister came up with the world's cutest proof.
@Unknown, I may look courageous on the blog, but asking my students to publicly post about what they don't like in my class is beyond my courage threshold. I might try google docs for other sorts of questions, though. (Although I'm not sure if enough of my students have access to computers. With email, they can use a cell phone. For google docs, I'm not sure...)
ReplyDeleteI will look for that Vi Hart video.
@Julia, I hope your comment keeps me looking for ways to get my students doing the proving.
@Denise, I'd bet those 400-level students have been seeing profs do proofs for years, but are only now being asked to do the proofs themselves. As much as I enjoy lecturing, I don't think what we say has much effect on the majority of students.
Ok, I looked. And of course she's fabulous. It's pretty much the same proof I know and love, but hands on this time. And she talks about making it general. I'll add the link to my post. Thank you!
ReplyDeleteproofs?
ReplyDeleteearly and often is... or was for me...
an enormous cultural advantage
(games, puzzles... what have you...
but that *knowing for sure*
that you're right thing... wow...).
geometry is the locus classicus
(google huxley's _young_archimedes_
or for that matter the pascal
apocrypha). but, for me, and
because i mean it...
new paragraph...
count!!
count sets!!
count *finite* sets!!
count *small* finite sets!
find two
(or more)
*ways*!
prove it, prove it, prove it!!!
(if that's not math, what is?)
((
this is *not* a commerial for
_math_ed_zine_ or its
"lectures without words"
diagrams about (the diagram
known as) "pascal's triangle".
that's why there's no link
)). still.
somewhere recently you
(sue v.) remarked that maybe
you were starting to grok my
trust-the-code-above-all vibe.
good. way good. but better still
would be some moment-of-clarity
around the whole compass-and-
straightedge approach to
real-number arithmetic.
i only know two "basic" ideas in math.
counting and measuring. i've known
'em since i was a kid; moreover i'm
convinced onto this day that my
contemporary peers've mainly had ideas
pretty similar to mine; one is a child
of one's culture (& it takes a university
to raise a university brat).
now, tying these ideas together
turns out to be enormously subtle.
and... here comes my actual
*practical hint*... it seems to me
(as of this moment) that it'll've
been a good idea (in the context
of early-exposure-to-proof)
to've tightly narrowed one's focus
to *one or the other* of these
("continuous versus discrete")
big-picture subject areas.
as is seldom done. thus "sine"
is a certain ratio (when its
argument is the measure
of an *acute* angle) or
(more generally) the
*signed* ratio of the
horizontal-and-vertical
"legs" of a triangle
(or possibly of a "degenerate"
line segment)... *or*
the (loosely-defined) sum
of a certain (infinite, ghod
save the mark) collection
of so-called "odd" functions.
all of which yadda-yadda
typing makes sense *only*
to them that've already
learned some trig.
and there's no getting out
of the whole "square root"
business right in here.
i say it's spinach.
many, maybe most, of the
math-for-poets texts...
the terminal-introductory
stuff... give plenty of
multi-colored instruction
in SOCAHTOA (or what have
you) with no hint that
what is on offer is a
*language for playing games with*
instead of, say, just another
way of pushing people around.
viva math-mama!