[math-fun] Novel ways to present proofs
What novel ways have been used to present proofs or (more generally) mathematical ideas? I've encountered people answer olympiad problems with proofs written as limericks and sonnets before; there's also the Socratic dialogue, used by one student in a selection test and later to popularise the elementary measure-theoretic proof of Poncelet's porism: https://cp4space.wordpress.com/2014/04/19/poncelets-porism-the-socratic-dial... I've also heard of lecture in which the lecturer wrote down the truth table for logical conjunction on an overhead projector transparency like so: t ^ t = t t ^ f = f f ^ t = f f ^ f = f He then flipped the entire OHP sheet over, transforming every AND into an OR whilst transforming the `f's into `t's and vice-versa. Indeed, this is not the only spectacular use of an overhead projector. The late Christopher Bradley famously used an overhead projector to demonstrate projective transformations in a beautifully literal way: he tilted the projector, demonstrating the following concepts: -- Interchangeability of different types of conics; -- Interchangeability of parallel lines and convergent lines; -- Preservation of collinearity and concurrency; -- Conservation of cross-ratio. Any other examples of novel presentations of mathematical ideas? Sincerely, Adam P. Goucher
The classic math theorem ode has to be Soddy's "The Kiss Precise" (1937) --- quoted, with later instalments of varying pedigree, at http://www.pballew.net/soddy.html http://mathbabe.org/category/becky-jaffe/ Again, not quite to the point I know --- and I may well anyway have inflicted this anecdote on the list before, in which case I apologise --- but I'm irresistably reminded of a disaster which befell the enthusiastic but hapless young mathematics teacher whose lot it was to present the "new math" (mostly binary arithmetic, as far as I can recall) to the parents of children attending a British secondary school in the 1970's. He had prepared his material painstakingly in advance, including hand-written OHP tables (high-tech, back then) illustrating addition and multiplication modulo two. To facilitate comprehension by the great unwashed innumerati, for binary "0" and "1" he had thoughtfully substituted "E" and "O", representing "odd" and "even" respectively. Unfortunately, it was customary in university algebra courses of the time --- which he had doubtless only recently attended as a student himself --- to represent by "o" and "e" the zero and unit members of a ring, as a result of which the subsequent course of his presentation proceeded to unfold with the remorseless inevitability of a Sophoclean tragedy. Up went his tables, the first entry "E + E = E --- oh dear, that's not right!" he interjected chattily, hastily altering E to O . "E + O = O --- no, no, that should be ---", crossly altering O to E . And so it continued, pressing on into an ever-deepening morass of perplexity and despondency for lecturer and audience alike. By the time I had managed to suppress my natural modesty and imminent hysterics sufficiently to attempt to intervene, the poor fellow was in a state of blind panic, and quite incapable of processing corrective input. Whether the disrepute which subsequently overtook this particular educational initiative can be laid entirely at his door is debatable; but if many of his colleagues suffered similar misfortunes on opening night, it surely won't have helped matters! Fred Lunnon On 4/24/14, Adam P. Goucher <apgoucher@gmx.com> wrote:
What novel ways have been used to present proofs or (more generally) mathematical ideas?
I've encountered people answer olympiad problems with proofs written as limericks and sonnets before; there's also the Socratic dialogue, used by one student in a selection test and later to popularise the elementary measure-theoretic proof of Poncelet's porism:
https://cp4space.wordpress.com/2014/04/19/poncelets-porism-the-socratic-dial...
I've also heard of lecture in which the lecturer wrote down the truth table for logical conjunction on an overhead projector transparency like so:
t ^ t = t t ^ f = f f ^ t = f f ^ f = f
He then flipped the entire OHP sheet over, transforming every AND into an OR whilst transforming the `f's into `t's and vice-versa.
Indeed, this is not the only spectacular use of an overhead projector. The late Christopher Bradley famously used an overhead projector to demonstrate projective transformations in a beautifully literal way: he tilted the projector, demonstrating the following concepts:
-- Interchangeability of different types of conics; -- Interchangeability of parallel lines and convergent lines; -- Preservation of collinearity and concurrency; -- Conservation of cross-ratio.
Any other examples of novel presentations of mathematical ideas?
Sincerely,
Adam P. Goucher
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Thu, Apr 24, 2014 at 6:55 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Any other examples of novel presentations of mathematical ideas?
Kevin Wald's presentation of a proof of the irrationality of Pi, set to a possibly familiar tune: http://www.math.uchicago.edu/~wald/lit/pi_proof.txt He also had a crossword in the Mathematical Intelligencer that proved the irrationality of Phi. Several of the clues are "first part of the proof", "second part of the proof", etc. The clever thing is that the crossword requires a diagram, as they generally do, and the proof requires a diagram, as they often do, but it turns out to be the same diagram! Andy Latto
The main (indeed, only!) theorem in my article "A Galois Connection in the Social Network" (Math. Mag. 85, No. 1, 34-36 (2012); http://jamespropp.org/galois.pdf) was written to be sung to the tune of "The Irish Washerwoman": (Oh,) the people who know all the people who know all the people you know all are people you know and the people you know all are people who know all the people who know all the people you know. However, the proof that I give is a traditional proof (not a musical one). Jim Propp On Thu, Apr 24, 2014 at 8:44 PM, Andy Latto <andy.latto@pobox.com> wrote:
On Thu, Apr 24, 2014 at 6:55 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Any other examples of novel presentations of mathematical ideas?
Kevin Wald's presentation of a proof of the irrationality of Pi, set to a possibly familiar tune:
http://www.math.uchicago.edu/~wald/lit/pi_proof.txt
He also had a crossword in the Mathematical Intelligencer that proved the irrationality of Phi. Several of the clues are "first part of the proof", "second part of the proof", etc. The clever thing is that the crossword requires a diagram, as they generally do, and the proof requires a diagram, as they often do, but it turns out to be the same diagram!
Andy Latto
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Oh, yeah, that crossword is immortal! It's available from Springer for a lot of money, unless you have access: < http://link.springer.com/article/10.1007/BF02985658 >. On Apr 24, 2014, at 5:44 PM, Andy Latto <andy.latto@pobox.com> wrote:
[Kevin Wald] also had a crossword in the Mathematical Intelligencer that proved the irrationality of Phi. Several of the clues are "first part of the proof", "second part of the proof", etc. The clever thing is that the crossword requires a diagram, as they generally do, and the proof requires a diagram, as they often do, but it turns out to be the same diagram!
I was the editor for Kevin's golden ratio cryptic in the Intelligencer, and have a copy of it. Send me email if you'd like a copy. (I'd send it to math-fun, but as I recall the mailing list doesn't allow attachments.) --Michael On Thu, Apr 24, 2014 at 10:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Oh, yeah, that crossword is immortal!
It's available from Springer for a lot of money, unless you have access: < http://link.springer.com/article/10.1007/BF02985658 >.
On Apr 24, 2014, at 5:44 PM, Andy Latto <andy.latto@pobox.com> wrote:
[Kevin Wald] also had a crossword in the Mathematical Intelligencer that proved the irrationality of Phi. Several of the clues are "first part of the proof", "second part of the proof", etc. The clever thing is that the crossword requires a diagram, as they generally do, and the proof requires a diagram, as they often do, but it turns out to be the same diagram!
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
Michael, Thanks -- I'd love to have a electronic copy of that. Much appreciated. --Dan On Apr 24, 2014, at 7:42 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
I was the editor for Kevin's golden ratio cryptic in the Intelligencer, and have a copy of it. Send me email if you'd like a copy. (I'd send it to math-fun, but as I recall the mailing list doesn't allow attachments.)
I'd love a copy, thanks! Andy On Thu, Apr 24, 2014 at 10:42 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
I was the editor for Kevin's golden ratio cryptic in the Intelligencer, and have a copy of it. Send me email if you'd like a copy. (I'd send it to math-fun, but as I recall the mailing list doesn't allow attachments.)
--Michael
On Thu, Apr 24, 2014 at 10:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Oh, yeah, that crossword is immortal!
It's available from Springer for a lot of money, unless you have access: < http://link.springer.com/article/10.1007/BF02985658 >.
On Apr 24, 2014, at 5:44 PM, Andy Latto <andy.latto@pobox.com> wrote:
[Kevin Wald] also had a crossword in the Mathematical Intelligencer that proved the irrationality of Phi. Several of the clues are "first part of the proof", "second part of the proof", etc. The clever thing is that the crossword requires a diagram, as they generally do, and the proof requires a diagram, as they often do, but it turns out to be the same diagram!
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
Here are a couple that I’ve used in the past. I find that the gimmicks that work best usually do so for no reasons connected to the concept I’m trying to get across. electoral college FFT In the week leading up to a US presidential election I use my laptop to perform the convolution of 51 Bernoulli distributions to get the distribution of electoral votes. It’s amazing how a slight change in the “margin of error” in the polling data affects the convergence to the normal distribution. coffee cup double cover As required of any lecturer introducing spin-1/2 and the topology of the 3D rotation group, I do the standard 4pi twist on a cup while not letting go of the handle — mocha with extra whipped cream for added suspense. beard bisection When word came down from on high, “teach them how to estimate”, I responded by estimating the number of hairs in a beard. At the first lecture I showed up with a full beard, the second with half shaved off, etc. each time with yet another half removed. When there was just a few square mm of beard left we counted the hairs. homework cycles In this one I was among the students, about 100 in discrete math for CS. Manuel Blum (then at Berkeley) decided he would hand back our homework at random. He then set about calculating the probability nobody got their own work handed back to them. On Apr 24, 2014, at 6:55 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
What novel ways have been used to present proofs or (more generally) mathematical ideas?
I've encountered people answer olympiad problems with proofs written as limericks and sonnets before; there's also the Socratic dialogue, used by one student in a selection test and later to popularise the elementary measure-theoretic proof of Poncelet's porism:
https://cp4space.wordpress.com/2014/04/19/poncelets-porism-the-socratic-dial...
I've also heard of lecture in which the lecturer wrote down the truth table for logical conjunction on an overhead projector transparency like so:
t ^ t = t t ^ f = f f ^ t = f f ^ f = f
He then flipped the entire OHP sheet over, transforming every AND into an OR whilst transforming the `f's into `t's and vice-versa.
Indeed, this is not the only spectacular use of an overhead projector. The late Christopher Bradley famously used an overhead projector to demonstrate projective transformations in a beautifully literal way: he tilted the projector, demonstrating the following concepts:
-- Interchangeability of different types of conics; -- Interchangeability of parallel lines and convergent lines; -- Preservation of collinearity and concurrency; -- Conservation of cross-ratio.
Any other examples of novel presentations of mathematical ideas?
Sincerely,
Adam P. Goucher
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
About three years ago I looked for a video of the coffee cup double cover. I couldn't find one, so I made one myself. I should have held the handle! Here it is https://www.youtube.com/watch?v=Rzt_byhgujg On Sun, Apr 27, 2014 at 9:14 AM, Veit Elser <ve10@cornell.edu> wrote:
Here are a couple that I’ve used in the past. I find that the gimmicks that work best usually do so for no reasons connected to the concept I’m trying to get across.
electoral college FFT
In the week leading up to a US presidential election I use my laptop to perform the convolution of 51 Bernoulli distributions to get the distribution of electoral votes. It’s amazing how a slight change in the “margin of error” in the polling data affects the convergence to the normal distribution.
coffee cup double cover
As required of any lecturer introducing spin-1/2 and the topology of the 3D rotation group, I do the standard 4pi twist on a cup while not letting go of the handle — mocha with extra whipped cream for added suspense.
beard bisection
When word came down from on high, “teach them how to estimate”, I responded by estimating the number of hairs in a beard. At the first lecture I showed up with a full beard, the second with half shaved off, etc. each time with yet another half removed. When there was just a few square mm of beard left we counted the hairs.
homework cycles
In this one I was among the students, about 100 in discrete math for CS. Manuel Blum (then at Berkeley) decided he would hand back our homework at random. He then set about calculating the probability nobody got their own work handed back to them.
On Apr 24, 2014, at 6:55 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
What novel ways have been used to present proofs or (more generally) mathematical ideas?
I've encountered people answer olympiad problems with proofs written as limericks and sonnets before; there's also the Socratic dialogue, used by one student in a selection test and later to popularise the elementary measure-theoretic proof of Poncelet's porism:
https://cp4space.wordpress.com/2014/04/19/poncelets-porism-the-socratic-dial...
I've also heard of lecture in which the lecturer wrote down the truth table for logical conjunction on an overhead projector transparency like so:
t ^ t = t t ^ f = f f ^ t = f f ^ f = f
He then flipped the entire OHP sheet over, transforming every AND into an OR whilst transforming the `f's into `t's and vice-versa.
Indeed, this is not the only spectacular use of an overhead projector. The late Christopher Bradley famously used an overhead projector to demonstrate projective transformations in a beautifully literal way: he tilted the projector, demonstrating the following concepts:
-- Interchangeability of different types of conics; -- Interchangeability of parallel lines and convergent lines; -- Preservation of collinearity and concurrency; -- Conservation of cross-ratio.
Any other examples of novel presentations of mathematical ideas?
Sincerely,
Adam P. Goucher
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
Spilling some coffee is a reasonable sacrifice for the sake of pedagogy. However, sometimes these things can go badly wrong. Take the "BICEP2 bracketology" I tried this year. As everyone knows, the fluctuations in the March Madness playoffs are so huge, the idea of making a prediction for the entire “bracket”, or even just the final teams, is just ridiculous. Not one to miss out on the madness in these predictions, this year I showed the class how I was going to use the BICEP2 cosmic microwave polarization pattern (on a patch of sky that just happened to match the NCAA official bracket graphic) to pick the final matchup. (The polarizations indicate which teams advance — was Einstein ever wrong?). So imagine my chagrin when it turned out the BICEP2 data correctly predicted UConn! Creighton, the other team predicted was eliminated early. Still, it makes you wonder … -Veit On Apr 27, 2014, at 1:18 PM, Thane Plambeck <tplambeck@gmail.com> wrote:
About three years ago I looked for a video of the coffee cup double cover. I couldn't find one, so I made one myself. I should have held the handle! Here it is
________________________________ From: Veit Elser <ve10@cornell.edu> To: math-fun <math-fun@mailman.xmission.com> Sent: Monday, April 28, 2014 9:16 AM Subject: Re: [math-fun] Novel ways to present proofs
Spilling some coffee is a reasonable sacrifice for the sake of pedagogy. However, sometimes these things can go badly wrong. Take the "BICEP2 bracketology" I tried this year.
As everyone knows, the fluctuations in the March Madness playoffs are so huge, the idea of making a prediction for the entire “bracket”, or even just the final teams, is just ridiculous.
Not one to miss out on the madness in these predictions, this year I showed the class how I was going to use the BICEP2 cosmic microwave polarization pattern (on a patch of sky that just happened to match the NCAA official bracket graphic) to pick the final matchup. (The polarizations indicate which teams advance — was Einstein ever wrong?). So imagine my chagrin when it turned out the BICEP2 data correctly predicted UConn! Creighton, the other team predicted was eliminated early. Still, it makes you wonder …
-Veit
------------------------------------------------------ But, of course, if this prediction had failed, you would not have made this post.
-- Gene
Ha! As a grad student at Berkeley, I cut off one half of my beard, just for fun, and left it that way for several months. Never thought of the pedagogical applications, though. --Dan P.S. Nu, so how many hairs *are* in a beard? Veit Elser wrote: ----- beard bisection When word came down from on high, “teach them how to estimate”, I responded by estimating the number of hairs in a beard. At the first lecture I showed up with a full beard, the second with half shaved off, etc. each time with yet another half removed. When there was just a few square mm of beard left we counted the hairs. -----
It's not as good as beard-cutting, but the birthday paradox makes a nice class discussion in a class of 30 or so. (Once I got unlucky in a class of 35 and they all had distinct birthdays... which only happens with probability 0.18.) Cris On Apr 27, 2014, at 2:02 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Ha! As a grad student at Berkeley, I cut off one half of my beard, just for fun, and left it that way for several months. Never thought of the pedagogical applications, though.
--Dan
P.S. Nu, so how many hairs *are* in a beard?
Veit Elser wrote:
----- beard bisection
When word came down from on high, “teach them how to estimate”, I responded by estimating the number of hairs in a beard. At the first lecture I showed up with a full beard, the second with half shaved off, etc. each time with yet another half removed. When there was just a few square mm of beard left we counted the hairs. ----- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
Cris Moore writes: (Once I got unlucky in a class of 35 and they all had distinct birthdays...
which only happens with probability 0.18.)
Actually, it's a consequence of Murphy's Law that the probability of this happening in a probability class is significantly higher than 0.18. Jim Propp
participants (10)
-
Adam P. Goucher -
Andy Latto -
Cris Moore -
Dan Asimov -
Eugene Salamin -
Fred Lunnon -
James Propp -
Michael Kleber -
Thane Plambeck -
Veit Elser