Something beautiful AND exciting, could get students interested in mathe

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Something beautiful AND exciting, could get students interested in mathe

GS Chandy
>From Scientific American:

> Polynomial Plot: Simple Math Expressions Yield
> Intricate Visual Patterns [Slide Show]
> Plotting the roots of run-of-the-mill polynomials yields
> dazzling results, by John Matson
http://www.scientificamerican.com/article.cfm?
id=math-> polynomial-roots
>
I provide this link to support my strong belief that an excellent way to get students who may be getting turned off by the 'traditional approach' to math would be to show him/her some of the beautiful/wonderful/striking/.... things that can be done with math.  Nowadays, with all these wonderful computer graphics available, we can really spark a firestorm in the student's mind!

That kind of exercise develops from an element located at or near the lowest level in one of the structures I drew up to guide me when I wanted to convince a student that math can be very exciting indeed:

"To stimulate student interest by showing them fascinating things that can be done with and through math: e.g. George A. Hart’s ‘mathematical sculptures’; ‘Moebius curves; fractals; the Yoshimoto cube..Etc, etc."  

(I had attached the structure to one of my earlier posts).  

Much later, this element worked very well indeed with my grand-daughter when she told me: "Oh, math is SO boring!"

(Interesting aside: John Baez - the physicist who first put up these plots for  at his website for Dan Christensen - happens to be the father of the renowned singer, Joan Baez.  In the world of physics, John Baez is probably as famous as is his daughter in the world outside).

GSC
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Re: Something beautiful AND exciting, could get students interested in mathe

Dave L. Renfro
GS Chandy wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=7009733

> (Interesting aside: John Baez - the physicist who first
> put up these plots for at his website for Dan Christensen -
> happens to be the father of the renowned singer, Joan Baez.
> In the world of physics, John Baez is probably as famous
> as is his daughter in the world outside).

For what it's worth, I've never heard of Joan Baez,
but as for John, he's far more of a mathematician than
a physicist. His Ph.D. is in math, his employment is
in a math dept., . . . Of course, more precisely he's
a mathematical physicist, but despite the ordering of
the words, this means he's primarily a mathematican
who investigates issues in physics.

Dave L. Renfro

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Re: Something beautiful AND exciting, could get students interested in mathe

Robert Hansen
In reply to this post by GS Chandy
I will go along with this GS, but I think what is important in these discussions is "What do you mean by firestorm?" I mean can you describe the vision of the firestorm. I think for these engagements to be successful in the context of these discussions that firestorm should translate to a curiosity about the math behind the pictures and then to the math itself. The firestorm should mean that kid goes home and starts looking at his textbook and really getting interested in it.

These engagements need to be followed with vigorous discussion. All too often the engagement is mostly show and little tell.

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Re: Something beautiful AND exciting, could get students interested in mathe

GS Chandy
In reply to this post by GS Chandy
Dave Renfro posted Mar 13, 2010 3:37 AM:
>
> For what it's worth, I've never heard of Joan Baez,
> but as for John, he's far more of a mathematician
> than a physicist. His Ph.D. is in math, his employment
> is in a math dept., . . . Of course, more precisely he's
> a mathematical physicist, but despite the ordering of
> the words, this means he's primarily a mathematican
> who investigates issues in physics.
>
Indeed you're right, Dave, about John Baez being "primarily a mathematician who investigates issues in physics", rather than a "physicist".

But I'm surprised you've not heard of his daughter, Joan.  She was once rather spectacularly famous to the world at large, particularly as a 'protest singer' (against the Vietnam war) in the early to late 'sixties. Early sixties was when I was at grad school in the US, in a math department - and we ALL knew about and listened to Joan Baez.  She was, in fact, quite as well-known in those days as Bob Dylan.  Very few, even in our math department, knew of John Baez.

I do not hear her or of her much, these days, alas.  She had a wonderful voice and she was a wonderful person.  (I guess it must have helped that she was also very good-looking).

GSC
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Re: Something beautiful AND exciting, could get students interested in mathe

Jonathan Groves-3
In reply to this post by GS Chandy
GS Chandy,

I take it that you referring to the same article that stimulated a brief
discussion on mathedcc in late January and up to about mid-February of
this year.  I cannot get the link to work, but one on mathedcc did
provide a link that works (assuming that this is the same article):

http://mathforum.org/kb/message.jspa?messageID=6954904&tstart=15.

Here is a link to a message of mine on that list that gives some of
my impressions of that article:

http://mathforum.org/kb/message.jspa?messageID=6980366&tstart=15.


Jonathan Groves  



On 3/12/2010 at 12:38 pm, GS Chandy wrote:

> >From Scientific American:
>
> > Polynomial Plot: Simple Math Expressions Yield
> > Intricate Visual Patterns [Slide Show]
> > Plotting the roots of run-of-the-mill polynomials
> yields
> > dazzling results, by John Matson
> http://www.scientificamerican.com/article.cfm?
> id=math-> polynomial-roots
> >
> I provide this link to support my strong belief that
> an excellent way to get students who may be getting
> turned off by the 'traditional approach' to math
> would be to show him/her some of the
> beautiful/wonderful/striking/.... things that can be
> done with math.  Nowadays, with all these wonderful
> computer graphics available, we can really spark a
> firestorm in the student's mind!
>
> That kind of exercise develops from an element
> located at or near the lowest level in one of the
> structures I drew up to guide me when I wanted to
> convince a student that math can be very exciting
> indeed:
>
> "To stimulate student interest by showing them
> fascinating things that can be done with and through
> math: e.g. George A. Hart’s ‘mathematical
> sculptures’; ‘Moebius curves; fractals; the Yoshimoto
> cube..Etc, etc."  
>
> (I had attached the structure to one of my earlier
> posts).  
>
> Much later, this element worked very well indeed with
> my grand-daughter when she told me: "Oh, math is SO
> boring!"
>
> (Interesting aside: John Baez - the physicist who
> first put up these plots for  at his website for Dan
> Christensen - happens to be the father of the
> renowned singer, Joan Baez.  In the world of physics,
> John Baez is probably as famous as is his daughter in
> the world outside).
>
> GSC
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Re: Something beautiful AND exciting, could get students interested in mathe

GS Chandy
In reply to this post by GS Chandy
Jonathan Groves had posted Mar 13, 2010 2:21 PM:
> I take it that you referring to the same article that
> stimulated a brief discussion on mathedcc in late
> January and up to about mid-February of this year. I
> cannot get the link to work, but one on mathedcc did
> provide a link that works (assuming that this is the
> same article):
> http://mathforum.org/kb/message.jspa?
> messageID=6954904&tstart=15

Yes, indeed, it is the same article.  The link you have posted contains a 'tinyurl' link, which is much better:  For convenience of those who might wish to see it, I paste the tinyurl link below (in case the longer links do not work well):

http://tinyurl.com/yfxh3j7.

Thanks and regards
GSC
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Re: Something beautiful AND exciting, could get students interested in mathe

kirby urner-4
In reply to this post by GS Chandy
On Fri, Mar 12, 2010 at 9:38 AM, GS Chandy <[hidden email]> wrote:

> >From Scientific American:
>
>> Polynomial Plot: Simple Math Expressions Yield
>> Intricate Visual Patterns [Slide Show]
>> Plotting the roots of run-of-the-mill polynomials yields
>> dazzling results, by John Matson
> http://www.scientificamerican.com/article.cfm?
> id=math-> polynomial-roots
>>
> I provide this link to support my strong belief that an excellent way to get students who may be getting turned off by the 'traditional approach' to math would be to show him/her some of the beautiful/wonderful/striking/.... things that can be done with math.  Nowadays, with all these wonderful computer graphics available, we can really spark a firestorm in the student's mind!
>

I'm somewhat skeptical that traditionalists never show intriguing eye
candy etc. i.e. no matter what your pedagogical approach, isn't it
somewhat expected that you will show some cool art?

The devil is in the details I'd say, i.e. how accessible, how
hands-on, how participatory is all this.

If the message is "if you study hard for twenty more years, you too
will be able to make these kinds of pictures" that's different from
"we will learn about how to do this today, as a project."

For my part, I continually and consistently bring up polyhedra as
visually exciting, and also accessible.

I also tell students they're on the front lines simply for daring to
use a tetrahedron for a unit of volume.  No need to study twenty years
to see some action in Math Wars (if that's what they're looking for).

The unit volume tetrahedron meme connects them to all kinds of
relevant thinking, about how humans might make a success of themselves
if they dared to be iconoclastic.

I think this is a primary distinction you need to make, when talking
about "traditional" versus "not traditional":  to what extent do you
portray mathematics as helping to challenge a status quo versus
helping to entrench a status quo?

I would put it to you that mathematics has often played a subversive
role, e.g. having those Hindu-Arabic algorithms flooding into Europe,
along with double-entry bookkeeping, is what gave rise to a merchant
class with independent means, an ability to become patrons of the arts
independently of church authorities.

Just telling history like that is already non-traditional vis-a-vis
how math is currently taught (as ahistorical, with tiny forays into
biography in this or that sidebar).

I think if you want to break out of a traditional mold you need to (a)
cast math as a change agent, as sometimes subversive (the story of RSA
would be apropos at this point) and (b) you need to share lore,
stories, meaningful ones, not just "story problems".

What happens here on math-teach when I mention tetrahedral mensuration
(tetrahedron as unit volume) is I'm simply ignored (not on other
issues perhaps, but on that one in particular).

I show this pattern to my students, as proof that I'm up against some
deeply conditioned reflexes.  This is very accessible to them, i.e.
they don't need a lot of background to see what I'm talking about.
They understand tetrahedral mensuration, and they understand what it
means to be ignored.  Looking over my shoulder is a piece of cake
then, quite enlightening.

If you're unwilling to entertain an alternative base volume and use
that in lesson plans (ala that NCTM lesson plan on Tetrahedral Kites),
then in my book you're a traditionalist by definition.  I may not
distinguish between Haim, Wayne or MPG in that sense -- three peas in
a pod.  You've seemed quite traditional yourself in this respect.

> That kind of exercise develops from an element located at or near the lowest level in one of the structures I drew up to guide me when I wanted to convince a student that math can be very exciting indeed:
>
> "To stimulate student interest by showing them fascinating things that can be done with and through math: e.g. George A. Hart’s ‘mathematical sculptures’; ‘Moebius curves; fractals; the Yoshimoto cube..Etc, etc."
>

George Hart has done Pavilion of Polyhedreality and is an enthusiastic
supporter of Zome, a geometric construction kit I've mentioned often.

> (I had attached the structure to one of my earlier posts).
>
> Much later, this element worked very well indeed with my grand-daughter when she told me: "Oh, math is SO boring!"
>
> (Interesting aside: John Baez - the physicist who first put up these plots for  at his website for Dan Christensen - happens to be the father of the renowned singer, Joan Baez.  In the world of physics, John Baez is probably as famous as is his daughter in the world outside).
>
> GSC
>

John Baez attended Princeton living at 2 Dickinson Street, as I did,
though not at the same time.

Here's a recent picture of Joan Baez with her arm around someone (not
me, sat next to him at dinner last Thursday and he was bragging about
this picture).

http://www.flickr.com/photos/17157315@N00/4429354327/

Note: their relationship (John and Joan) is "cousin", not father-daughter.

Kirby
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Re: Something beautiful AND exciting, could get students interested in mathe

GS Chandy
In reply to this post by GS Chandy
Kirby Urner posted  Mar 14, 2010 12:49 AM:

> On Fri, Mar 12, 2010 at 9:38 AM, GS Chandy
> <[hidden email]> wrote:
> > >From Scientific American:
> >
> >> Polynomial Plot: Simple Math Expressions Yield
> >> Intricate Visual Patterns [Slide Show]
> >> Plotting the roots of run-of-the-mill polynomials
> yields
> >> dazzling results, by John Matson
> > http://www.scientificamerican.com/article.cfm?
> > id=math-> polynomial-roots
> >>
> > I provide this link to support my strong belief
> that an excellent way to get students who may be
> getting turned off by the 'traditional approach' to
> math would be to show him/her some of the
> beautiful/wonderful/striking/.... things that can be
> done with math.  Nowadays, with all these wonderful
> computer graphics available, we can really spark a
> firestorm in the student's mind!
> >
>
> I'm somewhat skeptical that traditionalists never
> show intriguing eye candy etc. i.e. no matter what your
> pedagogical approach, isn't it somewhat expected that
> you will show some cool art?
>
By and large, the 'traditionalists' would prefer, I think, that their charges learn the rules, live by the rules, die for the rules.
>
> The devil is in the details I'd say, i.e. how accessible,
> how hands-on, how participatory is all this.
>
We should definitely aim at making all learning (not just of math) as participatory as possible, I would say.
>
> If the message is "if you study hard for twenty more
> years, you too will be able to make these kinds of
> pictures" that's different from "we will learn about how
> to do this today, as a project."
>
> For my part, I continually and consistently bring up
> polyhedra as visually exciting, and also accessible.
>
Polyhedra are, I agree, very exciting constructs for students to learn and understand math.  In fact, when I started a company called "Thinkit" (which, way back in the early 'seventies, aimed at designing and developing educational kits based on math), we started practically all our work with polyhedra, creating kits to help students construct and understand the wonders of polyhedra.  Now that company is no more, but I still do tell all teachers that I meet about the readily accessible wonders of polyhedra to help them get their students hooked on to math.  (I've mentioned the 'Yoshimoto cube' in several of my messages - and I believe you had responded to one of those messages).

>
> I also tell students they're on the front lines
> simply for daring to use a tetrahedron for a unit of
> volume.  No need to  study twenty years to see some
> action in Math Wars (if that's what they're looking for).
>
> The unit volume tetrahedron meme connects them to
> all  kinds of relevant thinking, about how humans
> might make a success of themselves if they dared to
> be iconoclastic.
>
The linkages you perceive between the tetrahedron, human success and iconoclastic thinking are not very clear to me, I'm afraid.  (Are you referring to Buckminster Fuller, perhaps?)
>
> I think this is a primary distinction you need to
> make, when talking about "traditional" versus "not
> traditional":  to  what extent do you portray
> mathematics as helping to challenge a status quo
> versus helping to entrench a status quo?
>
It is only the people who use a tool, say math, that can use it for "challenging/entrenching the status quo"; the tool itself has no such intent or interest.  For instance the powers-that-be would surely like to use math (or any other tool) to entrench themselves in power.  All too often, they misunderstand the historical reality and do not properly learn to use the tools available.  The people who want change are often in a position to learn to use the tools they have to better effect in their cause, and thereby they are able to change history.
>
> I would put it to you that mathematics has often
> played a subversive role, e.g. having those Hindu-
> Arabic algorithms flooding into Europe,
> along with double-entry bookkeeping, is what gave
> rise to a merchant class with independent means, an
> ability to become patrons of the arts independently of
> church authorities.
>
Here I believe you seriously misunderstand history.  In pre-Renaissance Europe, the nobility and the church were in fact the 'powers-that-be' of the time, seeking to entrench themselves in power.  The merchant class, the rising 'bourgeousie' was in fact part of the revolutionary surge that changed Europe (and the world) forever.  Why else would that period immediately following be called 'the Renaissance' ???!!!  
>
> Just telling history like that is already non-traditional
> vis-a-vis  how math is currently taught (as ahistorical,
> with tiny forays into biography in this or that sidebar).
>
See above.
> I think if you want to break out of a traditional
> mold you need to (a) cast math as a change agent, as sometimes subversive (the story of RSA would be
> apropos at this point)
>
See above - math itself, I believe, is only a tool, not itself the "change agent": it is the change agents who could possibly use the math (or any other tool), if they learn the HOWs? and the WHYs? of their tool, to their revolutionary ends.  (But I do not know what is "RSA" - would love to be informed).
>
> and (b) you need to share lore, stories, meaningful
> ones, not just "story problems".
>
I agree.  
> What happens here on math-teach when I mention
> tetrahedral mensuration (tetrahedron as unit volume) is
> I'm simply ignored (not on other issues perhaps, but on > that one in particular).
>
You have to address the teachers here at Math-teach on that.  As I've mentioned, I'm not myself a teacher of math - or of anything else except what I call the 'One Page Management System' (OPMS) - see attachment. [I do very much love math and believe it needs to be taught and learned MUCH better than is the case today.  The OPMS is - I believe I've amply proven this; and can demonstrate this claim - a potentially powerful instrument that could help people learn to teach math much better.  It has in fact helped one (college freshman) student to learn math much better (I gave that student no math tuition AT ALL - I merely showed him how to apply the tool to his objective: "To understand all topics of my syllabus thoroughly - and thereby to improve very significantly my results in my math exams".  He was highly successful in this Mission.  This was long ago.  Since then, I've been trying to get math teachers develop the OPMS to their specific purposes, to very little effect thus far.  Right now, a couple of math teachers have just approached me, but the work has not yet started here.

I note that the OPMS is the most powerful tool available today to help us understand the "HOWs?" and they "WHYs?" of every element that we may encounter in our system.  (See the structures attached herewith.  If you're interested, I can send you a 4MB presentation that discusses this aspect of OPMS in some detail - write me at:

gs (underscore) chandy (at) yahoo (dot) com

- that's my email id, remove the bracketed words and the spaces around them with the appropriate Internet address symbols).
>
> I show this pattern to my students, as proof that I'm
> up against some deeply conditioned reflexes.  
>
I know full well about those "deeply conditioned reflexes" - I see them operating all the time right here at Math-teach!!  Read, for instance, any of Robert Hansen's or Wayne Bishop's messages to me.
>
>This is very accessible to them, i.e. they don't need a
> lot of background to see what I'm talking about.
> They understand tetrahedral mensuration, and they
> understand what it means to be ignored.  Looking over > my shoulder is a piece of cake  then, quite
> enlightening.
>
I don't quite see the connection you are making here.
>
> If you're unwilling to entertain an alternative base
> volume and use that in lesson plans (ala that NCTM
> lesson plan on Tetrahedral Kites), then in my book
> you're a traditionalist by definition.  I may not
> distinguish between Haim, Wayne or MPG in that
> sense -- three peas in a pod.  You've seemed quite
> traditional yourself in this respect.
>
As mentioned in many of my posts, I am not at all a teacher of math or of any other discipline myself (except of OPMS as a fundamental 'thought tool').  I did create the OPMS concept, which is something that (in the minds and hands of people who really want change in the way things are) could help them bring about such change.  So I do dispute your classing me as a 'traditionalist'.  

Here, by the way, is a suggestion for you:  try out the OPMS on your Mission; you will be remarkably surprised at the significantly better grip you get on it thereby.

> > That kind of exercise develops from an element
> located at or near the lowest level in one of the
> structures I drew up to guide me when I wanted to
> convince a student that math can be very exciting
> indeed:
> >
> > "To stimulate student interest by showing them
> fascinating things that can be done with and through
> math: e.g. George A. Hart’s ‘mathematical
> sculptures’; ‘Moebius curves; fractals; the Yoshimoto
> cube..Etc, etc."
> >
>
> George Hart has done Pavilion of Polyhedreality and
> is an enthusiastic supporter of Zome, a geometric
> construction kit I've mentioned often.
>
I pass on herewith as an attachment the rest of that structure the single element of which you've quoted above, in the hope that it might help remove some of your misapprehensions.  See attachment "Helping students learn math - detail".  Also attached, a 'student model' - a reconstruction of one of the many models that a (freshman college) student made to help him with his Mission noted above.
> > (I had attached the structure to one of my earlier
> > posts).
> >
Not attaching it right there was obviously a mistake on my part!

> >
> > Much later, this element worked very well indeed
> > with my grand-daughter when she told me: "Oh,
> > math is SO boring!"
> >
> > (Interesting aside: John Baez - the physicist who
> first put up these plots for  at his website for Dan
> Christensen - happens to be the father of the
> renowned singer, Joan Baez.  In the world of physics,
> John Baez is probably as famous as is his daughter in
> the world outside).
> >
> > GSC
> >
>
> John Baez attended Princeton living at 2 Dickinson
> Street, as I did, though not at the same time.
>
> Here's a recent picture of Joan Baez with her arm
> around someone (not me, sat next to him at dinner last
> Thursday and he was bragging about this picture).
> http://www.flickr.com/photos/17157315@N00/4429354327/
>
> Note: their relationship (John and Joan) is "cousin",
> not father-daughter.
>
You may well be correct on this.  Joan Baez was a good friend of an acquaintance of mine those days (early 'sixties), and she would visit him whenever she was in/around Boston.  Late 'eighties or so, I met John Baez at a conference here in Bangalore, and he was interested to know that I was acquainted with Joan - but I must clarify here (as I did to him) that that acquaintance was very slight indeed, even tenuous!

GSC

01_OPMS___in_Outline.doc (206K) Download Attachment
02 The power of the word.doc (263K) Download Attachment
03 How a child learns2.doc (58K) Download Attachment
04 OPMS Deep Logic.doc (91K) Download Attachment
Helping students learn math - detail.doc (112K) Download Attachment
improve math results4.JPG (160K) Download Attachment
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Re: Something beautiful AND exciting, could get students interested in mathe

Jonathan Groves-3
In reply to this post by GS Chandy
On 3/13/2010 at 2:19 pm, Kirby Urner wrote:


> On Fri, Mar 12, 2010 at 9:38 AM, GS Chandy
> <[hidden email]> wrote:
> > >From Scientific American:
> >
> >> Polynomial Plot: Simple Math Expressions Yield
> >> Intricate Visual Patterns [Slide Show]
> >> Plotting the roots of run-of-the-mill polynomials
> yields
> >> dazzling results, by John Matson
> > http://www.scientificamerican.com/article.cfm?
> > id=math-> polynomial-roots
> >>
> > I provide this link to support my strong belief
> that an excellent way to get students who may be
> getting turned off by the 'traditional approach' to
> math would be to show him/her some of the
> beautiful/wonderful/striking/.... things that can be
> done with math.  Nowadays, with all these wonderful
> computer graphics available, we can really spark a
> firestorm in the student's mind!
> >
>
> I'm somewhat skeptical that traditionalists never
> show intriguing eye
> candy etc. i.e. no matter what your pedagogical
> approach, isn't it
> somewhat expected that you will show some cool art?
>
> The devil is in the details I'd say, i.e. how
> accessible, how
> hands-on, how participatory is all this.
>
> If the message is "if you study hard for twenty more
> years, you too
> will be able to make these kinds of pictures" that's
> different from
> "we will learn about how to do this today, as a
> project."
>
> For my part, I continually and consistently bring up
> polyhedra as
> visually exciting, and also accessible.
>
> I also tell students they're on the front lines
> simply for daring to
> use a tetrahedron for a unit of volume.  No need to
> study twenty years
> to see some action in Math Wars (if that's what
> they're looking for).
>
> The unit volume tetrahedron meme connects them to all
> kinds of
> relevant thinking, about how humans might make a
> success of themselves
> if they dared to be iconoclastic.
>
> I think this is a primary distinction you need to
> make, when talking
> about "traditional" versus "not traditional":  to
> what extent do you
> portray mathematics as helping to challenge a status
> quo versus
> helping to entrench a status quo?
>
> I would put it to you that mathematics has often
> played a subversive
> role, e.g. having those Hindu-Arabic algorithms
> flooding into Europe,
> along with double-entry bookkeeping, is what gave
> rise to a merchant
> class with independent means, an ability to become
> patrons of the arts
> independently of church authorities.
>
> Just telling history like that is already
> non-traditional vis-a-vis
> how math is currently taught (as ahistorical, with
> tiny forays into
> biography in this or that sidebar).
>
> I think if you want to break out of a traditional
> mold you need to (a)
> cast math as a change agent, as sometimes subversive
> (the story of RSA
> would be apropos at this point) and (b) you need to
> share lore,
> stories, meaningful ones, not just "story problems".
>
> What happens here on math-teach when I mention
> tetrahedral mensuration
> (tetrahedron as unit volume) is I'm simply ignored
> (not on other
> issues perhaps, but on that one in particular).
>
> I show this pattern to my students, as proof that I'm
> up against some
> deeply conditioned reflexes.  This is very accessible
> to them, i.e.
> they don't need a lot of background to see what I'm
> talking about.
> They understand tetrahedral mensuration, and they
> understand what it
> means to be ignored.  Looking over my shoulder is a
> piece of cake
> then, quite enlightening.
>
> If you're unwilling to entertain an alternative base
> volume and use
> that in lesson plans (ala that NCTM lesson plan on
> Tetrahedral Kites),
> then in my book you're a traditionalist by
> definition.  I may not
> distinguish between Haim, Wayne or MPG in that sense
> -- three peas in
> a pod.  You've seemed quite traditional yourself in
> this respect.
>
> > That kind of exercise develops from an element
> located at or near the lowest level in one of the
> structures I drew up to guide me when I wanted to
> convince a student that math can be very exciting
> indeed:
> >
> > "To stimulate student interest by showing them
> fascinating things that can be done with and through
> math: e.g. George A. Hart’s ‘mathematical
> sculptures’; ‘Moebius curves; fractals; the Yoshimoto
> cube..Etc, etc."
> >
>
> George Hart has done Pavilion of Polyhedreality and
> is an enthusiastic
> supporter of Zome, a geometric construction kit I've
> mentioned often.
>
> > (I had attached the structure to one of my earlier
> posts).
> >
> > Much later, this element worked very well indeed
> with my grand-daughter when she told me: "Oh, math is
> SO boring!"
> >
> > (Interesting aside: John Baez - the physicist who
> first put up these plots for  at his website for Dan
> Christensen - happens to be the father of the
> renowned singer, Joan Baez.  In the world of physics,
> John Baez is probably as famous as is his daughter in
> the world outside).
> >
> > GSC
> >
>
> John Baez attended Princeton living at 2 Dickinson
> Street, as I did,
> though not at the same time.
>
> Here's a recent picture of Joan Baez with her arm
> around someone (not
> me, sat next to him at dinner last Thursday and he
> was bragging about
> this picture).
>
> http://www.flickr.com/photos/17157315@N00/4429354327/
>
> Note: their relationship (John and Joan) is "cousin",
> not father-daughter.
>
> Kirby



Kirby,

I think that having students study polyhedra is a good way to make the
mathematics exciting for them.  And it will help them to learn how to
generalize some of what they already know and extend it to new situations.
They will need that kind of thinking to extend the idea of measuring
volume via cubes to measuring volume via polyhedra.  

Sometimes we do need to be daring enough to challenge the status quo since
what is generally accepted may not necessarily be the best approach, and
what is ridiculed may be the best or one of the best approaches.  The
best mathematicians and scientists were the ones who dared to challenge
conventional thinking, and they revolutioned our thinking about science and
mathematics.  For examples, let us consider Georg Cantor, Albert Einstein,
Hermann Grassmann, Berhhard Riemann, Srinivasa Ramanujan, and many others.
Not all of them contributed to mathematics education, but they did dare
to challenge conventional thinking.  And many such thinkers were ignored
or ridiculed until their work was later appreciated.  


Jonathan Groves
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Re: Something beautiful AND exciting, could get students interested in mathe

Kirby Urner-5
In reply to this post by GS Chandy
> Kirby Urner posted  Mar 14, 2010 12:49 AM:

<< SNIP >>

> By and large, the 'traditionalists' would prefer, I
> think, that their charges learn the rules, live by
> the rules, die for the rules.

Captains of industry have needed a steady supply of
"computers" (human number crunchers) who would do
accurate work without asking a lot of questions or
thinking too deeply about what they were doing.

I would hypothesize that much that is dull and dreary
about Factory Math is intentionally so.

> > For my part, I continually and consistently bring
> up
> > polyhedra as visually exciting, and also
> accessible.
> >
> Polyhedra are, I agree, very exciting constructs for
> students to learn and understand math.  In fact, when
> I started a company called "Thinkit" (which, way back
> in the early 'seventies, aimed at designing and
> developing educational kits based on math), we
> started practically all our work with polyhedra,
> creating kits to help students construct and
> understand the wonders of polyhedra.  Now that
> company is no more, but I still do tell all teachers
> that I meet about the readily accessible wonders of
> polyhedra to help them get their students hooked on
> to math.  (I've mentioned the 'Yoshimoto cube' in
> several of my messages - and I believe you had
> responded to one of those messages).

Yes, I'm familiar with the Yoshimoto Cube, and a huge
range of exciting spatial geometry toys, many from
companies that have gone out of business because
Factory Math has no need of any of them.

> >
> > I also tell students they're on the front lines
> > simply for daring to use a tetrahedron for a unit
> of
> > volume.  No need to  study twenty years to see some
> > action in Math Wars (if that's what they're looking
> for).
> >
> > The unit volume tetrahedron meme connects them to
> > all  kinds of relevant thinking, about how humans
> > might make a success of themselves if they dared to
> > be iconoclastic.
> >
> The linkages you perceive between the tetrahedron,
> human success and iconoclastic thinking are not very
> clear to me, I'm afraid.  (Are you referring to
> Buckminster Fuller, perhaps?)

Yes of course.

> >
> > I think this is a primary distinction you need to
> > make, when talking about "traditional" versus "not
> > traditional":  to  what extent do you portray
> > mathematics as helping to challenge a status quo
> > versus helping to entrench a status quo?
> >
> It is only the people who use a tool, say math, that
> can use it for "challenging/entrenching the status
> quo"; the tool itself has no such intent or interest.

Right.  You can use a ball point pen to write poetry
or to scratch in the dirt.  However tools are suggestive.
To those who only know how to use a hammer, every problem
seems like a nail, as the saying goes.

What world map do you use (a mathematical artifact) and
what data does it show?  Google Earth has changed the
equations, opened new possibilities, as have other Google
tools (e.g. the translator).

> For instance the powers-that-be would surely like to
> use math (or any other tool) to entrench themselves
> in power.  All too often, they misunderstand the
> historical reality and do not properly learn to use
> the tools available.  The people who want change are
> often in a position to learn to use the tools they
> have to better effect in their cause, and thereby
> they are able to change history.

A concern I often hear is that all the best tools are
secret, proprietary, because no one wants "the
competition" to learn of one's advantage.  As a result,
students are left out in the cold, don't get to assemble
a mental picture of what tools are available.  Were they
able to do that, they might have a more encouraging
picture of what humans might do for themselves to avert
catastrophe.  Keeping everything too secret is perhaps
a great way to end a civilization, as it proves too
prohibitive and intimidating to its own young.  This
fear was expressed to me by an engineer working deep in
the Silicon Forest.  I've heard it echoed by others.
 

> >
> > I would put it to you that mathematics has often
> > played a subversive role, e.g. having those Hindu-
> > Arabic algorithms flooding into Europe,
> > along with double-entry bookkeeping, is what gave
> > rise to a merchant class with independent means, an
> > ability to become patrons of the arts independently
> of
> > church authorities.
> >
> Here I believe you seriously misunderstand history.
> In pre-Renaissance Europe, the nobility and the
> church were in fact the 'powers-that-be' of the
> time, seeking to entrench themselves in power.  The
> merchant class, the rising 'bourgeousie' was in fact
> part of the revolutionary surge that changed Europe
> (and the world) forever.  Why else would that period
> immediately following be called 'the Renaissance'
> ???!!!

I think we both just told the same story in different
words.  Where is the disagreement?
 

> >
> > Just telling history like that is already
> non-traditional
> > vis-a-vis  how math is currently taught (as
> ahistorical,
> > with tiny forays into biography in this or that
> sidebar).
> >
> See above.
> > I think if you want to break out of a traditional
> > mold you need to (a) cast math as a change agent,
> as sometimes subversive (the story of RSA would be
> > apropos at this point)
> >
> See above - math itself, I believe, is only a tool,
> not itself the "change agent": it is the change
> agents who could possibly use the math (or any other
> tool), if they learn the HOWs? and the WHYs? of their
> tool, to their revolutionary ends.  (But I do not
> know what is "RSA" - would love to be informed).

Rivest Shamir Adleman (RSA).

One of our goals along Track 2 (the emergent more
discrete / computational math track) is to get into this
tool more, and its story.  

There's this text I've mentioned a number of times
'Mathematics for the Digital Age' (Skylit Publishing)
gets into it (the tool and the math behind it, not the
lore so much).

The lore, the story, is really excellent.  But Factory
Math is mostly bereft of lore.  That Phil Zimmerman would
defy authorities and release PGP from New Zealand
instead... that was radical.

> > and (b) you need to share lore, stories, meaningful
> > ones, not just "story problems".
> >
> I agree.  

High schoolers learning the story of RSA will feel they're
being taken more seriously as young adults.  This is more
contemporary history, feels less dumbed down than a lot
of that sidebar stuff that passes for "stories" currently.

> > What happens here on math-teach when I mention
> > tetrahedral mensuration (tetrahedron as unit
> volume) is
> > I'm simply ignored (not on other issues perhaps,
> but on > that one in particular).
> >
> You have to address the teachers here at Math-teach
> on that.  

I do that frequently.

> As I've mentioned, I'm not myself a teacher
> of math - or of anything else except what I call the
> 'One Page Management System' (OPMS) - see attachment.

I've looked it over, yes.  I don't deny the relevance of
management workshop type stuff in schooling.  Many
schools neglect to teach effective study practices,
time management, other life skills.  Urbanized kids aren't
learning how to cook for themselves, put together a
healthy diet.  Lack of physical exercise is a concern.
Ignorance of public transportation...

In my recent posting on "off your duff" math, I write a
lot about how Track 2 might include a lot more physical
components, breaking the stereotype that exercising one's
mathematical skills is an exclusively indoor / sedentary
activity.  Once we get a less passivist curriculum, then
the kinds of workshops and seminars your materials
suggest will become even more necessary, including for
teachers and administrators.

> [I do very much love math and believe it needs to be
> taught and learned MUCH better than is the case
> today.  The OPMS is - I believe I've amply proven
> this; and can demonstrate this claim - a potentially
> powerful instrument that could help people learn to
> teach math much better.  It has in fact helped one
> (college freshman) student to learn math much better
> (I gave that student no math tuition AT ALL - I
> merely showed him how to apply the tool to his
> objective: "To understand all topics of my syllabus
> thoroughly - and thereby to improve very
> significantly my results in my math exams".  He was
> highly successful in this Mission.  This was long
> ago.  Since then, I've been trying to get math
> teachers develop the OPMS to their specific purposes,
> to very little effect thus far.  Right now, a couple
> of math teachers have just approached me, but the
> work has not yet started here.
>

My mission is to encourage more math teachers to tackle
and share what I call Verboten Math, i.e. mathematics
that is currently considered out of bounds.  A co-teacher
and I gave a class on precisely this topic just yesterday
in fact, with young adults our primary audience.

http://worldgame.blogspot.com/2010/03/radical-math.html

> I note that the OPMS is the most powerful tool
> available today to help us understand the "HOWs?" and
> they "WHYs?" of every element that we may encounter
> in our system.  (See the structures attached
> herewith.  If you're interested, I can send you a 4MB
> presentation that discusses this aspect of OPMS in
> some detail - write me at:
>
> gs (underscore) chandy (at) yahoo (dot) com
>

I wonder how OPMS will assist math teachers who wish to
join in the struggle to preserve our heritage and share
it with coming generations.  

To just bleep over the concentric hierarchy of polyhedra,
based around a unit volume tetrahedron, is to me a
terrifyingly cowardly decision, is a threat to our
shared future (OPMS uses the word "threat" I notice).
"How could humans do this to themselves?" I find myself
asking.

No one is suggesting this curriculum segment be used to
supplant centuries of cubism, as if that were even a
current possibility.  And yet the level of defensiveness
I sometimes encounter assures me that, like it our not, I
must be a radical.

You'd think I'd not have to be one (a radical), given
NCTM has that Tetrahedral Kites lesson plan featuring a
non-standard unit of volume (a tetrahedron) and given
buckminsterfullerene has completely permeated the
chemical literature.  But these dots go unconnected.

> - that's my email id, remove the bracketed words and
> the spaces around them with the appropriate Internet
> address symbols).
> >
> > I show this pattern to my students, as proof that
> I'm
> > up against some deeply conditioned reflexes.  
> >
> I know full well about those "deeply conditioned
> reflexes" - I see them operating all the time right
> here at Math-teach!!  Read, for instance, any of
> Robert Hansen's or Wayne Bishop's messages to me.

You might need to be a radical too then.  It's not about
wasting all your time trying to get agreement from
everyone you meet.  I've learned to admire John Saxon
thanks to Wayne.  Hansen shares an interest in computing.

Wayne and I used to argue whether buckminsterfullerene
was a mathematically significant topic, with me posting
various lesson plans relating to the concepts of
"frequency", "hexapents" also V + F = E + 2.  From the
standpoint of lore, of storytelling, I regard these as
essential threads.  The discovery of a 3rd allotrope of
carbon as late as the 1980s is a clear breakthrough,
resulting in a Nobel (Curly, Smalley, Kroto), and forming
a basis for nanotechnology (includes nanotubes,
dendrimers...).

Sometimes the best way to get the information out there is
to argue with someone.  This is a time tested format in
philosophical and theological works, going back to the
ancient Greeks and before:  present the material in the
form of a dialog, a debate.  Readers follow more easily
when they encounter a "dialectic".

So my advice to you (to me as well) is to use opposing
viewpoints as opportunities.  They're not really threats.
Neither of us needs permission or approval from other
posters here to continue with our work, right?

> >
> >This is very accessible to them, i.e. they don't
> need a
> > lot of background to see what I'm talking about.
> > They understand tetrahedral mensuration, and they
> > understand what it means to be ignored.  Looking
> over > my shoulder is a piece of cake  then, quite
> > enlightening.
> >
> I don't quite see the connection you are making here.
>

I'm saying I'm able to use the math-teach archive as an
exhibit space.  If someone asks me "well, if you think
these ideas are important, have you been sharing them
with other math teachers?"  I can point to math-teach or
geometry-precollege to show I have not been lazy in this
regard.  

That the other math teachers often come across as asleep
at the switch, daydreaming about other things,
listless, clueless, not engaged, is par for the course.
Of course that's just to turn tables, on how many
students get typecast.  In point of fact, these are busy
adults with professional pride and obviously minority
viewpoints can't always be bothered with.  There's that
"bigger war" to worry about.  You and I are but marginal
figures.

So then when I go off and vent by making a little video
suggesting I'm just dealing with quacks, readers looking
over my shoulder can at least see where I'm coming from.
It drives them crazy too (some of them) as I'm not the
only die-hard in my little camp.

> >
> > If you're unwilling to entertain an alternative
> base
> > volume and use that in lesson plans (ala that NCTM
> > lesson plan on Tetrahedral Kites), then in my book
> > you're a traditionalist by definition.  I may not
> > distinguish between Haim, Wayne or MPG in that
> > sense -- three peas in a pod.  You've seemed quite
> > traditional yourself in this respect.
> >
> As mentioned in many of my posts, I am not at all a
> teacher of math or of any other discipline myself
> (except of OPMS as a fundamental 'thought tool').  I
> did create the OPMS concept, which is something that
> (in the minds and hands of people who really want
> change in the way things are) could help them bring
> about such change.  So I do dispute your classing me
> as a 'traditionalist'.  

Well, I have this real world challenge of getting
more intelligently designed spatial geometry segments
phased in along Track 2, where we use more computers.

This includes focusing on the Mite as a minimum space-
filler, in that here we have an irregular tetrahedron
without handedness that fills space (Aristotle was
right in that sense).  Giving it a simple name helps
students tune it in.  This is not a silly or immature
design.  The Mite has a simple volume of 1/8 relative
to our unit volume tetrahedron (of course any number
will do, but we have this canonical arrangement called
the concentric hierarchy of polyhedra... has a volume
6 rhombic dodecahedron that I bet Kepler would have
liked, accessible to 3rd graders).

> Here, by the way, is a suggestion for you:  try out
> the OPMS on your Mission; you will be remarkably
> surprised at the significantly better grip you get on
> it thereby.
>

I'm back to looking at those threats.

OPMS has not proved sufficiently powerful to overcome
the threats and barriers to OPMS being widely accepted
or adopted, at least by that name.  Most management
heuristics I've seen involve some kind of circle or
cycle -- you probably have one of those too, perhaps
several.

In the case of the concentric hierarchy, it's published
in book form, as color posters, the toys are out there.
Fuller had lots of awards, patents, degrees.  It takes
concerted work, strong intent, to paint him as a kook,
but that's what many resort to, as a way of delaying much
focus on even these simplest geometric ideas.  Keeping
verboten math verboten takes real time and energy.

> > > That kind of exercise develops from an element
> > located at or near the lowest level in one of the
> > structures I drew up to guide me when I wanted to
> > convince a student that math can be very exciting
> > indeed:
> > >
> > > "To stimulate student interest by showing them
> > fascinating things that can be done with and
> through
> > math: e.g. George A. Hart’s ‘mathematical
> > sculptures’; ‘Moebius curves; fractals; the
> Yoshimoto
> > cube..Etc, etc."
> > >
> >
> > George Hart has done Pavilion of Polyhedreality and
> > is an enthusiastic supporter of Zome, a geometric
> > construction kit I've mentioned often.
> >
> I pass on herewith as an attachment the rest of that
> structure the single element of which you've quoted
> above, in the hope that it might help remove some of
> your misapprehensions.  See attachment "Helping
> students learn math - detail".  Also attached, a
> 'student model' - a reconstruction of one of the many
> models that a (freshman college) student made to help
> him with his Mission noted above.
> > > (I had attached the structure to one of my
> earlier
> > > posts).
> > >
> Not attaching it right there was obviously a mistake
> on my part!
> > >
> > > Much later, this element worked very well indeed
> > > with my grand-daughter when she told me: "Oh,
> > > math is SO boring!"
> > >
> > > (Interesting aside: John Baez - the physicist who
> > first put up these plots for  at his website for
> Dan
> > Christensen - happens to be the father of the
> > renowned singer, Joan Baez.  In the world of
> physics,
> > John Baez is probably as famous as is his daughter
> in
> > the world outside).
> > >
> > > GSC
> > >
> >
> > John Baez attended Princeton living at 2 Dickinson
> > Street, as I did, though not at the same time.
> >
> > Here's a recent picture of Joan Baez with her arm
> > around someone (not me, sat next to him at dinner
> last
> > Thursday and he was bragging about this picture).
> >
> http://www.flickr.com/photos/17157315@N00/4429354327/
> >
> > Note: their relationship (John and Joan) is
> "cousin",
> > not father-daughter.
> >
> You may well be correct on this.  Joan Baez was a
> good friend of an acquaintance of mine those days
> (early 'sixties), and she would visit him whenever
> she was in/around Boston.  Late 'eighties or so, I
> met John Baez at a conference here in Bangalore, and
> he was interested to know that I was acquainted with
> Joan - but I must clarify here (as I did to him) that
> that acquaintance was very slight indeed, even
> tenuous!
>
> GSC

I have neither met either one.  The "cousin" relationship
was revealed to me on Wikipedia.

Kirby

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Re: Something beautiful AND exciting, could get students interested in mathe

GS Chandy
In reply to this post by GS Chandy
Kirby Urner posted: Mar 15, 2010 5:27 AM - GSC's responses interspersed:

> > Kirby Urner posted  Mar 14, 2010 12:49 AM:
>
> << SNIP >>
>
> > By and large, the 'traditionalists' would prefer, I
> > think, that their charges learn the rules, live by
> > the rules, die for the rules.
>
> Captains of industry have needed a steady supply of
> "computers" (human number crunchers) who would do
> accurate work without asking a lot of questions or
> thinking too deeply about what they were doing.
>
> I would hypothesize that much that is dull and dreary
> about Factory Math is intentionally so.
>
Possibly you are right.  In which case, it is surely for us to demonstrate clearly that all math is not 'Factory Math'.  As to how this might be done, I believe we need to use effectively the inherent learning capacities that are part of all children; and they are, I believe, still part of most of us - though forced into dormancy by pressures exerted on our minds by the systems within which we live and work.

> > > For my part, I continually and consistently bring
> > up
> > > polyhedra as visually exciting, and also
> > accessible.
> > >
> > Polyhedra are, I agree, very exciting constructs
> for
> > students to learn and understand math.  In fact,
> when
> > I started a company called "Thinkit" (which, way
> back
> > in the early 'seventies, aimed at designing and
> > developing educational kits based on math), we
> > started practically all our work with polyhedra,
> > creating kits to help students construct and
> > understand the wonders of polyhedra.  Now that
> > company is no more, but I still do tell all
> teachers
> > that I meet about the readily accessible wonders of
> > polyhedra to help them get their students hooked on
> > to math.  (I've mentioned the 'Yoshimoto cube' in
> > several of my messages - and I believe you had
> > responded to one of those messages).
>
> Yes, I'm familiar with the Yoshimoto Cube, and a huge
> range of exciting spatial geometry toys, many from
> companies that have gone out of business because
> Factory Math has no need of any of them.
>
I must confess that my company went out of business NOT because 'Factory Math' had no need of what I was planning to create, but because of my own several deficiencies in running a business.  We did make some money - but my whole system failed, which caused me to question what I was doing. (*See attachment, which also explains something about how OPMS came into being)
<snip>

> For instance the powers-that-be would surely like
> to
> > use math (or any other tool) to entrench themselves
> > in power.  All too often, they misunderstand the
> > historical reality and do not properly learn to use
> > the tools available.  The people who want change
> are
> > often in a position to learn to use the tools they
> > have to better effect in their cause, and thereby
> > they are able to change history.
>
> A concern I often hear is that all the best tools are
> secret, proprietary, because no one wants "the
> competition" to learn of one's advantage.  As a
> result, students are left out in the cold, don't get to
> assemble a mental picture of what tools are available.  > Were they able to do that, they might have a more
> encouraging picture of what humans might do for
> themselves to avert
> catastrophe.  Keeping everything too secret is
> perhaps a great way to end a civilization, as it proves
> too prohibitive and intimidating to its own young.  This
> fear was expressed to me by an engineer working
> deep in the Silicon Forest.  I've heard it echoed by
> others.
>
Indeed you're correct.  Tools like the OPMS help users to  construct pretty sound mental pictures of what's going on within the systems they are working with.

> > >
> > > I would put it to you that mathematics has often
> > > played a subversive role, e.g. having those
> Hindu-
> > > Arabic algorithms flooding into Europe,
> > > along with double-entry bookkeeping, is what gave
> > > rise to a merchant class with independent means,
> an
> > > ability to become patrons of the arts
> independently
> > of
> > > church authorities.
> > >
> > Here I believe you seriously misunderstand history.
> > In pre-Renaissance Europe, the nobility and the
> > church were in fact the 'powers-that-be' of the
> > time, seeking to entrench themselves in power.  The
> > merchant class, the rising 'bourgeousie' was in
> fact
> > part of the revolutionary surge that changed Europe
> > (and the world) forever.  Why else would that
> period
> > immediately following be called 'the Renaissance'
> > ???!!!
>
> I think we both just told the same story in different
>
> words.  Where is the disagreement?
>  
I TOTALLY misread/misunderstood what you had written when I read it first time around!  Sorry about that.

> > >
> > > Just telling history like that is already
> > non-traditional
> > > vis-a-vis  how math is currently taught (as
> > ahistorical,
> > > with tiny forays into biography in this or that
> > sidebar).
> > >
> > See above.
> > > I think if you want to break out of a traditional
> > > mold you need to (a) cast math as a change agent,
> > as sometimes subversive (the story of RSA would be
> > > apropos at this point)
> > >
> > See above - math itself, I believe, is only a tool,
> > not itself the "change agent": it is the change
> > agents who could possibly use the math (or any
> other
> > tool), if they learn the HOWs? and the WHYs? of
> their
> > tool, to their revolutionary ends.  (But I do not
> > know what is "RSA" - would love to be informed).
>
> Rivest Shamir Adleman (RSA).
>
Thanks for the RSA explanation - I've found plenty to read about RSA, and shall do so as soon as I can.

> One of our goals along Track 2 (the emergent more
> discrete / computational math track) is to get into
> this
> tool more, and its story.  
>
> There's this text I've mentioned a number of times
> 'Mathematics for the Digital Age' (Skylit Publishing)
>
> gets into it (the tool and the math behind it, not
> the
> lore so much).
>
> The lore, the story, is really excellent.  But
> Factory
> Math is mostly bereft of lore.  That Phil Zimmerman
> would
> defy authorities and release PGP from New Zealand
> instead... that was radical.
>
> > > and (b) you need to share lore, stories,
> meaningful
> > > ones, not just "story problems".
> > >
> > I agree.  
>
> High schoolers learning the story of RSA will feel
> they're
> being taken more seriously as young adults.  This is
> more
> contemporary history, feels less dumbed down than a
> lot
> of that sidebar stuff that passes for "stories"
> currently.
>
> > > What happens here on math-teach when I mention
> > > tetrahedral mensuration (tetrahedron as unit
> > volume) is
> > > I'm simply ignored (not on other issues perhaps,
> > but on > that one in particular).
> > >
> > You have to address the teachers here at Math-teach
> > on that.  
>
> I do that frequently.
>
> > As I've mentioned, I'm not myself a teacher
> > of math - or of anything else except what I call
> the
> > 'One Page Management System' (OPMS) - see
> attachment.
>
> I've looked it over, yes.  I don't deny the relevance
> of
> management workshop type stuff in schooling.  
>
The OPMS goes quite a way beyond the 'management workshop type stuff'.  In fact, I have a running battle going on with the management types because of what I note briefly below.  In OPMS, there are 3 major steps to resolving an issue, any issue:

1) Collecting relevant data/information
This is pretty well done by the management schools in their workshops, etc.

2) Organizing the information in terms of relationships appropriate to the system being explored.
This is scarcely done by the management schools - they only explore "precedence of events and activities" which is, in fact, one of the things to be explored in a system only AFTER you have come to understand pretty well the system and its functioning.  Thus, for nearly a century, the management types have been exploring, via "precedence" systems that they have scarcely any understanding of.  Naturally, our systems do not function optimally at all - indeed they fail catastrophically a good bit of the time.  

In OPMS, it is recommended that available information should first of all be organised in terms of the "contribution" relationship (which is something we've all learned to do when we were children - but since then taken for granted).  "Contribution" is probably the most important of the systems relationships to explore (along its opposite "Hinders").  I've found that true clarity in thinking about a system comes ONLY after one has explored the relationships between its elements using "contributes to".

3) Integrating everything into a 'working system' - this is, in fact, the OPMS for the Mission.

Management schools know very little about No. 2 above, and practically nothing at all about No. 3.

I've plenty more material available on this - but the first of that material is in a PowerPoint presentation of about 3.5 MB size, and I'm not certain whether Math-teach moderators would appreciate putting up such large attachments.  I can send it to you direct if you're interested.

> Many schools neglect to teach effective study
> practices, time management, other life skills.
> Urbanized kids aren't learning how to cook for
> themselves, put together a healthy diet.  Lack of
> physical exercise is a concern.
> Ignorance of public transportation...
>
Most of such 'good system practices' would come naturally when one uses the OPMS approach.

>
> In my recent posting on "off your duff" math, I write
> a lot about how Track 2 might include a lot more
> physical components, breaking the stereotype that
> exercising one's mathematical skills is an exclusively
> indoor / sedentary activity.  Once we get a less
> passivist curriculum, then the kinds of workshops and
> seminars your materials suggest will become even
> more necessary, including for teachers and
> administrators.
>
Some excellent insights here, which I shall be using.

> >
> > [I do very much love math and believe it needs to
> be taught and learned MUCH better than is the case
> > today.  The OPMS is - I believe I've amply proven
> > this; and can demonstrate this claim - a
> > potentially powerful instrument that could help
> > people learn to teach math much better.  It has in
> > fact helped one (college freshman) student to learn
> > math much better
> > (I gave that student no math tuition AT ALL - I
> > merely showed him how to apply the tool to his
> > objective: "To understand all topics of my syllabus
> > thoroughly - and thereby to improve very
> > significantly my results in my math exams".  He was
> > highly successful in this Mission.  This was long
> > ago.  Since then, I've been trying to get math
> > teachers develop the OPMS to their specific
> purposes,
> > to very little effect thus far.  Right now, a
> couple
> > of math teachers have just approached me, but the
> > work has not yet started here.
> >
>
> My mission is to encourage more math teachers to
> tackle and share what I call Verboten Math, i.e.
> mathematics that is currently considered out of
> bounds.  A co-teacher and I gave a class on precisely
> this topic just yesterday in fact, with young adults our
> primary audience.
> http://worldgame.blogspot.com/2010/03/radical-math.htm

I shall explore this in due course of time

>
> >l I note that the OPMS is the most powerful tool
> > available today to help us understand the "HOWs?"
> > and the "WHYs?" of every element that we may
> > encounter in our system.  (See the structures
> > attached herewith.  If you're interested, I can send
> > you a 4MB presentation that discusses this aspect of
> > OPMS in some detail - write me at:
> >
> > gs (underscore) chandy (at) yahoo (dot) com
> >
>
> I wonder how OPMS will assist math teachers who
> wish to join in the struggle to preserve our heritage and
> share it with coming generations.  
>
It will help them to understand their systems better, and thereby help them to accomplish their Missions better.
>
> To just bleep over the concentric hierarchy of
> polyhedra, based around a unit volume tetrahedron, is
> to me a terrifyingly cowardly decision, is a threat to our
> shared future (OPMS uses the word "threat" I notice).
> "How could humans do this to themselves?" I find
> myself asking.
>
Well, we ARE doing it all to ourselves!  That IS the sad reality of things.

>
> No one is suggesting this curriculum segment be used
> to supplant centuries of cubism, as if that were even a
> current possibility.  And yet the level of defensiveness
> I sometimes encounter assures me that, like it our
> not, I must be a radical.
>
> You'd think I'd not have to be one (a radical), given
> NCTM has that Tetrahedral Kites lesson plan featuring
> a non-standard unit of volume (a tetrahedron) and
> given buckminsterfullerene has completely permeated
> the chemical literature.  But these dots go
> unconnected.
>
As noted, the OPMS is a good way to "connect the dots" - the best I've seen.  The only desideratum is that one comes to understand it properly only if one uses it systematically on a Mission - which requires some 5-10 minutes of (OPMS)-work on the Mission, after one has begun to understand the OPMS process itself, which takes a few hours.

> > - that's my email id, remove the bracketed words
> and
> > the spaces around them with the appropriate
> Internet
> > address symbols).
> > >
> > > I show this pattern to my students, as proof that
> > I'm
> > > up against some deeply conditioned reflexes.  
> > >
> > I know full well about those "deeply conditioned
> > reflexes" - I see them operating all the time right
> > here at Math-teach!!  Read, for instance, any of
> > Robert Hansen's or Wayne Bishop's messages to me.
>
> You might need to be a radical too then.  It's not
> about wasting all your time trying to get agreement
> from everyone you meet.  I've learned to admire John
> Saxon thanks to Wayne.  Hansen shares an interest in
> computing.
>
I'm not too worried about convincing Wayne Bishop or  Robert Hansen specifically.  But every argument I put up provides me with some useful 'elements' for my main Mission "to propagage OPMS".

As for being a radical - yes indeed, I am that.  Except that I'd like to clarify that what I'm after is what one might describe as "constructive radicalism" (rather than the destructive variety we see too often in this sad world of ours).

> Wayne and I used to argue whether
> buckminsterfullerene
> was a mathematically significant topic, with me
> posting various lesson plans relating to the concepts of
> "frequency", "hexapents" also V + F = E + 2.  From
> the standpoint of lore, of storytelling, I regard these
> as essential threads.  The discovery of a 3rd allotrope
> of carbon as late as the 1980s is a clear breakthrough,
> resulting in a Nobel (Curly, Smalley, Kroto), and
> forming a basis for nanotechnology (includes
> nanotubes, dendrimers...).
>
I think it very much is a mathematically significant topic - though I must confess I personally do not as yet know HOW to go about bringing out that significance
>.
> Sometimes the best way to get the information out
> there is to argue with someone.  This is a time tested
> format in philosophical and theological works, going
> back to the ancient Greeks and before:  present the
> material in the form of a dialog, a debate.  Readers
> follow more easily when they encounter a "dialectic".
>
Indeed. That is a good part of the reason why I argue with Robert Hansen and Wayne Bishop.
>
> So my advice to you (to me as well) is to use
> opposing viewpoints as opportunities.  They're not
> really threats.
> Neither of us needs permission or approval from other
> posters here to continue with our work, right?
>
Absolutely!

> > >
> > >This is very accessible to them, i.e. they don't
> > need a
> > > lot of background to see what I'm talking about.
> > > They understand tetrahedral mensuration, and they
> > > understand what it means to be ignored.  Looking
> > over > my shoulder is a piece of cake  then, quite
> > > enlightening.
> > >
> > I don't quite see the connection you are making
> here.
> >
>
> I'm saying I'm able to use the math-teach archive as
> an exhibit space.  If someone asks me "well, if you
> think these ideas are important, have you been sharing
> them with other math teachers?"  I can point to math-
> teach or geometry-precollege to show I have not been
> lazy in this regard.  
>
That's a good bit of my reasoning too.

>
> That the other math teachers often come across as
> asleep at the switch, daydreaming about other things,
> listless, clueless, not engaged, is par for the
> course.
> Of course that's just to turn tables, on how many
> students get typecast.  In point of fact, these are
> busy
> adults with professional pride and obviously minority
> viewpoints can't always be bothered with.  There's
> that "bigger war" to worry about.  You and I are but
> marginal figures.
>
> So then when I go off and vent by making a little
> video suggesting I'm just dealing with quacks, readers
> looking over my shoulder can at least see where I'm
> coming from It drives them crazy too (some of them)
> as I'm not the only die-hard in my little camp.
>
Very nice, indeed!

> > >
> > > If you're unwilling to entertain an alternative
> > base
> > > volume and use that in lesson plans (ala that
> NCTM
> > > lesson plan on Tetrahedral Kites), then in my
> book
> > > you're a traditionalist by definition.  I may not
> > > distinguish between Haim, Wayne or MPG in that
> > > sense -- three peas in a pod.  You've seemed
> quite
> > > traditional yourself in this respect.
> > >
> > As mentioned in many of my posts, I am not at all a
> > teacher of math or of any other discipline myself
> > (except of OPMS as a fundamental 'thought tool').
>  I
> > did create the OPMS concept, which is something
> that
> > (in the minds and hands of people who really want
> > change in the way things are) could help them bring
> > about such change.  So I do dispute your classing
> me
> > as a 'traditionalist'.  
>
> Well, I have this real world challenge of getting
> more intelligently designed spatial geometry segments
> phased in along Track 2, where we use more computers.
>
Well, I can pretty surely guarantee OPMS can help to a significant extent.  I can help you apply OPMS to any issue you choose (even when I know little or even nothing about the issue at hand).  Of course, it does turn out that I generally come to learn a little something about the issue at hand in showing how the OPMS should be used.

>
> This includes focusing on the Mite as a minimum
> space-filler, in that here we have an irregular
> tetrahedron without handedness that fills space
> (Aristotle was
> right in that sense).  Giving it a simple name helps
> students tune it in.  This is not a silly or immature
> design.  The Mite has a simple volume of 1/8 relative
> to our unit volume tetrahedron (of course any number
> will do, but we have this canonical arrangement
> called the concentric hierarchy of polyhedra... has a
> volume 6 rhombic dodecahedron that I bet Kepler
> would have liked, accessible to 3rd graders).
>
I would like to learn more about this

> >
> > Here, by the way, is a suggestion for you:  try out
> > the OPMS on your Mission; you will be remarkably
> > surprised at the significantly better grip you get
> > on it thereby.
> >
>
> I'm back to looking at those threats.
> OPMS has not proved sufficiently powerful to overcome the threats and barriers to OPMS being widely
> accepted or adopted, at least by that name.  
>
You're wrong: it has overcome a good number of those barriers and threats.  For instance, I did manage to get the OPMS software done in prototype, which was a major breakthrough.  Now, I'm awaiting the opportunity to take the "next step".

>Most management heuristics I've seen involve some
> kind of circle or cycle -- you probably have one of those
> too, perhaps several.
>
Indeed, there are a whole lot of these 'vicious cycles': I've identified a good number of them, and am working on them.  They do take quite a while to work out - but that is because the human mind does take a good while to open up to new ideas.  And when there are fairly powerful vested interests (like the 'institutes of management studies', well they do have a huge power that they can exert, against which it's not easy to struggle.  Let me give you an an analog in the field you are familiar with - this may be only approximately true: but you would be better able to fill in the details that would make it exactly true.  

If I were to say that Linux is not powerful enough to overcome the threats and barriers to its progress put up by Microsoft - well I would be wrong.  It's taken time, and it may take quite some more time yet - but the trend is unmistakable.  It's all happening, and it will continue to happen, but there's a long, LONG way to go as yet.  But there's no question at all, I think, that Linux (in one or many of its avatars - Ubuntu is the one that I got installed recently by a knowledgeable friend of mine) will rule much of the Microsoft world one day.

>
> In the case of the concentric hierarchy, it's
> published in book form, as color posters, the toys are
> out there.
> Fuller had lots of awards, patents, degrees.  It
> takes concerted work, strong intent, to paint him as a
> kook,  but that's what many resort to, as a way of
> delaying much
> focus on even these simplest geometric ideas.
>  Keeping verboten math verboten takes real time and
> energy.
>
But they're willing and able to spend all that time and energy!!!
 

> > > > That kind of exercise develops from an element
> > > located at or near the lowest level in one of the
> > > structures I drew up to guide me when I wanted to
> > > convince a student that math can be very exciting
> > > indeed:
> > > >
> > > > "To stimulate student interest by showing them
> > > fascinating things that can be done with and
> > through
> > > math: e.g. George A. Hart’s ‘mathematical
> > > sculptures’; ‘Moebius curves; fractals; the
> > Yoshimoto
> > > cube..Etc, etc."
> > > >
> > >
> > > George Hart has done Pavilion of Polyhedreality
> and
> > > is an enthusiastic supporter of Zome, a geometric
> > > construction kit I've mentioned often.
> > >
> > I pass on herewith as an attachment the rest of
> >that structure the single element of which you've
> >quoted above, in the hope that it might help remove
> >some of your misapprehensions.  See attachment
> > "Helping students learn math - detail".  Also
> > attached, a 'student model' - a reconstruction of one
> > of the many models that a (freshman college)
> > student made to help him with his Mission noted
> > above.
> > > > (I had attached the structure to one of my
> > earlier posts).
> > > >
> > Not attaching it right there was obviously a
> > mistake on my part!
> > > >
> > > > Much later, this element worked very well
> indeed
> > > > with my grand-daughter when she told me: "Oh,
> > > > math is SO boring!"
> > > >
> > > > (Interesting aside: John Baez - the physicist
> who
> > > first put up these plots for  at his website for
> > Dan
> > > Christensen - happens to be the father of the
> > > renowned singer, Joan Baez.  In the world of
> > physics,
> > > John Baez is probably as famous as is his
> daughter  in the world outside).
> > > >
> > > > GSC
> > > >
> > >
> > > John Baez attended Princeton living at 2
> Dickinson
> > > Street, as I did, though not at the same time.
> > >
> > > Here's a recent picture of Joan Baez with her arm
> > > around someone (not me, sat next to him at dinner
> > last
> > > Thursday and he was bragging about this picture).
> > >
> >
> http://www.flickr.com/photos/17157315@N00/4429354327/
> > >
> > > Note: their relationship (John and Joan) is
> > "cousin",
> > > not father-daughter.
> > >
> > You may well be correct on this.  Joan Baez was a
> > good friend of an acquaintance of mine those days
> > (early 'sixties), and she would visit him whenever
> > she was in/around Boston.  Late 'eighties or so, I
> > met John Baez at a conference here in Bangalore,
> and
> > he was interested to know that I was acquainted
> with
> > Joan - but I must clarify here (as I did to him)
> that
> > that acquaintance was very slight indeed, even
> > tenuous!
> >
> > GSC
>
> I have neither met either one.  The "cousin"
> relationship was revealed to me on Wikipedia.
>
I had planned to write to John Baez sometime soon.  When I do, I shall ask him specifically what the relationship is!

Best
GSC
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Re: Something beautiful AND exciting, could get students interested in mathe

Dave L. Renfro
In reply to this post by GS Chandy
GS Chandy wrote (at [2], responding to me at [1]):

[1] http://mathforum.org/kb/message.jspa?messageID=7009845

[2] http://mathforum.org/kb/message.jspa?messageID=7010210


> Indeed you're right, Dave, about John Baez being "primarily
> a mathematician who investigates issues in physics", rather
> than a "physicist".
>
> But I'm surprised you've not heard of his daughter, Joan.
> She was once rather spectacularly famous to the world at
> large, particularly as a 'protest singer' (against the
> Vietnam war) in the early to late 'sixties. Early sixties
> was when I was at grad school in the US, in a math
> department - and we ALL knew about and listened to
> Joan Baez. She was, in fact, quite as well-known in
> those days as Bob Dylan. Very few, even in our math
> department, knew of John Baez.

I saw your post Sunday when I was on the internet briefly,
but didn't have time to reply then or research the issue,
but my initial reaction was that surely John Baez isn't
old enough to have a daughter who was a singer in the early
1960s. I didn't get around to looking them up then, but
just now I did and it seems she's probably a bit too old
to be his daughter:

"Joan Chandos Baez (born January 9, 1941)" (from [3])

"John Carlos Baez (born June 12, 1961)" (from [4])

[3] http://en.wikipedia.org/wiki/Joan_Baez

[4] http://en.wikipedia.org/wiki/John_C._Baez

According to [3] and [4] (search for "John" in [3] and
search for "Joan" in [4]), Joan and John are first cousins.
Still, that's pretty closely related, and no, I still don't
recall having ever heard of her. However, I'm not much of
a pop music fan, having never even watched part of an episode
of "American Idol", for example.

Dave L. Renfro
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Re: Something beautiful AND exciting, could get students interested in mathe

GS Chandy
In reply to this post by GS Chandy
A confusion about the (deservedly very distinguished) Baez family, confusion probably initiated by me:
Dave Renfro posted Mar 16, 2010 3:26 AM responding to me (at 2):
> GS Chandy wrote (at [2], responding to me at [1]):
> [1]
> http://mathforum.org/kb/message.jspa?messageID=7009845
> [2]
> http://mathforum.org/kb/message.jspa?messageID=7010210
> > Indeed you're right, Dave, about John Baez being
> >"primarily a mathematician who investigates issues in
> > physics", rather than a "physicist".
> >
You have the right Joan Baez about whom I was writing.  

The John Baez I had originally been thinking and writing about is/was (possibly) the father of the John Chandos Baez you are referring to.  Who, by the way, I have just looked up and who appears to be a fascinating young person.  I'm trying to find references to the John Baez I had been thinking about, but I've been unsuccessful thus far.  Possibly he has passed away - he was, when I met him in the late 'eighties, old enough to be the father of the Joan Baez.

> > But I'm surprised you've not heard of his daughter,
> >Joan. <snip>
>
> I saw your post Sunday when I was on the internet
> briefly, but didn't have time to reply then or research
> the issue,  but my initial reaction was that surely John
> Baez isn't old enough to have a daughter who was a
> singer in the early 1960s. I didn't get around to looking
> them up then, but just now I did and it seems she's
> probably a bit too old to be his daughter:
>
> "Joan Chandos Baez (born January 9, 1941)" (from [3])
> "John Carlos Baez (born June 12, 1961)" (from [4])
> [3] http://en.wikipedia.org/wiki/Joan_Baez
> [4] http://en.wikipedia.org/wiki/John_C._Baez
>
> According to [3] and [4] (search for "John" in [3]
> and search for "Joan" in [4]), Joan and John are first
> cousins.
> Still, that's pretty closely related, and no, I still
> don't  recall having ever heard of her. However, I'm not
> much of a pop music fan, having never even watched
> part of an episode of "American Idol", for example.
>
Joan Baez was NOT a pop singer by any means!  (Even in my callow early adolescence I never listened to pop).  She could be described as, I believe, a 'folk singer' - and she was very much worth listening to, if you can get hold of some of her earlier recordings - http://www.joanbaez.com/ 

I myself have never watched "American Idol" ... not for that matter its Indian avatar, "Indian Idol".

I shall now (for the sake of completeness) try and find the John Baez I had been thinking about when I wrote earlier - I'm pretty sure  now that he was in fact a (theoretical/mathematical) physicist, who must have passed on - and he was probably a brother or a cousin of the father of the John Baez you are referring to.

Who, by the way, is definitely a person one should know about - I was particularly intrigued by the statement he put out when he 'retired'  from mathematical physics (at a VERY early age)...  
 ""I want to shift the focus of my research away from fancy abstract n-categorical math to slightly more practical things. My job at the CQT (Centre for Quantum Technologies) will give me a chance to explore computer science, microtraps, and quantum optics. What I really want to do is help save our beleagured planet."  

Somewhere else he wrote, I think, that he wants to do a bit to 'battle the forces of evil'.  Very difficult to do effectively, maybe impossible - but still one must try...

...SO THANK YOU for helping clear up my error!! - I shall have a good bit to do reading up on this (younger) John Baez and his works - which at first glance seem to be quite mind-boggling.  

GSC