Ways of figuring things out in Math

[m...@ms.lt]

Maria, Milo, Kirby,

Thank you for your ideas, questions and encouragement! about the
multiplication models. I look forward to thinking and replying.

Meanwhile, I'm rewriting my thoughts about the "deep structures" with
which our minds work on math problems. I want to send them to some
professors I met at the University of Chicago who might direct me to
researchers in the foundations of mathematics.

I realized that I should rely more on "visual language" for expressing
these ideas. I send my first sheet, how I analyze an "illustrative
example" of Euclid's which Polya organized in his book "Mathematical
Discovery". It gives the essence of what I'm trying to show in Math.
I'll make a map of the 24 kinds of math structure that I found. The
"powerset lattice of conditions" is just like rows and columns on a
chessboard. If you know the row AND you know the column THEN you know the
square. The chessboard could be three-dimensional, four-dimensional, but
the basic idea is the same. And it's a way of thinking about
multiplication, too (!)

As a side note about rescaling, I've found in teaching the quadratic
equations (and searching for a use for them) that the parabola is arguably
an ideal curve for learning to graph. That's because all parabolas have
one and the same shape, if you discount zooming in and out. Each
parabola, if you zoom in, will look flat, and if you zoom out, will look
narrow, in exactly the same proportions. You can see this if you
substitute x -> ax and y->ay and thereby you can transform x = y**2 to xa
= y**2 a**2 so x = a y**2 effectively where a can be as you like. This
means that all parabolas look the same and their graphs differ only in how
you move them around, up and down, left or right, negative or positive,
zoom in or zoom out. Also, I teach my students to draw too graphs because
most never realize how the parabola completely flattens out at the bottom
where -1 < x < 1. It's a bit like filming a movie where you have to
combine full length shots of people with head shots. One shot won't do
it.

I look forward to reading, writing, thinking and replying. Oh, I would
like to find a way to make a living from such research and/or from
creating related learning materials and/or teaching people genuinely or
even potentially interested. I thought there were grant opportunities for
Public Domain textbooks, etc. I wonder where to find them. Thank you for
keeping me in mind, I am grateful.

Andrius

Andrius Kulikauskas
m...@ms.lt
(773) 306-3807

[kirby urner]

On Fri, May 27, 2011 at 2:39 PM, <m...@ms.lt> wrote:

Maria, Milo, Kirby,

Thank you for your ideas, questions and encouragement! about the
multiplication models.  I look forward to thinking and replying.

Meanwhile, I'm rewriting my thoughts about the "deep structures" with
which our minds work on math problems.  I want to send them to some
professors I met at the University of Chicago who might direct me to
researchers in the foundations of mathematics.

There is no one stop shopping for foundations of mathematics work,

and if the past is any guide, you'll find groupthink in U of Chicago 

that's special to that institution.  Not unusual for a school to have

a signature stamp.  London School of Economics comes to mind

as another one.

There are many "secret wars" going on (secret mainly because

they're abstruse and few care or have a means of joining the 

conversation).  You have neuroscientists and cognitive 

psychologists making all sorts of claims and/or hypotheses 

about brain circuitry and how that must related to such 

"phenomena" as "understanding".  

Then you have a branch of analytic philosophers, mostly fans 

of Wittgenstein, who vigorously oppose casting "meaning" in 

terms of hypothesized "brain events", as that way lies too 

much grammatical confusion.  We weren't aware enough of 

our own language, our ethnography, to wade into that meme 

pool without drowning in misconceptions, thereby entering 

another dark age of quasi-science.  

Ray Monk would be in the latter group, and he list Wittgenstein's

Remarks on the Foundations of Mathematics as on of the 

top 10 philo books of the 20th century.  I doubt U of Chicago 

philosophy is similarly biased, though I could be mistaken.

http://www.guardian.co.uk/books/2001/nov/01/bestbooks.philosophy

http://forum-network.org/lecture/philosophy-circa-1905  

(lots of discussion of this I can patch you into)

I realized that I should rely more on "visual language" for expressing
these ideas.  I send my first sheet, how I analyze an "illustrative
example" of Euclid's which Polya organized in his book "Mathematical
Discovery".  It gives the essence of what I'm trying to show in Math.
I'll make a map of the 24 kinds of math structure that I found.  The
"powerset lattice of conditions" is just like rows and columns on a
chessboard.  If you know the row AND you know the column THEN you know the
square.  The chessboard could be three-dimensional, four-dimensional, but
the basic idea is the same.  And it's a way of thinking about
multiplication, too (!)

Maria has done a lot of nice diagrams.  I certainly appreciate timelines

myself and don't think we do nearly enough with them.  With 

computerized timelines which should be able to explore for overlaps

that had been unsuspected.  More and better detective work.  That 

life a few people have, of exploring in archives for rare papers, should

be a joy in more peoples lives.  Having the Internet helps, though one

should travel also, and visit these historical sites in person.

Here in Portland, on Hawthorne Boulevard, we feature the boyhood

home of Linus Pauling, x2 Nobel Prizes (unshared).  His and Ava's

papers are not there however, but at Oregon State University.

All I'm saying is I'd like to see more diagrams that relate mathematicians

as personalities to their time and place.  Nor need we confine the 

focus to "mathematicians" particularly.  

Like I'm fascinated by the story of The Turk, that seeming-automaton 

that beat Napoleon at chess.  What mathematicians might have 

been influenced by the idea of "machine intelligence"?  I still think 

the article 'As We May Think' by Vannevar Bush is one of the more 

prescient pieces of the 1900s.  He envisioned search engines

(the fact that we call them "engines" is culturally significant).

 

As a side note about rescaling, I've found in teaching the quadratic
equations (and searching for a use for them) that the parabola is arguably
an ideal curve for learning to graph.  That's because all parabolas have
one and the same shape, if you discount zooming in and out.  Each
parabola, if you zoom in, will look flat, and if you zoom out, will look
narrow, in exactly the same proportions.  You can see this if you
substitute x -> ax and y->ay and thereby you can transform x = y**2 to xa
= y**2 a**2 so x = a y**2 effectively where a can be as you like.  This
means that all parabolas look the same and their graphs differ only in how
you move them around, up and down, left or right, negative or positive,
zoom in or zoom out.  Also, I teach my students to draw too graphs because
most never realize how the parabola completely flattens out at the bottom
where -1 < x < 1.  It's a bit like filming a movie where you have to
combine full length shots of people with head shots.  One shot won't do
it.

Polynomial curves get introduced in many contexts.  Some would argue we

should keep a physical context, such as by rotating a parabola to make a 

dish, as in dish receiver.  One can read maths for a year and never make the

connection to a satellite dish.  You can hear some teachers rant about 

this, as if throwing a temper tantrum, in a five minute format.

From my point of view, I like to have students visit Oak's Park (near 

here) and ride a roller coaster.  We need to talk about velocity and 

acceleration, as 2nd powering comes alive when you experience it

for real.  Linking 2nd powering to areal growing/shrinking (of balls,

of shapes), and 3rd powering to volumetric growing/shrinking, is about

as fundamental as it gets.  Rescaling and the difference that makes

to line, surface and volume:  that's the divergent core for so many 

trains of thought, a kind of Grand Central Station.

10 * F * F + 2 is important here.  It's the sequence of successive

shells of balls, packing around a nuclear ball and fanning outward,

preserving even density and maximizing inter-tangency with 

neighbors (12-around-1 if not on a boundary).  This is the familiar

FCC or face-centered-cubic lattice of chemistry and crystallography

ala Linus Pauling, as well as the CCP (cubic closest packing) of

the sphere packing nerds (J.H. Conway being one 'em). 

You can find 10 * F * F + 2 written up by Coxeter in his paper on

Polyhedral Numbers.  He passes credit back to Bucky Fuller, who

say his discovery printed on the front page of the New York Harold

Tribune, back when the geometric structure of the virus was first

being guessed at.

http://mybizmo.blogspot.com/2007/01/gnu-math-memo.html

I look forward to reading, writing, thinking and replying.  Oh, I would
like to find a way to make a living from such research and/or from
creating related learning materials and/or teaching people genuinely or
even potentially interested.  I thought there were grant opportunities for
Public Domain textbooks, etc. I wonder where to find them.  Thank you for
keeping me in mind, I am grateful.

The economics of all this are somewhat tumultuous as you know.  

People have prayed for automation to take the drudgery out of life,

but when a lot of it goes away, people are suddenly at a loss as to

why they deserve to live, if not "working".  It's a conundrum.  

http://unemploymentisgood.wordpress.com/

http://unemploymentisgood.wordpress.com/2009/08/04/unemployment-is-a-distribution-problem-not-a-production-problem/

Mathematicians could be more help, but they're not trained to be

all that useful.  The idea of a mathematician as a "logical thinker"

has somewhat fallen by the wayside, and it's a mantle few want 

to pick up except maybe the analytic philosophers referred to 

above, and computer scientists perhaps.

Hyper-specialization is more about creating holding patterns and 

inducing quasi-paralysis.  Don't expect much help from the already 

straitjacketed.

A main point to weigh in on is whether you're trying to embrace

different content, or whether you're embracing the same content

as X (where X == national standards or some published curriculum)

while offering more effective methods for implementation.

In my case, I am highly suspicious of the status quo math track

and have moved to embrace a lot of alternative content.

Kirby

 

Andrius

Andrius Kulikauskas
m...@ms.lt
(773) 306-3807

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[for...@ozonline.com.au]

Thanks Kirby, Andrius and others.

"I'm rewriting my thoughts about the "deep structures" with
which our minds work on math problems."

"There is no one stop shopping for foundations of mathematics work...
There are many "secret wars" going on ... secret mainly because
they're abstruse"

This interests me

What is the relationship between:
thinking mathematically
computer programming
higher order thinking
problem solving

What do we mean by higher order thinking? It is often mentioned but
rarely defined. Bloom's Taxonomy defines a pyramid of learning but its
a better fit to the humanities than the sciences.

Whether mentally rotating a parabola to create an antenna or
predicting the output of for i=1 to 10 print i
it seems that you are exercising similar skills, creating mental
models, running them and observing the output. Some of the thinking is
visual and some is not but still feels like visual thinking.

A possible analogy: reading to yourself or out loud exercises similar
functions even though you do not sound out the words in both cases.

If computer programming and maths are using similar ways of thinking,
then you can expect some kind of transfer of skills between them.

If we better understood the nature of these common skills then we
could better use math modelling tools like virtual manipulatives, math
aplets, Geogebra, Etoys, Scratch for thinking mathematically.

Tony