[Math 2.0] posting up a storm...
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kirby urner
May 10, 2010, 3:12:48 AM5/10/10
to mathf...@googlegroups.com
... or more a tempest in a teapot.
OREGON CURRICULUM NETWORK
Some obscure writer in Portland, Oregon, on
the war path (yet a Quaker, so presumably
not too violently), keeps putting it out there.
Here are some links:
http://mybizmo.blogspot.com/2010/05/math-reform.html
I should also mention Another Alien Curriculum
which Dr. Richard Hake thought was pretty cool:
http://www.4dsolutions.net/ocn/alien.html
The 'Four Axes for Reformers' essay, published
in two editions, provides some useful dichotomies
to consider:
I. Cardinality vs. Ordinality
2. Lexical vs. Graphical
3. Geometry vs. Geography
4. Logic vs. History
I'm lifting the last from Hegel, as the basis of his
idea of a dialectic.
THE IMPORTANCE OF HISTORY
Restoring a time axis or time line to math teaching
is a core objective, along with making that time line
"place based" i.e. local history forms a starting
context, reaching out to larger narratives.
I illustrate this approach in my concluding section,
where I focus on regional geography, the Columbia
Gorge in particular.
No, I'm not the originator of the "place based education"
idea -- that's as old as the hills, until (more recently)
educators put a name to it.
A RENAISSANCE FOR POLYHEDRA
Continuing threads in this archive, I'll quote a paragraph
re Geometry vs. Geography:
"""
Beginning with the Platonic Five, the tetrahedron
is self dual whereas the others form two pairs of
duals. Combining these duals gives us additional
polyhedra: a good beginning for spatial geometry
segments, including many animations (anime).
The space-filling rhombic dodecahedron gets
into the mix early this way, whereas it's mostly
missing from the lame texts of our day (another
litmus test if you're evaluating competing curricula:
does it include a rhombic dodecahedron?).
"""
That's probably less than clear even to many a
math teacher.
The Platonic Five are the tetrahedron, cube,
octahedron, pentagonal dodecahedron and
icosahedron. To take the dual of a polyhedron
is to exchange its faces for corners, keeping edge
numbers the same, and the Platonic Five include
all their own duals, with the tetrahedron self-dual.
Then one may combine a polyhedron with its own
dual to generate another polyhedron. Doing this
with the Platonic Five nets a "2nd generation" of:
tetrahedron + tetrahedron = cube (already a Platonic)
cube + octahedron = rhombic dodecahedron (yes!)
pentagonal dodeca + icosahedron = rhombic triacontahedron
FUTURISTIC TOPIC: SPHERE PACKING
The rhombic dodecahedron is a space-filler and
favorite of J. Kepler's. If one fills space with it, while
placing a ball (sphere) in each one, the resulting
matrix is known by various names, such as CCP,
FCC or octet truss. CCP refers to the ball packing
in particular whereas FCC more refers to the ball
centers in isolation and "octet truss" refers to
the skeleton or scaffolding one gets when
connecting these centers with equi-length edges.
You'll find pictures / animations at this old web
page of mine, but also all over the Internet of course
i.e. this is not really esoteric knowledge in any way,
and deserves coverage pre-college.
http://www.4dsolutions.net/ocn/xtals101.html
UNIT VOLUME TETRAHEDRON
I don't go into a lot of detail with the tetrahedral
accounting, having done so in other posts. My
school of thought likes to see at least some
exposition of this concept, with associated lore.
The rhombic dodecahedron has a volume of 6
according to this canon, with the dual pair that
define it (cube + octahedron) having volumes 3
and 4 respectively. Not sharing this information,
anywhere along the K-12 time line, is proof positive
of a corrupted curriculum (is what we assert --
obviously a minority / elitist view, which doesn't
mean indefensible).
AN ALGEBRA OF TOMORROW
I'm also fairly light on the computer language
aspect in these most recent writings, having
dwelt on that elsewhere, with more to come.
Jonathan Groves is in agreement that more of
what we call "abstract algebra" might usefully be
imported into what we call "elementary algebra",
with computer languages fomenting (by making
more concrete) an awareness of "types".
Ideas about closure, inverse, identity elements
relate to our typology or taxonomy of numbers
(e.g. N < Z < Q < R < C). When you ask for
the additive inverse of 3, i.e. 3 + x = 0, you
leave the set N for set Z. Asking for multiplicative
inverses gets one into Q. The circle of numbers
has widened over the centuries, plus given rise
to other "math objects" (types of object).
The types we have on machines do not
correspond exactly to the traditional types.
There's no "real number" type (not really).
Rather than consider this a limitation, we
look at the philosophy mathematics more
deeply. Floating point numbers need not
be "approximations" for anything -- they're
a technology in their own right. This topic
came up at our Oregon Math Summit way
back in 1997. And I quote:
"""
My response, expressed in the breakout
session (as the official note taker, I managed
to interject twice — once about the
wierdnesses in floating point math as
implemented in computers, reason
enough to learn the algorithms on paper
as well) was that if teachers wanted to
fight the standard bearers and take
back control of the curriculum (they all
agreed that these tests from on high
were severely limiting their freedoms
and creativity as teachers on the
front lines) they could use the ammo
I was supplying via Beyond Flatland.
Here was basic, low level, primary
school material that every kid should
know, and yet isn’t part of the standard
— clear evidence that the mathematics
department knows relevant content
far better than whatever officials charged
with concocting these tests.
"""
http://grunch.net/archives/130
To me, these seem to be futuristic ideas.
Actual changes to content is somewhat
distinct from using Internet technologies to
relay essentially the same content.
Back to the rhombic dodecahedron:
"""
The space-filling rhombic dodecahedron gets
into the mix early this way, whereas it's mostly
missing from the lame texts of our day (another
litmus test if you're evaluating competing curricula:
does it include a rhombic dodecahedron?).
"""
(from Four Axes of Reform)
Obviously I'm stirring up controversy by
coming up with criteria whereby most in-place
curricula would be judged deficient and/or
inferior. Clearly I'm some kind of radical and
should not expect too much of a hearing.
That being said, fostering debate about the
future of mathematics could generate some
interesting threads -- has already.
Kirby Urner
Portland, Oregon
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To post to this group, send email to mathf...@googlegroups.com.
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OREGON CURRICULUM NETWORK
Some obscure writer in Portland, Oregon, on
the war path (yet a Quaker, so presumably
not too violently), keeps putting it out there.
Here are some links:
http://mybizmo.blogspot.com/2010/05/math-reform.html
I should also mention Another Alien Curriculum
which Dr. Richard Hake thought was pretty cool:
http://www.4dsolutions.net/ocn/alien.html
The 'Four Axes for Reformers' essay, published
in two editions, provides some useful dichotomies
to consider:
I. Cardinality vs. Ordinality
2. Lexical vs. Graphical
3. Geometry vs. Geography
4. Logic vs. History
I'm lifting the last from Hegel, as the basis of his
idea of a dialectic.
THE IMPORTANCE OF HISTORY
Restoring a time axis or time line to math teaching
is a core objective, along with making that time line
"place based" i.e. local history forms a starting
context, reaching out to larger narratives.
I illustrate this approach in my concluding section,
where I focus on regional geography, the Columbia
Gorge in particular.
No, I'm not the originator of the "place based education"
idea -- that's as old as the hills, until (more recently)
educators put a name to it.
A RENAISSANCE FOR POLYHEDRA
Continuing threads in this archive, I'll quote a paragraph
re Geometry vs. Geography:
"""
Beginning with the Platonic Five, the tetrahedron
is self dual whereas the others form two pairs of
duals. Combining these duals gives us additional
polyhedra: a good beginning for spatial geometry
segments, including many animations (anime).
The space-filling rhombic dodecahedron gets
into the mix early this way, whereas it's mostly
missing from the lame texts of our day (another
litmus test if you're evaluating competing curricula:
does it include a rhombic dodecahedron?).
"""
That's probably less than clear even to many a
math teacher.
The Platonic Five are the tetrahedron, cube,
octahedron, pentagonal dodecahedron and
icosahedron. To take the dual of a polyhedron
is to exchange its faces for corners, keeping edge
numbers the same, and the Platonic Five include
all their own duals, with the tetrahedron self-dual.
Then one may combine a polyhedron with its own
dual to generate another polyhedron. Doing this
with the Platonic Five nets a "2nd generation" of:
tetrahedron + tetrahedron = cube (already a Platonic)
cube + octahedron = rhombic dodecahedron (yes!)
pentagonal dodeca + icosahedron = rhombic triacontahedron
FUTURISTIC TOPIC: SPHERE PACKING
The rhombic dodecahedron is a space-filler and
favorite of J. Kepler's. If one fills space with it, while
placing a ball (sphere) in each one, the resulting
matrix is known by various names, such as CCP,
FCC or octet truss. CCP refers to the ball packing
in particular whereas FCC more refers to the ball
centers in isolation and "octet truss" refers to
the skeleton or scaffolding one gets when
connecting these centers with equi-length edges.
You'll find pictures / animations at this old web
page of mine, but also all over the Internet of course
i.e. this is not really esoteric knowledge in any way,
and deserves coverage pre-college.
http://www.4dsolutions.net/ocn/xtals101.html
UNIT VOLUME TETRAHEDRON
I don't go into a lot of detail with the tetrahedral
accounting, having done so in other posts. My
school of thought likes to see at least some
exposition of this concept, with associated lore.
The rhombic dodecahedron has a volume of 6
according to this canon, with the dual pair that
define it (cube + octahedron) having volumes 3
and 4 respectively. Not sharing this information,
anywhere along the K-12 time line, is proof positive
of a corrupted curriculum (is what we assert --
obviously a minority / elitist view, which doesn't
mean indefensible).
AN ALGEBRA OF TOMORROW
I'm also fairly light on the computer language
aspect in these most recent writings, having
dwelt on that elsewhere, with more to come.
Jonathan Groves is in agreement that more of
what we call "abstract algebra" might usefully be
imported into what we call "elementary algebra",
with computer languages fomenting (by making
more concrete) an awareness of "types".
Ideas about closure, inverse, identity elements
relate to our typology or taxonomy of numbers
(e.g. N < Z < Q < R < C). When you ask for
the additive inverse of 3, i.e. 3 + x = 0, you
leave the set N for set Z. Asking for multiplicative
inverses gets one into Q. The circle of numbers
has widened over the centuries, plus given rise
to other "math objects" (types of object).
The types we have on machines do not
correspond exactly to the traditional types.
There's no "real number" type (not really).
Rather than consider this a limitation, we
look at the philosophy mathematics more
deeply. Floating point numbers need not
be "approximations" for anything -- they're
a technology in their own right. This topic
came up at our Oregon Math Summit way
back in 1997. And I quote:
"""
My response, expressed in the breakout
session (as the official note taker, I managed
to interject twice — once about the
wierdnesses in floating point math as
implemented in computers, reason
enough to learn the algorithms on paper
as well) was that if teachers wanted to
fight the standard bearers and take
back control of the curriculum (they all
agreed that these tests from on high
were severely limiting their freedoms
and creativity as teachers on the
front lines) they could use the ammo
I was supplying via Beyond Flatland.
Here was basic, low level, primary
school material that every kid should
know, and yet isn’t part of the standard
— clear evidence that the mathematics
department knows relevant content
far better than whatever officials charged
with concocting these tests.
"""
http://grunch.net/archives/130
To me, these seem to be futuristic ideas.
Actual changes to content is somewhat
distinct from using Internet technologies to
relay essentially the same content.
Back to the rhombic dodecahedron:
"""
The space-filling rhombic dodecahedron gets
into the mix early this way, whereas it's mostly
missing from the lame texts of our day (another
litmus test if you're evaluating competing curricula:
does it include a rhombic dodecahedron?).
"""
(from Four Axes of Reform)
Obviously I'm stirring up controversy by
coming up with criteria whereby most in-place
curricula would be judged deficient and/or
inferior. Clearly I'm some kind of radical and
should not expect too much of a hearing.
That being said, fostering debate about the
future of mathematics could generate some
interesting threads -- has already.
Kirby Urner
Portland, Oregon
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To post to this group, send email to mathf...@googlegroups.com.
To unsubscribe from this group, send email to mathfuture+...@googlegroups.com.
For more options, visit this group at http://groups.google.com/group/mathfuture?hl=en.
Maria Droujkova
May 10, 2010, 9:49:26 AM5/10/10
to mathf...@googlegroups.com
On Mon, May 10, 2010 at 3:12 AM, kirby urner <kirby...@gmail.com> wrote:
The 'Four Axes for Reformers' essay, published
in two editions, provides some useful dichotomies
to consider:
I. Cardinality vs. Ordinality
2. Lexical vs. Graphical
3. Geometry vs. Geography
4. Logic vs. History
Kirby,
I am preparing two online courses, one for family and community math, one for classroom math. Both will have a topic about finding your balance in the "flow channels" between the opposites. The most frequent example is probably the challenge-skill channel between anxiety and boredom (diagram attached).
I am trying to relate your categories to that (reform?) work, from this version of the essay. Immediately I need to broaden them.
1. Qualitative and quantitative understanding
2. Symbolic and iconic(?)
I need a better term here. Line and surface graphs, mind maps, GapMinder, Prezi zoom-able visualizations, and other such visuals aren't really iconic.
3. I am not sure I can relate to Geometry vs. Geography. Some interesting people are working on 3d immersive environments, for example: http://immersiveeducation.org/ The division that came up in a recent geometry project with students was Measurement vs. Abstraction, in our video, ruler and protractor or Scratch steps and angles vs. straightedge and compass or Origami. This has to do with qualitative vs. quantitative gradient.
4. Situated vs. general is the gradient name I like here. There are cool studies of situated mathematics and its differences from generalized mathematics, for example, Hoyles and Noss work with nurses. Through maintaining the balance, you want to achieve high levels of context-specific math knowledge, in many contexts, AND the ability to see "similarities over differences" (Alan Kay).
Cheers,
Maria Droujkova
http://www.naturalmath.com
Make math your own, to make your own math.
Jon Awbrey
May 10, 2010, 12:20:17 PM5/10/10
to mathf...@googlegroups.com
Re: "The most frequent example is probably
the challenge-skill channel between
anxiety and boredom" ...
There is a stream of research by Sorrentino and Roney on what they
call the "uncertainty orientation" vs. the "certainty orientation"
that may bear on this dimension.
Here's a survey in book form:
| Sorrentino, Richard M., and Roney, Christopher J.R. (2000),
| The Uncertain Mind : Individual Differences in Facing the Unknown,
| (Essays in Social Psychology, Miles Hewstone (ed.)), Taylor and Francis,
| Philadelphia, PA.
|
| http://books.google.com/books?id=P8Nrf0isk0wC
We had been discussing this at The Wikipedia Review
in connection with some issues affecting Wikipedia:
http://wikipediareview.com/index.php?showtopic=15318
Regards,
Jon Awbrey
Maria Droujkova wrote:
> On Mon, May 10, 2010 at 3:12 AM, kirby urner <kirby...@gmail.com> wrote:
>
>>
>> The 'Four Axes for Reformers' essay, published
>> in two editions, provides some useful dichotomies
>> to consider:
>>
>> I. Cardinality vs. Ordinality
>> 2. Lexical vs. Graphical
>> 3. Geometry vs. Geography
>> 4. Logic vs. History
>>
>
> Kirby,
>
> I am preparing two online courses, one for family and community math, one
> for classroom math. Both will have a topic about finding your balance in the
> "flow channels" between the opposites. The most frequent example is probably
> the challenge-skill channel between anxiety and boredom (diagram attached).
>
> I am trying to relate your categories to that (reform?) work, from this
> version of the essay <http://groups.yahoo.com/group/synergeo/message/58784>.
mwb: http://www.mywikibiz.com/Directory:Jon_Awbrey
knol: http://knol.google.com/k/-/-/3fkwvf69kridz/1
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
the challenge-skill channel between
anxiety and boredom" ...
There is a stream of research by Sorrentino and Roney on what they
call the "uncertainty orientation" vs. the "certainty orientation"
that may bear on this dimension.
Here's a survey in book form:
| Sorrentino, Richard M., and Roney, Christopher J.R. (2000),
| The Uncertain Mind : Individual Differences in Facing the Unknown,
| (Essays in Social Psychology, Miles Hewstone (ed.)), Taylor and Francis,
| Philadelphia, PA.
|
| http://books.google.com/books?id=P8Nrf0isk0wC
We had been discussing this at The Wikipedia Review
in connection with some issues affecting Wikipedia:
http://wikipediareview.com/index.php?showtopic=15318
Regards,
Jon Awbrey
Maria Droujkova wrote:
> On Mon, May 10, 2010 at 3:12 AM, kirby urner <kirby...@gmail.com> wrote:
>
>>
>> The 'Four Axes for Reformers' essay, published
>> in two editions, provides some useful dichotomies
>> to consider:
>>
>> I. Cardinality vs. Ordinality
>> 2. Lexical vs. Graphical
>> 3. Geometry vs. Geography
>> 4. Logic vs. History
>>
>
> Kirby,
>
> I am preparing two online courses, one for family and community math, one
> for classroom math. Both will have a topic about finding your balance in the
> "flow channels" between the opposites. The most frequent example is probably
> the challenge-skill channel between anxiety and boredom (diagram attached).
>
> I am trying to relate your categories to that (reform?) work, from this
> Immediately I need to broaden them.
>
> 1. Qualitative and quantitative understanding
>
> 2. Symbolic and iconic(?)
> I need a better term here. Line and surface graphs, mind maps, GapMinder,
> Prezi zoom-able visualizations, and other such visuals aren't really iconic.
>
> 3. I am not sure I can relate to Geometry vs. Geography. Some interesting
> people are working on 3d immersive environments, for example:
> http://immersiveeducation.org/ The division that came up in a recent
> geometry project with students was Measurement vs. Abstraction, in our
> video, ruler and protractor or Scratch steps and angles vs. straightedge and
> compass or Origami. This has to do with qualitative vs. quantitative
> gradient.
>
> 4. Situated vs. general is the gradient name I like here. There are cool
> studies of situated mathematics and its differences from generalized
> mathematics, for example, Hoyles and Noss work with
> nurses<http://www.lkl.ac.uk/rnoss/papers/ProportionalReasoningNursingJRME.pdf>.
>
> 1. Qualitative and quantitative understanding
>
> 2. Symbolic and iconic(?)
> I need a better term here. Line and surface graphs, mind maps, GapMinder,
> Prezi zoom-able visualizations, and other such visuals aren't really iconic.
>
> 3. I am not sure I can relate to Geometry vs. Geography. Some interesting
> people are working on 3d immersive environments, for example:
> http://immersiveeducation.org/ The division that came up in a recent
> geometry project with students was Measurement vs. Abstraction, in our
> video, ruler and protractor or Scratch steps and angles vs. straightedge and
> compass or Origami. This has to do with qualitative vs. quantitative
> gradient.
>
> 4. Situated vs. general is the gradient name I like here. There are cool
> studies of situated mathematics and its differences from generalized
> mathematics, for example, Hoyles and Noss work with
> Through maintaining the balance, you want to achieve high levels of
> context-specific math knowledge, in many contexts, AND the ability to
> see "similarities
> over differences<http://learningevolves.wikispaces.com/nonUniversals#simDiff>"
> context-specific math knowledge, in many contexts, AND the ability to
> see "similarities
> (Alan Kay).
>
>
> Cheers,
> Maria Droujkova
> http://www.naturalmath.com
>
> Make math your own, to make your own math.
>
--
inquiry list: http://stderr.org/pipermail/inquiry/
>
>
> Cheers,
> Maria Droujkova
> http://www.naturalmath.com
>
> Make math your own, to make your own math.
>
--
mwb: http://www.mywikibiz.com/Directory:Jon_Awbrey
knol: http://knol.google.com/k/-/-/3fkwvf69kridz/1
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
Maria Droujkova
May 10, 2010, 2:29:52 PM5/10/10
to mathf...@googlegroups.com
Thank you very much for the link. As a person with high level of "novelty-seeking" and uncertainty tolerance, I sometimes make others uncomfortable organizing group events according to my own tastes, or have to "bring my own uncertainty" to find comfort at events organized by others. I am reading the discussion and the book now. It will help with the project.
Managing uncertainty becomes increasingly important as "multisubculturalism" (an awesomely funny word) increases in information spaces.
Managing uncertainty becomes increasingly important as "multisubculturalism" (an awesomely funny word) increases in information spaces.
kirby urner
May 10, 2010, 5:27:46 PM5/10/10
to mathf...@googlegroups.com
On Mon, May 10, 2010 at 6:49 AM, Maria Droujkova <drou...@gmail.com> wrote:
> On Mon, May 10, 2010 at 3:12 AM, kirby urner <kirby...@gmail.com> wrote:
>>
>>
>> The 'Four Axes for Reformers' essay, published
>> in two editions, provides some useful dichotomies
>> to consider:
>>
>> I. Cardinality vs. Ordinality
>> 2. Lexical vs. Graphical
>> 3. Geometry vs. Geography
>> 4. Logic vs. History
>
> Kirby,
>
> I am preparing two online courses, one for family and community math, one
> for classroom math. Both will have a topic about finding your balance in the
> "flow channels" between the opposites. The most frequent example is probably
> the challenge-skill channel between anxiety and boredom (diagram attached).
>
That's an interesting diagram.
> On Mon, May 10, 2010 at 3:12 AM, kirby urner <kirby...@gmail.com> wrote:
>>
>>
>> The 'Four Axes for Reformers' essay, published
>> in two editions, provides some useful dichotomies
>> to consider:
>>
>> I. Cardinality vs. Ordinality
>> 2. Lexical vs. Graphical
>> 3. Geometry vs. Geography
>> 4. Logic vs. History
>
> Kirby,
>
> I am preparing two online courses, one for family and community math, one
> for classroom math. Both will have a topic about finding your balance in the
> "flow channels" between the opposites. The most frequent example is probably
> the challenge-skill channel between anxiety and boredom (diagram attached).
>
Getting stressed out in A3 might be addressed by raising skills or lowing the
challenge. However the arrows suggest an evolutionary pressure to move
towards more skills and higher challenges. Lowering the challenge is
implicitly frowned upon.
I consider boredom a high state, an invitation to reflect, to look inward.
Maybe the bored student needs to focus less on technical skills
and more on lore (story, context). More on why, less on what.
What foments boredom in many a math class is a shortage of lore,
and absence of storytelling.
The stories I tell around polyhedra (geometry) links them to energy
planning (geography) and therefore to living standards. I tend to get
polemical in this context (raising the stakes, raising the challenge).
> I am trying to relate your categories to that (reform?) work, from this
> version of the essay. Immediately I need to broaden them.
>
> 1. Qualitative and quantitative understanding
>
myself to four gave me plenty to yak about.
Or are you saying Qualitative vs. Quantitative is a way of broadening
my Cardinal versus Ordinal?
I could see where these would be in the same ballpark.
Before one can answer "how many?" one needs to know what's
being counted versus what's not being counted.
Before you can compare apples to apples, oranges to oranges
(ordinality), you need to be able to identify oranges apart from
apples (cardinality).
> 2. Symbolic and iconic(?)
> I need a better term here. Line and surface graphs, mind maps, GapMinder,
> Prezi zoom-able visualizations, and other such visuals aren't really iconic.
>
like your flow channel diagram. The components may be symbolic, like
the words and arrows that comprise it.
What I think about as a sometime teacher of computer programming skills,
is how lexical this activity is. One plays with symbols, interacts with glyphs.
Yet a student may be used to a lot of graphical stimulation. On being
challenged with such a lexical activity, the student becomes anxious or
bored.
They want to "write games" which is code for wanting more graphics, fewer
lexemes. Some games (challenges, puzzles, simulations) may be lexical
though -- a place to start.
The dichotomy is akin to reading books with lots of pictures (with comic
books and TV at one extreme), versus reading books with no pictures at all.
In the latter case, once is supposed to fill in with one's own imagination.
Learning to enjoy programming sometimes involves learning to imagine
what you're programming about.
> 3. I am not sure I can relate to Geometry vs. Geography. Some interesting
> people are working on 3d immersive environments, for example:
> http://immersiveeducation.org/ The division that came up in a recent
> geometry project with students was Measurement vs. Abstraction, in our
> video, ruler and protractor or Scratch steps and angles vs. straightedge and
> compass or Origami. This has to do with qualitative vs. quantitative
> gradient.
>
in this case.
The "geography" I'm talking about is *all* special case experience, so is
comprehensive. All time scales are involved, meaning stars and microbes
are all "geographic" (are objects in time).
Geometry lets us think in terms of patterns and generalizations
independently of where/when.
A tetrahedron may be investigated without specifying its size or
specific location.
Geometry generalizes about triangles whereas any actual triangle
with some physical / temporal existence is geographic in nature.
Geometry subsumes philosophy and mathematics.
You could think of this as a way of trying to get everything we
call "content" divided into just two subjects (which may then
be further subdivided -- like the quadrivium / trivium of old,
but here starting with just the one dichotomy why not?).
> 4. Situated vs. general is the gradient name I like here. There are cool
> studies of situated mathematics and its differences from generalized
> mathematics, for example, Hoyles and Noss work with nurses. Through
> maintaining the balance, you want to achieve high levels of context-specific
> math knowledge, in many contexts, AND the ability to see "similarities over
> differences" (Alan Kay).
>
Domain-specific data and knowledge makes the math come alive for
professionals immersed in that domain, yes.
This book 'Who Is Fourier?' by the LEX Institute is a good example
of a math book about something general (Fourier analysis and the
calculus behind it) supplied with a domain-specific context (wanting
to study vowel sounds in Japanese).
So many math teaching texts focus on purposely meaningless "story
problems" that fail to connect to student experience (by design).
On other lists, I advocate more "rich data structures" meaning when
we teach a math-programming mix, we need to use more significant
examples. Google Earth is a good example.
Google Earth might be too graphical in some contexts (not enough of a
challenge?). Maybe just a Python dict of some cities and their lat/long
coordinates would help with a Geometry-Geography segment.
http://www.4dsolutions.net/ocn/python/gis.py
I'm not against graphical content by any means, or iconic. The
O'Reilly 'Head First' books continue a lot of the techniques in
'Who is Fourier?' i.e. supply a lot of glyphs, graphics of mnemonic
value.
Crafting mnemonics is both an art and a science. How to
design curriculum materials that are both memorable and
accessible: this is at the core of effective pedagogy.
Graphs in the sense of hyperlinks between web pages, in
the sense of edges between vertexes of polyhedra, remind us
of what it means to "connect the dots". The idea of neural
nets but also spider webs, enters the lore.
Polyhedra connect around circumferentially every which way,
connoting systematic, cross-referenced, organized dot
connecting, versus a radially dispersive "information
explosion" wherein everything "gets away" (unordered,
unconnected). Good stories, strong mnemonics, help us
systematize. Polyhedra as mind maps, or symbols of
mind maps.
For mathematics to serve as a "glue language" it needs
to connect the geographic to the geometric i.e. give us
ways to think about and generalize from the exploding
array of special cases we encounter.
Thank you for elaborating on my posting and interjecting some
interesting thinking and observations.
Kirby
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