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Re: List of 'good' geometry textbooks
Posted:
Aug 11, 2007 1:27 PM
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> I agree with much of this as a approach to learning
> geometry!
>
> A very old source (1840's0 which started with some
> 3-D explorations, well before anything that was
> primarily 2D, was Froebel - the man who
> developed a revolutionary education program he
> called kindergarten (the children all had small
> gardens)....
Yes, the first time I learned of Froebel and his
curriculum was form Stu Quimby, CEO of the now
defunct Design Science Toys.
> Foebel's work played with symmetry, with shapes
> creating by spinning, etc. as well as the way
> carefully designed decompositions came apart
> and reassembled into patterns.
One of the toys I share in classroom settings is
CubeIt!, a magnetic cube that decomposes into 24
identical right tetrahedra, each defined by
a face center, cube center, two adjacent corners.
Pictures of CubeIt! in my blog:
http://worldgame.blogspot.com/2006/11/journey-home.html
In our curriculum, these right tetrahedra are
called MITEs for MInimum TEtrahedra, so-called
because they (a) fill space and (b) are the
simplest shape to do so (in the sense of having
only four facets).
These MITEs in turn deconstruct into what we call
A and B modules (also tetrahedra).
What's so helpful to children is not only are we
dealing with spatial shapes, but we're exploring
volumetric relationships using simple fractions
and whole numbers.
Shape - Volume:
A module - 1/24
B module - 1/24
MITE - 1/8 (= AAB)
Tetrahedron - 1
Coupler - 1 **
Cube - 3
Octahedron - 4
Rhombic Dodecahedron - 6
Cuboctahedron - 20
The ratio of the cube to rhombic dodecahedron is
therefore 3:6 or 1:2, as might be shown by pouring two
cube-fulls of something into the rhombic dodeca (see
below). The rhombic dodeca is a space-filler by the
way, much studied by Kepler.
We're also talking about symmetry, left and right handed,
as the A and B modules fold into either left or right
handed versions of themselves from the same plane net,
depending on how you fold the creases.
I'm compressing several grade levels here. It only
seems like an overwhelming amount of info when
compressed into a short post like this -- true of
*any* substantive curriculum I would think.
> There is a lot of evidence that all of us learn first
> to 'see' in 3-dimensions, and find that cognitively
> easier in many circumstances.
Yes, we're indigenous to space. The idea of "two
dimensional beings" per Abbott's 'Flatland' is very
counter-intuitive and non-experiential, as even a
simple line segment is surrounded by a blank canvas
or background, plus our seeing it implies distance
from an observer. Already, space is conceptually
present (viewer + canvas).
The more formally defined 2D Euclidean stuff needs
to be saved for when students are more mature, ready
for non-experiential abstractions such as "planes to
infinity with no thickness" (never found in reality,
requires imagination and a leap of faith).
Artists of the Renaissance taught themselves techniques
for translating this world to a flat canvas, using the
"rules of perspective." This often involved drawing
grids and faithfully mapping contents from the visual
field to each square.
From such innovations came our current concept of the
XY grid and Cartesian/Fermatian Coordinates.
> This even appears to be true among blind people - so
> the 'seeing' here is our cognition, with input from
> the eyes, and the hands. See for example the book:
> Brosterman, Norman. "Inventing Kindergarten" New York:
> Harry N. Abrams, Inc., 1997.
>
> Walter Whiteley
At the the early Montessori level, I introduce properly
proportioned polyhedra (described above), hollow, but
with one face open.
The children sometimes spontaneously identify these as
"measuring cups" as they use them to pour a dry granular
material from one to another, thereby exploring volume.
I use dried beans, as I've found finer grains like cornmeal
get everywhere, including in the cracks between floor
boards. Even though these volume relationships are exact,
we're not trying to "prove" that with the beans, just
get a feel for the them, plus work on vocabulary such as
"octahedron".
A clear plastic version of the set using colored water
would make an attractive salable kit I should think.
Kirby
** Coupler, like A and B module, is unfamiliar
nomenclature to many. It's relationship to the
cube, and the MITE, is suggesting in the graphic
at the top of this web page:
http://www.4dsolutions.net/ocn/pymath.html
(what my curriculum looks like once we get to
the high school level -- lots of computer use).
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