> 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.
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.
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.
** 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: