Those of us familiar with the Montessori design for a pre-school, featuring off-the-shelf "jobs" with which the children discover (both alone and in groups) know that "playing with blocks" is an important aspect of this curriculum.
When it comes to Fractions, one may use various senses of "conservation" to demonstrate equivalence. Pouring water or dry granular material from one container to another is a good way to demonstrate relative volume: four cups make a quart, four quarts make a gallon, eight ounces make a cup.
DISTINGUISHING VOLUME FROM WEIGHT
There's a blend of weights and volumes you'll notice, because once the density is set, and presumably the substance, you're able to use volumes to compare weights and vice versa. Twice as much of something is both twice the volume and twice the weight.
More formally, however, we distinguish between volume and weight as different "dimensions" and start to tease out these differences in discussions of physical principles. The same volumes will have different weights on the moon i.e. the gravitational field is a variable.
Decartes and his contemporaries referred to pure extension, minus any secondary characteristics of temperature, weight, color, density, as Res Extensa. The XYZ coordinate system is its own best example, in spanning a volume without contributing any mass or material of its own. The XYZ system is a "reference frame" meaning we impose it as an imaginary ruler or grid. Even when we construct our rulers out of physical materials (as we often do), we keep this sense of the "purely imaginary" and refer to that vista when doing "solid" and/or "spatial" geometry.
I think many students start having philosophical confusions right about here. Many teachers don't have the philosophical training to tease apart the various nuances and a student may lose track of how a "dimensionless point" differs from a "dimensionless cube", as neither one has any physical existence to speak of (conflating two meanings of "dimensionless").
In spatial geometry, we may eventually want to model the weight of something, in which case it turns into physics and/or trading in the market place (weights and measures). What's important to get is this isn't the starting position i.e. the XYZ coordinate system is not a physical scaffolding and therefore has no physical dimensions as these might relate to time and/or space (space-time) in the sense of having energetic content. We speak of energy-based mathematics as "applied" and/or "about the real world". The no-energy world of res extensa is frequently referred to as "Platonic".
FRACTIONS AND POLYHEDRA
We've been discussing here on math-teach about how the linear, areal and volumetric rates of change, of a shape with fixed angles, are related as first, second and third power quantities respectively.
Starting with a tetrahedron of volume one, and doubling all its edges, results in a tetrahedron of relative volume eight. Halving the edges from the same starting position gives a tetrahedron of relative volume 1/8.
Surface area varies as a power of two, as we see in the many formulas using exponential notation (i.e. most all of them).
Most 1900s textbooks, as well as most early Montessori jobs, introduced these concepts almost exclusively with little cubes as units. A rod of cubes would represent an edge or line. An n x n square of cubes represented a flat surface (growing as a 2nd power of n), and the 3rd power, or volume, was made of stacked squares. n x n x n was modeled as a cube with sides n.
This modeling technique isn't going to fade away, but nor is it sacrosanct. Mathematics is a flexible discipline, and what better way to illustrate this fact than to switch bases? Just as we go back and forth between decimal (10) and hexadecimal (16), so might we switch between triangular, 60-degree based thinking, and the more orthodox 90-degree based thinking. As a mnemonic, we note that 10 is a triangular number, whereas 16 is a square one. Mathematics allows this kind of diversity. Sometimes (often) two ways of looking is better than one.
One shortcoming of cubes is they don't play with others that well, only with themselves. The volume formulas for almost all polyhedra generate irrational numbers, because little thought was given to a more integrated design.
Another problem with cubes is they're topologically more complicated than tetrahedra, being hexahedra.
One might easily imagine a tropical island paradise where the natives pack together three coconuts (as idealized equal-radius spheres) and call that a unit of area, pack together four coconuts (as a tetrahedron) and call that a unit of volume. Why not study this island culture then, explore the consequences of this more primitive topological beginning? Might this civilization turn out to be more advanced than our own? Or perhaps we should look at Martian Math? Here in Oregon, we're looking at both of these storyboards as useful backdrops for future mathcasts (animations, cartoons).
Cubes do have the advantage of filling space however, thereby generating our imaginary XYZ coordinate system (once you pick an origin and basis edges), so of course we don't want to lose them or abandon centuries of useful algorithms (that'd be wasteful). Cubes are cool, but not the whole story, is what these segments on Fractions will make clear.
The NCTM lesson plan on tetrahedral kites goes through the logic needed to prove how the regular octahedron of edges N, has 4x the volume of the regular tetrahedron of edges N. We start talking about V + F = E + 2 at this point, building these shapes from ping pong balls, and spinning them around their various axes -- all branches to other curriculum topics. The kite itself has an historical dimension connecting us back to Alexander Graham Bell, as most math teachers will know, if aware of this lesson plan in the first place.
Using the regular tetrahedron and octahedron together, we are able to fill space (same as with XYZ). We have a relative population of 2:1 (twice as many tetrahedra) and a volume ratio of 1:4 (per tetrahedral kites). That's an excellent beginning for discussing fractional relationships, but where is our cube? A regular tetrahedron embedded as face diagonals in a cube has one third its volume, another math fact easy to prove visually, as well as algebraically. Or just pour water or dry grain if doing a Montessori job (a self directed activity). The cube contains the unit tetrahedron and has a relative volume of three.
This shape gets special attention in any spatially aware geometry curriculum. Kepler gets a lot of credit for "tuning it in" (a radio metaphor). Given it's neither a Platonic nor an Archimedean polyhedron, it was languishing in obscurity, not appreciated for its marvelous properties.
What are these marvelous properties?
Well, for one thing, it fills space completely, without gaps. You can put a sphere inside each one, and they'll just "kiss" one another at the diamond face centers. That's twelve spheres around any nuclear sphere. Do we have the models in class? If not, shall we make them? That's what Linus Pauling did, and he won two Nobel Prizes, one for chemistry.
For another thing, the long face diagonals define an octahedron, and the short face diagonals define a cube. These two shapes are "dual" to one another, a concept we explore.
Taking both of these (the cube and octahedron) to be the same as those described above (volumes 3 and 4 respectively), and we have a great set of geometry blocks to play with: a tetrahedron and octahedron that fill space together, and a rhombic dodecahedron that fills space by itself, for example by surrounding each corner where the first two share a vertex.
In sum, here we have two space-filling matrices or lattices, interpenetrating, the one of two complementary shapes (tetrahedron + octahedron), the other of all the same shape (rhombic dodecahedron).
That's a lot to get across in words, but on a bright, sharp computer screen, you're looking at enlightening cartoons (animations, mathcasts). You may also build these as hands-on models, as is done in the NCTM lesson plan on tetrahedral kites.
Note that "kites" is the visual motif for the next NCTM annual session. How might we relate kites to math topics? In many ways, with tetrahedral kites being one of them (Penrose kites is another, with links to Kepler again).
Our 2000s curriculum is no longer restricted to only sharing about cubes when it comes to describing space and the relative volumes of polyhedra. Per the NCTM lesson plan on tetrahedral kites, we have the option to build with a different set and some curricula, including mine (pioneering in Oregon), take advantage of this option.
There's no presumption of either/or, and in showing how our regular tetrahedron might be used to calibrate this alternative set of blocks (volumes 1, 3, 4, 6 for tetrahedron, cube, octahedron, rhombic dodecahedron respectively), we're not letting go of the more traditional cubical mensuration system.
The fractions 1/3, 1/4, 1/6, 3/4, 1/2 derive easily from this new set of blocks. As we spiral through in future segments, we will add to this vocabulary, both by splintering the regular tetrahedron and octahedron into smaller tetrahedra of volume 1/24, and by adding additional polyhedra, including some from the five-fold symmetric family, such as the rhombic triacontahedron, icosahedron, and pentagonal dodecahedron These latter three have a relationship that parallels that of the rhombic dodecahedron to its embedded shapes, as defined by its rhombic facets. Our rhombic triacontahedron of volume 5 splinters into 120 tetrahedral modules equal to 1/24.
Sometimes teachers question whether doing the work to introduce these blocks is going to grow with the student, or immediately become back burner and forgotten as one of these elementary school topics that's likely a bridge to nowhere.
Lets reassure these teachers that rhombic dodecahedra in closest packing is likewise chemistry. The octahedron- tetrahedron lattice is likewise architecture. Those branching topics mentioned earlier, derived from spinning, packing with ping pong balls, looking at V + F = E + 2 and the concept of dual, are all gateways to a huge and multi-disciplinary literature.
Not only that, but spatial geometry includes all the flat lander stuff as a special case. The trigonometry of flat surfaces, as well as curved ones, is embedded in this context, so none of the Euclidean material leaks away.
So the real question is whether spatial geometry as a whole is relevant, as once we admit that it is, then we're somewhat obliged to acknowledge its core content and would be foolish to bleep over its most streamlined encapsulations.
Spatial geometry underwent a lot of changes in the 1900s and just sticking with the awkward volume formulas and those mostly irrational number treatments (except where the cube is concerned) is a sure way to fall behind, relative to more enlightened approaches, already available.
The more 60-degree based way of thinking described above is consistent with findings about naturally occurring geometries, which take advantage of the triangle's (tetrahedron's) superior stability vis-a-vis the square's (cube's). The emergence of corresponding mathematical treatments has been a long term trend. One could list many published titles at this point, such as 'Beyond the Cube'. Dr. Arthur Loeb of Harvard and MIT helped get the ball rolling in 1966, in some issue of Math Teacher. None of this is new information at this point. Its phase-in is long overdue and above professional reproach, given its secure mathematical basis.
LEVERAGE FOR TEACHERS
Teachers chafing under the restrictive nature of state standards might want to assert their professional status and call into question the authority of any standards body that is still mired in 1900s cubes-only thinking. Even at the elementary school level, such "rule by committee" is manifestly without integrity and is doing much to hold back innovation and academic success.
Of course it may not be wise to question authority that overtly. A teacher may prefer to quietly phase in this material, using some of the lesson plans already on-line, just to establish a track record for future review. It's always nice, in hindsight, to be seen as an "early adopter" in some relative sense, even though in a conservative discipline such as mathematics, upgrading is often a slow process.
Our K-12 curriculum is something of a relic as early math mostly changes in geological time, although a punctuated equilibrium model may pertain.
Even just a dash of this more contemporary thinking goes a long way, like a spicy hot sauce, so it's OK to use it sparingly, experimentally. Find your peers and collaborate on a "place based" approach. Don't wait for a wood pulp textbook to spell it all out. Those dino days are likely over.
To *not* use the more sophisticated blocks in any way, shape or form is to settle for bland "business as usual" a somewhat risky (and vapid) choice in today's climate, if hoping to seem in any way current (up to date). Let students vote with their feet?
At least project some of the Internet material and see if they perk up a little. In my experience, they usually do, especially when they pick up on the subversive flavor, of challenging some crufty-dusty status quo -- there's the scent of a hunt, a game, a sport.
Mass-published elementary school textbooks of course have nothing like this, high school textbooks precious little, so they become exhibits in our "museum of retro". It's fun taking these apart in class, a kind of archeology, and pointing out the strong cubic bias -- a lot like investigating racism, in terms of finding the tell tale signs.
Remember: all mathematics is ethno-mathematics.
I would encourage teachers to engage student interest at this level, as "critical thinking" applies not just to what you see on television, but what you read in the paper and find in your textbooks. Unthinking acceptance of media, of mathematics included, is the antithesis of developing those critical thinking skills we seek to encourage (those of us in the liberal arts tradition, who value philosophy and its freedoms -- not saying everyone does).