On Mon, Jul 15, 2013 at 12:46 PM, Robert Hansen <firstname.lastname@example.org> wrote:
> > On Jul 15, 2013, at 1:30 PM, Joe Niederberger <email@example.com> > wrote: > > In this particular case, though, I'd have to suspect that Robert just > doesn't see it. > > > > Is it too much to ask that if you claim that I don't see "it" that you say > what "it" is? >
What "it" is, is that it makes as much sense to say "5 tetrahedroned" as it makes to say "5 cubed".
The choice of shape was cultural / conventional, not dictated by any deities nor by logic, and in fact we may develop a consistent mathematics by following this other branch, and in fact we have.
Which is another point to make: that mathematics is a process of branching and sometimes merging (like in Git).
> > Look, I stacked 1 bag of sand here, and then I stacked 8 bags here, and > then 27 bags... Do you see how stacking bags of sand is cubing? > > You are stacking them in a cube shape in the picture. You've seemed keen to just throw away the picture as if visualizations can't mean anything:
The single unit cube starts us out and then we need what is called a gnomon, which in 2D means putting an L of three squares around a square to get a 2x2 square. But in 3D, we affix cubes to just three of the faces and then fill in the blanks to get a next bigger sized cube of 8 blocks (see picture).
We had to add 7 to the 1 to get 8 (see picture).
Good example addition as well as shape preservation (self-similarity).
Now the cube of 8 is our starting point and we want to take it to the next level. We pad with cubes on three sides adding a total of 4 to each of three faces, three segments of 2 for edges, and a new corner: 12 + 6 + 1 = 19. 19 + the 8 we started with = 27.
It's easier to see with an animation (or see picture).
Strewing sacks of hay all over the field, higgledly-piggledly is adding lots of noise and confusion, seems a kind of deliberate deconstruction and information loss. Of course they won't "get it" then.
And now, after doing the cube, we do the same game with a tetrahedron.
The starting shape is different and the "gnomon" (what preserves self similarity) affixes to just one face.
Adding the next triangular number of balls (if we're sphere packing) adds a next layer, and its volume will be 7, an octahedron (4) with three tetrahedra (1) = 7 (see picture).
And the game continues. We have a growing tetrahedron of volume 1, 8, 27...
The tetrahedron shows the same algebraic relationships, it just we splays the originating vectors differently.
With a cube, the vectors from the origin are mutually orthogonal.
The corner of a tetrahedron is sharper. So what? We can still play the game of "growing gnomons" and the numbers stay the same too.
An extreme example? How does that differ from stacking tetrahedrons if the > students can't connect it to "cubing" other than it happens to be 1, 8, > 27... > > We *do* connect it to cubing, very definitely.
But then we don't reach a full stop like the Common Core does.
We who write STEM curriculum for the better-than-mediocre schools take this opportunity to say "wait kids, one of the most famous Americans, a transcendentalist like Emerson and Thoreau, showed in the mid-1900s that tetrahedron-as-unit-volume is an interesting branch. Lets investigate..."
There's been lots of development along that branch ever since, a rich and growing set of interconnected books. It's high time at least the better schools woke up to recent history and stop pretending the 1900s never happened. "He won the Medal of Freedom from Ronald Reagan and universities just gave him top degrees to help themselves stay relevant."
Like, kids in Baton Rouge should know about that Union Carbide dome, even if it's gone. American History, ya know.
> Sometimes I think you (Joe) are talking about advanced students who > already know cubing (and algebra and calculus) and this tetrahedron > stacking an engaging activity. Like figuring out why the sequence 1 + 3 + 5 > + 7 ... results in perfect squares. But Kirby produced this as an example > of teaching "cubing", only because it happens to result in a sequence of > cubes. So does the sequence 1 + 7 + 19 + 37 + ... But neither are very > intuitive ways to teach cubing. > > I think you completely missed my point. My point is we want kids / students to have a relatively mature understanding of mathematics and that means understanding that conventions / culture / ethnicity plays a role. All math is ethno-math.
At some point, the cube became a chief geometrical model for 3rd powering and now we say "cubing" without thinking about it, never stopping to question, even though much significant work has been done on "tetrahedroning" as another branch.
Why miss this golden opportunity to develop a more sophisticated understanding of what math is like?
Like everything else in the curriculum, it's a spiral staircase. If the kids are pre-verbal we don't start with calculus.
Where we can start pretty young though is having them recognize the difference between a face, corner or edge. This takes some abstraction because it might be a wire frame i.e. the "face" is more a "window" or a "hole".
They learn to recognize and count faces (windows), edges and corners (nodes, vertexes). Yes it's early graph theory. Topology. That used to be like in college.
But then when you and I went to elementary school, they didn't have a World Wide Web of pages (URLs) connected by hyperlinks.
The idea that some ideal curriculum is already out there, set in stone, is of course pure poppycock.
> We teach cubing with cubes because it is easy to connect the two. It is > easy to go from line to area and from area to volume. It is easy to show > how increasing the dimensions increases the base of x^3. > > It is also easy to show that a growing / shrinking tetrahedron demonstrates 3rd power change relative to 1st power linear change, true of any 3D volume that stays self-similar.
What's more, the tetrahedron is topologically simpler than the cube and plays well with others to give whole number volumes for many more shapes, a subject of Arthur Loeb's article for The Math Teacher in the 1960s, don't have the volume:page on me.
Just that the 1900s *did* happen, and the better schools will factor that in even if the Common Core does not.
Bob Hansen > >  connect: determine through reason that two things are mathematically > identical at some level. >