On Sun, Jul 14, 2013 at 10:04 AM, Robert Hansen <firstname.lastname@example.org> wrote:
> > On Jul 14, 2013, at 12:48 PM, kirby urner <email@example.com> wrote: > > If you shaped the bags into uniform-sized polyhedrons and fit them > together seamlessly, like bricks, to form this growing tetrahedron, then > you would be showing what the cube people show. The cube people show > smaller bricks making the bigger cube. Here, we have an octahedron and > tetrahedron that together fill space. > > Another demo would be to take a glass or clear plastic tetrahedron and > bury the tip in the sand so it stays upside down. Pour liquid into it > using a unit tetrahedron measuring cup. Mark the positions 1, 8, 27, 64... > and notice they are equally spaced up the side. > > Unlike a cube, when you slice a regular tetrahedron parallel to any side, > you still have a regular tetrahedron. Nice property. Slice a cube that > way and you have a cube no longer. > > > No, there is a lot more to it than that Kirby. That is probably why just > stacking stuff never caught on with teaching mathematics. Like I said, you > are confusing intuition with rhetoric. What point is there in > displaying examples that the student couldn't possibly understand the > underlying reasons for? The point was to teach about cubing, not to perform > a magic show. I think I stick with the old fashioned approach of teaching > cubing first, and several other things, and then later we can really dive > into the analytic geometry behind tetrahedrons. > > Bob Hansen >
The point of the Common Core section was to teach about 3rd powers and 3rd roots. Not just around the number 5 either -- that was an example.
The almost-unconscious hard-wired never-questioned link to "cubing" was embedded in the Common Core's language. Peter was taking issue with the phrasing, I was taking issue with the "cubing".
There is no point anywhere along K-16 where the curriculum goes "Wait! You could use a tetrahedron for 3rd powering!" except in ultra-obscure (at the moment) corners, like in the late Arthur Loeb's class at MIT, where I sat in for a day some years ago.
I'm saying: lets not have another generation that clueless, and instead lets have curricula which promise to cover Common Core and go better / beyond on many of these standard points. Alignment is the minimum.
The mediocre schools will continue to not mention the tetrahedron as a model of third powering or talk about what a mathematics might be like that went in this direction. What would other volumes be, if the tetrahedron is 1. We get a nice cube of volume 3. More water pouring.
Lets talk about how mathematics is a combination of conventions, definitions and logical deductions based on some rules. That's already the gold standard, to have those discussions. I'm talking about adding grist for the mill. Accessible stuff, not hold-your-head hard stuff.
All the material we need to be better than mediocre is out there, published, colorful, on computer and so on, so there's no reason to hold the students back.
>  intuition: to sense that something is true >  rhetoric: to say that something is true >
Always quick with your personal definitions of everything. For you, logic is a "feeling" if I'm not mistaken.