## Wednesday, November 2, 2011

### What Do You Make of This? (Build What I Have, Redux)

In my recent post Conversational Math: Part Two I described a game that my daughter and I like to play with Cuisenaire rods called "Build What I Have".  It's really fun.  So fun, in fact, that my daughter asks for it every other day or so.
Here's a funny thing that happened today:

It was my turn to give directions.  To start, take a look at how my design looked when I was finished giving directions.  I should point out that there is only one layer of blocks laid out on the board we were using.  I should also point out that we hid our work from each other.  She is trying to replicate my design using only my spoken words as a guide.  Ironically, no gestures are allowed in this game!

 Mine.
Here's how I started: "Take three light green rods and put them together so they are all touching and all perpendicular to the bottom of the board.  It should look like a square that is three units across.  Now add a fourth light green rod on top of the square.  It should be placed perpendicular to the other green rods and parallel to the bottom of the board.  It should look like a rectangle now."

So far so good!

Me: "Take an orange rod and place it on top of the green rectangle."
Kid: "Like a teter totter."
Me: "Yeah, I guess so! Now, take two white blocks and put them on top of each end of the orange rod" (etc.)

Then it was time for the big reveal...tah dah!

 Hers.
Geez Louise, what happened?!?  Her orange rod IS on top of the light green rectangle, but it appears the kid built up while I built...up?  I suspect we were both right, but I am at a loss to explain why or how.  I think it is accurate to call the kid's work three dimensional, but so is mine because it is built with polyhedra and not 2D lines on a flat surface.

Hmmm....any ideas?﻿

1. I don't have any ideas, but I find that really cool!

2. What a great game!!!

So your up is "north" Your obviously don't live in the southern hemisphere. It's a great opening to a conversation about what is up?

3. Its obvious that Iz thinks in a 3D universe, like looking at everything from all directions at once. As a painter/quilter I find it hard to see anything but flat surfaces moving around a flat surface. She has no preconceptions as to direction. If there was a demonstrable 4th dimension I'm sure she would be using it also. Have her look at the Cubist paintings of Juan Gris and ask her to describe or draw her observations. It might be just very interesting.
Youer father.

4. Yeah, what is 'up' anyhow? This is turning into a great conversation about maps vs. experience, orientation, perspective...

I guess the question is: Where are you starting from? In Math in Your Feet we start in the center of our squares and literally move forward, back, R, L, diagonal, etc. from there -- if I had been dancing that design I would have used the word forward. Instead, because I was essentially 'drawing' design (albeit with 3D blocks) I labeled that direction as 'on top' which, I guess, also means up.

Now that I'm thinking this through, is there a difference between direction and location? It seems to me there is and that is possibly why I might use different words in different situations. Direction is *where you're going* and location is *where you are*?

Thoughts?

5. The words that caused the problem were "on top of," rather than "up." I think that's worth attending to. And the fact that you were in 3-space doesn't quite work. You were only in 3-space in the sense that your elements were 3-D. But you dealt with them pretty much as if they were restricted to/embedded in the plane. Nothing wrong with that, but having not made an agreement to restrict instructions to the plane, you opened up the possibility your daughter chose. She ALMOST rotated your figure out of the plane into the third dimension, but not quite, possibly because the "on top of" phrase didn't enter in soon enough to send her out of the plane until the green rods were all arranged.

Frankly, this is a rather serendipitous outcome, in my opinion. Reminds me of a case where I wanted teachers to think about "inverting" coins (changing heads to tails or tails to heads) but kept slipping into using the word "flipping" (as in "flip them over"), unintentionally reinforcing the notion that some had that the problem could only be solved probabilistically, when in fact I kept specifying, much to the consternation of those teachers, that probability had nothing to do with it. Go know.

6. Hey Mike,

Thanks for your thoughts; I think I'm getting closer to understanding this whole thing. Now that I think of it, this is probably not the first time this has happened, although this is by far the most spectacular example of it, lol!

Sooo...how would we proceed in future games? If I want it to keep it in the plane, I'm thinking something along the lines of:

"Take a light green rod and lay it *flat* on the board, perpendicular to the bottom edge of the board..."??

And if I wanted to intentionally build in 3D I would say:

"Take a light green rod and place it on its end, perpendicular to the board space?"

What I think you're also saying is that 'up' is relative the space (2D vs. 3D)?

7. I would suggest that you and your daughter need a common vocabulary. How mathy you want that to get is up to the two of you, of course. Not sure if you want to start talking about x, y, and z axes (let alone other sorts of coordinates that aren't rectangular like cylindrical or spherical!) I think the fun here for the two of you is to see what comes up to begin with, then to test the system by trying to break it (that is, to use the rules and vocabulary you have and see if it's possible to get the follower to legitimately come up with something different from the leader. If that happens, you know there's a weakness that needs fixing).

In a way, you have a chance to do something like what Lakatos describes in his brilliant PROOFS AND REFUTATIONS. In that book, he uses an analogy to something that really happened in mathematics. The real thing was the struggle to make the definition of what a function is to be truly rigorous (a lot of mathematics developed before the 19th century focus on rigor was pretty seat-of-the-pants, including the development of calculus, which Newton and Leibniz invented independently and without what modern mathematicians would consider to be real rigor. Nonetheless, it worked for the purposes for which it was designed). In the struggle to get functions clearly and rigorously defined and grounded, mathematicians would produce examples that caused problems. Some folks wanted to dismiss them, and there was a dialectical process that forced everyone involved to grapple with their understanding of function on a deeper level.

In Lakatos' book, he uses an easier-to-grasp example about what counts as a polyhedron in his exploration of Euler's formula that links edges, vertices, and faces of polyhedra. Turns out that there are figures that cause problems for that relationship if you accept them as polyhedra, and Lakatos has various characters in his book (which he structures as arguments among students (and a professor) in a very advanced math class, where different students take viewpoints that parallel actual positions the mathematicians who argued about functions and various "monsters" that some embraced and some wished to banish.)

Not suggesting you and your daughter want to wade through Lakatos quite yet. But you don't have to. You get to develop your own mathematical language and then, if motivated, check to see how the pros do it. :^) Hope that's helpful, not confusing. It's a little late and I'm fading. ;^)

8. thanks mike -- this has been *really* helpful on numerous levels. I'll let you know how it goes! ;-)

9. Update: I tried to set up a system of a simple coordinate grid using tape on our playing surface. The kid balked, saying, essentially, that we wouldn't have anymore good laughs over the results if we got too formal about our game plans/rules! Ah well, the best laid plans...

10. Malke, I finally tried this game with my son. I was actually very surprised at how it went. First of all, he loved the challenge (usually he is not all that excited about working with small Lego pieces). Second, he was very good at following the directions. Third, it was very tough to give directions and stay consistent with the terminology throughout the build. Finally, once we finished building and compared our structures, he figured out that since they were symmetrical, he could combine them into a super-structure for his Star Wars pretend play. We are definitely going to be playing this game again and again. Thank you!

11. Yelena -- so you used lego pieces instead of cuisennaire rods? Interesting! That sounds a lot harder to me, but I'm apparently 3D challenged, lol! It also sounds like using legos it would be easier to eventually determine a shared, working terminology for distinguishing between a 2D word and a 3D word. Thanks for sharing -- can you believe we have no legos in our house, anywhere? I think I'll have to go get some. :)

12. I like this game! (Another thing I'm going to "steal" from you!) And where there's "misunderstandings" - that's going to be something interesting to talk about, even if there's no answers!