## Thursday, October 17, 2013

### Allegory of the (Math) Cave

Is this math?

No. It's a bike rack that, with the sun's help, created a shadow of the symbol for phi.

Is the symbol itself math?

No. Plato says a shadow is not reality.  Here's a description of his cave allegory (bolding and brackets mine):
"Plato has Socrates describe a gathering of people who have lived chained to the wall of a cave all of their lives, facing a blank wall. The people watch shadows projected on the wall [symbols] by things passing in front of a fire behind them, and begin to ascribe names to these shadows. According to Plato's Socrates, the shadows are as close as the prisoners get to viewing reality. He then explains how the philosopher [math teacher?] is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall do not make up reality at all, as he can perceive the true form of reality rather than the mere shadows seen by the prisoners." [Wikipedia]
A recent post over at Christopher Danielson's blog Overthinking My Teaching got me thinking about this image of the shadows at the back of Plato's cave, especially this comment by Sue VanHattum:
"Have I reached Malke’s beautiful standard of embodied math? Nah. But it’s in my head as a goal, whenever I can make sense of it.
"Malke, here’s one body thing I do that probably doesn't count. The graph of y=x^2 is a parabola. I like to think of it as having both arms up. I like to think of the graph of y=x^3 as having one arm (left) down and the other (right) up. We are working with graphing more complicated polynomial functions, and I ask them to show me (what I call) the big picture with their arms. I’m hoping to get them to hold this visual information more firmly in place while they work through the detailed parts, so that their graph will reflect their thinking on both the big picture and the details. I’m using body movement just as a cuing device, really."
"Sue, I think making sense of what math is in the body is a lengthy ongoing proposition for most of us, even the dancers.  My goal is to at least raise some questions: Is the graph the math? Are the arms doing the graph the math? My answers: The graph is the representation of the idea and the arms are the representation of the graph.  That is why my blog is called 'the map is not the territory'.  Those teachers Christopher mentioned are part of the legions of people who, by no fault of their own, mistake the textbook (map) for the math (actual experience of thinking and doing math) because they've never had a chance to travel the actual terrain of math land.  Personally I think there's no problem with a 'cuing device' or mnemonic as long as you are aware of where it lies in the continuum of math learning and your teaching goals."
I repeat: "...the philosopher is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall do not make up reality at all..."

Most of us (myself include) have learned shadow math in the caves they call school.  Here's a question from Christopher that might get us out of the cave:
"Math comes from, and lives within, textbooks. I am not OK with this.

"So what can we do in every lesson every day to represent mathematics as a subject that comes from, and lives within, the minds (and bodies) of our students?"
These are the things I am thinking as I try to make a worthwhile case toward using the body in math learning for something other than re-drawing the 2D representations in our textbooks.

1. Sue's arm functions remind me of my nutrition prof at Houghton who demonstrated the tetrahedron shape of some carbon based molecules by lunging her legs and raising her arms. The 3d effect became clear at once, also the way tetrahedrons can twist and turn. Usually we think of them as pyramids because that's the way the blocks sit on a desk.

The solids have become more interesting to me lately because I've been dabbling with embroidering temari balls - when I read Barbara Suess's books on design, she refers to the solids all the time.

2. That's great Christine! I am in no way critical of using the body as an illustration but I feel compelled these days to keep asking: Would it have been similarly or more effective to give you toothpicks and balls of clay so you could have constructed that structure yourself? Or make a tetrahedron and hang it from some string at various points and let it spin so you could observe the structure? My quest right now is centered around the question: What is it that the body has that these other math manipulatives don't? What are the possibilities for using our bodies in math learning to see or understand something completely different than a version of the standard math manipulatives, computer modelling, etc?

I think your temari ball experience is an example of how using the body to make with and explore math ideas creates a completely different learning result and insights than simply turning someone else's work in your hand and evaluating the result. I think this is the most important distinction -- the body can be used in literally *making* the meaning, connecting the mind to the senses. Hmmm...I so appreciate your comments because now I feel like I'm on to something!! THANKS!!!

3. You are SO on to something. Though not specifically about movement, this reminds me very much of some reading I've done lately about the differences between visual and symbolic ways of thinking, specifically the scientist Michael Faraday.

Faraday was a notoriously poor mathematician and yet one could argue that his influence continues to be equal to Einstein. It's not that he wasn't trained or avoided it, even simple calculations were a real challenge to him. It seems his mind was just not wired to turn mathematical concepts into symbols. It look a great mathematical physicist James Maxwell to turn Faraday's work into formulas. Maxwell said Faraday was " in reality a mathematician of a very high order – one from whom the mathematicians of the future may derive valuable and fertile methods."

To continue the metaphor: Faraday was "stuck" in reality while all the mathematicians of his time were still in the cave. It took Maxwell's extraordinary mind that could move freely between cave and reality to bring Faraday's work back to the mathematicians of the cave.

4. Love the photo by the way! Makes me think of descriptive geometry.

5. Hi Malke...

Thanks for a thought-provoking blog post! You have me thinking and I almost didn't write a comment here because I fear I won't be able to capture my ideas in writing, but I wanted to try to leave some thoughts.

I have always noticed the title of your blog...is it by any chance a reference to the Borges short story "On Exactitude in Science"? If not, you should check it out. I think you'll like it.
http://www.sccs.swarthmore.edu/users/08/bblonder/phys120/docs/borges.pdf
and
http://en.wikipedia.org/wiki/On_Exactitude_in_Science

I appreciated the way your post here reminds us that meaning does not reside IN symbols. The only thing that is nagging at me is the use of Plato's allegory as a metaphor for mathematics. I think of mathematical Platonism as identifying a certain transcendent mathematics (the discovery "beyond" the shadows of the cave)....that there is AN idea to be represented. Rather, I try to force myself to realize that I can never know the ways of thinking of an other and that it is likely that we all "know" mathematics is unique and individual ways (maybe check out the "Simulacra and Simulation" link at the bottom of the Wikipedia tab). If not forced to think a certain way in our own mathematics education, perhaps the development of even more diverse and unique mathematical ways of knowing might develop?

I also just recently read this article, which made me think of you and your post here. Some very interesting ideas about the dynamic nature of motion and embodied cognition versus the static nature of mathematical formalism.
http://www.cogsci.ucsd.edu/~nunez/web/Nunez_FFF.pdf

Thanks again for a thought-provoking post!

-Bryan

6. Hi Bryan!

The title of my blog goes all the way back to college where I was highly influenced by the writings of Gregory Bateson who was a contemporary of Alfred Korzybski who coined the phrase. Both were scientists and philosophers -- having first studied the real and actual in the world, I can only imagine how meaningful 'the map is not the territory' was for them. I've reinstated my "What's this about a map?" page -- up at the top right of this blog. Even before my personal mathematics renaissance I had the hunch that we *can't* know unless we *do* which relates directly to my math/dance work.

I appreciate your comments about how I used Plato's cave allegory in this post and I knew from the outset I might be missing something(s) on a bunch of levels, but I have been thinking about symbols not being the math for a long time. When I saw the shadow of the bike when I was in Cambridge this summer I knew I had a piece of my argument (I've always been a bit critical of people posting pictures of mathematical symbols under #foundmath). When a memory of the cave allegory popped into my mind, I realized I had finally found a use for the picture and a way to frame my thoughts.

I am, as you can see, trying to make a case for the textbook/symbols not being confused with the real mathematics. I do this b/c pretty much everyone (*everyone*) I meet in my work (public school teachers and students, mostly) thinks that what is on the page *is* the math. This is why the CC 'mathematical practices' section is such a puzzle to many, specifically, if math is about right answers, what's there to think about? I do not fault them because I learned math this way as a child -- and, now that I know a little differently, maybe that's why I feel so strongly about challenging this misconception. And, if you have read the last week or so of posts on this blog you'll see that I am really on a push to clarify my understanding of how the body is a partner in math learning, not some distraction from the 'real work' -- especially for elementary students, but more and more I am finding research that shows adult learners need it too.

I appreciate all the links as well. I don't know how far I can dig into mathematical Platonism, but I'm glad to know it's there. The Simulacra and Simulation stuff looks fascinating and relevant for even this present day of virtual realities, especially the fourth stage. I need to take a closer look very soon. And I have been aware of the work of Rafael Nunez but I would need another lifetime to fully understand it. ;-) The article you found looks like it was written for a more general audience (meaning me!) so thanks!

And thanks so much for deciding to share your thoughts -- they've been extremely helpful to me.