I have been surfing around in the polymer clay folks flikr pages, and I came across an old blog, called "Dora's Explorations." Dora made a terrific shaded polymer cane showing the Bhaskara's Behold! proof of Pythagoras' Theorem. (She also made a tutorial; the link is below the image.)

Making the clay blend so that the shading in this cane went the way she wanted it to lead her to experience mathematics in the very way that mathematicians experience it in the course of research. She wrote:

Then we give up on our wistful thoughts of greater insight, and chop up paper triangles until we can make some sense of it. Sometimes it takes a long time. And sometimes we get headaches. And then sometimes, if we are very lucky, we figure it out, and behold! There is that elation, woo hoo! I figured it out! Followed by that sinking suspicion that someone somewhere could have figured it out instantly. (Maybe the German mathematician Johann Carl Friedrich Gauss already did it in the 1800's?)

We feel that the part of math we understand is easy, and as we add to that body of understanding, we transition how we think about the newly understood ideas from hard to easy. As an idea makes that transition, we feel frustrated, as Dona suggests, because it now seems like the idea should always have been easy. Hindsight in mathematics is 20/20 with a pair of binoculars and a magnifying glass at the ready.

I think what distinguishes the mathematicians is that we revel in the necessity to cut out paper triangles, and are more willing to tolerate the feelings of inadequacy during exploration in exchange for the elation brought about by the (very) occasional Eureka! You bead-y folks who have 16 preliminary versions of a piece of beadwork, each representing several hours of sew-it-rip-it labor, which you made in search of the perfectly engineered and aesthetic design, have much in common with the mathematicians, with paper triangles littering our tables and floors.

P.S. Thanks, Dora, for your insights.

http://dorasexplorations.wordpress.com/2009/07/07/cane-of-the-week-7509-behold-part-ii/ |

"I had to make a Skinner block that would shade diagonally from the corner. Another 2 or 3 frustrating hours were spent in trying to figure out how to do this. If I had better mathematical or mechanical skills, I would have been able to mentally ’track back’ from the finished shaded triangles to the initial Skinner blend. But instead I had to cut out little shaded paper triangles and attempt to recreate the steps. Anyway, I did finally come up with a solution (and a headache!), although I am sure there is a better, easier way to do it. No doubt there are many people out there who could figure out in a few minutes what took me several hours."That is a wonderful description of what mathematics feels like to the mathematician and non-mathematician alike. Every time we work on a problem, we feel like if we just had one more inkling of an idea, or a tiny bit more experience, or if we only remembered how those ideas ever came together last time we worked with these tools, if only... then the problem would be easily solved.

Then we give up on our wistful thoughts of greater insight, and chop up paper triangles until we can make some sense of it. Sometimes it takes a long time. And sometimes we get headaches. And then sometimes, if we are very lucky, we figure it out, and behold! There is that elation, woo hoo! I figured it out! Followed by that sinking suspicion that someone somewhere could have figured it out instantly. (Maybe the German mathematician Johann Carl Friedrich Gauss already did it in the 1800's?)

We feel that the part of math we understand is easy, and as we add to that body of understanding, we transition how we think about the newly understood ideas from hard to easy. As an idea makes that transition, we feel frustrated, as Dona suggests, because it now seems like the idea should always have been easy. Hindsight in mathematics is 20/20 with a pair of binoculars and a magnifying glass at the ready.

I think what distinguishes the mathematicians is that we revel in the necessity to cut out paper triangles, and are more willing to tolerate the feelings of inadequacy during exploration in exchange for the elation brought about by the (very) occasional Eureka! You bead-y folks who have 16 preliminary versions of a piece of beadwork, each representing several hours of sew-it-rip-it labor, which you made in search of the perfectly engineered and aesthetic design, have much in common with the mathematicians, with paper triangles littering our tables and floors.

P.S. Thanks, Dora, for your insights.

Thanks for mentioning my blog,Florence, what a wonderful description of the 'mathematical process' !!! Many years ago, when I was in high school, math was taught as an isolated skill unrelated to the real world. I couldn't imagine how any of that 'stuff' would be of any use to anyone other than a mathematician or an engineer. With age came the realization that math is all around us and we use it daily whether or not we are aware of it. I can only hope that today's students are being taught to use math as a tool to solve life's problems and make sense of the world.

ReplyDeleteThat is a really good description of research, especially the transition from hard/impossible/arcane to obvious. I think it also implies strongly that in order to think about a subject creatively, be it beads, math, quilts or oceanography, you have to embed yourself in the material so that you can see the possible connections, relationships. The cutouts, trial projects, repeated model runs, whatever free your entire brain to work on the subject. Those who can see the answer in a few minutes have usually already done the deep thinking and tinkering in another context.

ReplyDeleteCan't say it better than Dora did. Wonderful article. Those 'feelings of inadequacy' plague us all as "analytical crafters" (so to speak), but it's nice to know we're not alone :)

ReplyDeleteWow, what a great cane! This will make a great homeschooling project! Thank you! It's also nice to know that I'm not alone in the analytical process. : )

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