What's The Difference?!?

02/11/2012

This one has been bugging me and I really wish I had more readers because I think there is a valuable discussion here:

What is the perimeter?

How long is the coastline?

The perplexing question here is essentially the same. This guy talks a lot about the quickness with which we move to abstraction. I'm wondering if maybe that applies here. I like the abstract because, in terms of creating a unit with a nice resolution, it easily comes full circle. I'm not sure you can say the same about the coastline. So…"what's the difference?!?"

Comments

Dan Meyer link

02/12/2012 8:08pm

Yeah, provocative pairing here, Bryan. Let me suggest that both of these are images are "concrete," at least in the sense I intend it. The abstraction happens as problem solvers decide "What information, tools, and resources do I need to answer my question?" So for the Koch triangle, they'll need to know the length of the original segment and how many iterations it underwent. For the coastline, they'll need to know the scale of that map. They'll need to know that the city names and the colors are irrelevant.

<a href="http://mrmeyer.com/blog/wp-content/uploads/concreteabstract.png">Here is the abstract form of both of the concrete problems you've posed</a>: http://bit.ly/wraYlr

The thing I yell about on my blog a lot is that students don't get much of a chance to participate in the abstraction of a problem. It's generally done for them, which is a shame, both because it's hard, important work and because the abstract form of these problems has a terrifying effect on a lot of learners.

Bryan Meyer link

02/13/2012 12:38pm

I think the thing that interests me most about this is engagement. Essentially, the perplexity here comes from the same place: the infinite fractal nature of each image creates a paradox about finite measurement. I wonder if students will respond more positively to one versus the other even though the questions are, in essence, the exact same? I think there is a lot more to discuss here but I'm sure you get the general idea.

Breedeen link

02/12/2012 8:18pm

A few things come to mind re: "what's the difference?" here. The main issue that springs to mind is the background knowledge needed to approach both pictures. Is the idea to show this to students who already know about Koch's snowflake? I think most students would recognize the coastline of CA, not so many the snowflake. You can make some good comparisons between measurable/infinite, how do you measure something accurately, levels of precision that are necessary to do ___________. Lots of cool stuff.

blaw0013

02/14/2012 7:57pm

Thought of this post when seeing this TED Talk http://bit.ly/yP1pXN

Mr K link

08/07/2012 8:34am

> "what's the difference"

That seems like a great basis for a lesson, in itself. Well, and the complimentary "what are the similarities".

I'm not sure I would agree with Dan that each image is concrete - they are both already abstractions. There may be some enlightenment in looking at what they are abstractions of - moving *down* the ladder, and seeing what the differences are in how you have to do that.

Angela

08/07/2012 8:43am

I teach an online HS Geometry course. My students start the fractals unit by doing this activity with this applet : http://polymer.bu.edu/java/java/coastline/ . They do this before they learn about finding the perimeter of the Koch snowflake. This post has me thinking about it again...