## A Problem in Self-Similar Tilings

April 14, 2009 by Michael

This [pdf] is the note on that remarkable series of lessons I mentioned – it has pictures, which is why it’s not posted directly here. The link takes you to the draft currently under review for publication in *The Mathematics Teacher*.

**Update (3/16/10)**: *The Mathematics Teacher* inexplicably did not accept the note for publication. In my view, this is just evidence of how revolutionary and ahead of its time the piece is, so it should just make you want to read it more.

Here are the first two of its three concluding paragraphs, not so much as a spoiler as for a sense of the article’s substance and approach (a sort of abstract/excerpt hybrid):

“…the students had the exhilarating and empowering experience of really doing mathematics – of stepping outside the realm of formulaic exercises with pre-ordained answers, to the dark rooms Andrew Wiles described in recounting his work on Fermat’s Last Theorem, where one has no idea where the furniture is, let alone the light switch, until one feels his way around.

I believe it is extremely valuable for students to see the role that creativity plays in mathematics; pursuing open problems allows the special brand of creativity that is the mathematician’s trade to step into the classroom, something that happens less often than it might. But there is also a more pedestrian benefit to these open problems, one perhaps even more important: seeing mathematics’ influence on creativity. Yes, students are asked to be creative in most of their classes; but the rigor and discipline imposed in a well-defined open problem demands intense creativity within a precisely-defined structure. The skills required to formulate creative solutions in a world ruled by highly precise definitions are invaluable in science, history, and the ability to write cogent and incisive arguments. Exposing students to open problems makes them better not just at mathematics, but all subjects with an analytical component.”

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on April 14, 2009 at 11:43 pm |zonean inspiring story…although i thought you said these were going to be short essays?…

i am unclear of the purpose of your article. is your hope that other six grade math teachers will incorporate this exercise into their lesson plans? and im assuming you actually knew the solutions that your students “discovered.” were you worried that they ultimately wouldn’t figure out the “square test” and LCM formula? if they hadn’t how would you have let them in on the fun? gentle hints, further prodding/encouragement?

certainly a good way to start a blog!

on April 15, 2009 at 12:09 am |MichaelThanks, I’m glad you thought so! (and yes, I’m pretty sure that will be the longest post on the blog)

The purpose of the article is to tell a neat story about (1) a cool, and substantively rich, series of lessons, and (2) a

wayof teaching some lessons — with open problems.I wasn’t worried that my students wouldn’t figure out the square test or the geometric representation of LCM, because I wasn’t aware of them myself (!). I didn’t have any specific answers in mind for the question I posed — I just thought it was interesting. Attacking (mathematical) questions to which you don’t yet know the answer is what it means to do mathematics, and I thought it would be good to both share that experience with my students, and model it for them.

Now, it’s simple to teach with the open problem method even if the teacher knows the answers, and that needs to be done if one is to be sure of teaching some specific topic. But the goal of the project was to have an authentic experience of actually doing mathematics — I wanted the problem to be open for me too, so I could be an authentic model.

So in sum, the project was as much about experiencing what it means to do mathematics as it was to teach the underlying content. And the goal of the article is to communicate how that experience can work, and how valuable it can be.