In 2012 I was fortunate enough to listen to a talk by Bob Bosch, Oberlin College, at the MAA Pacific Northwest Section Meeting, held at the University of Portland. The conference was superb, and I learned much from my experience there. Here's a LINK to Bosch's art.
One of the ideas I came home with was to develop a unit based on Bob Bosch's Domino Art work. Fast forward to January 2013 around MLK day, I was working with Pacheco Elementary, San Luis Coastal USD, and the 6th grade teachers let me throw a math + art unit into their curriculum for about 4 days. Thank you Mr. Deutsch and Mrs. Irion, and to Jamie Coxon at the Linker Workshop for preparing the materials and framing it for final display!!
The setup is that the 12 complete sets of double-nine dominos are used to make a mosaic of MLK. (Bob Bosch has other images besides MLK.) The mosaic is created from a B&W image, where the image is turned into a gray scale image of squares with 10 grey-scale levels from black to white. Double nine dominoes have approximately these same 10 steps from black to white. Thus an image can be made into an array of numbers, where each "cell" has a number from 0 to 9. That's the mathematization of the image into a mosaic.
The big question is "how do you arrange the dominos to approximate the perfect mosaic?" This is a challenge, because one of the rules of the game is to use all of the pieces in the 12 sets of dominos. No replacements are allowed. Generally, the perfect mosaic is not attainable, so we are left to find the best approximation of it.
This means we have to accept some error. One way to try and find the best possible approximate mosaic, is to find a way to measure error and then to minimize that error. That's where modern math comes in, because the number of possible combinations is astronomical. Bob Bosch sorted that, and I'll send you to his site (and papers) to get the nitty gritty details.
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| The MLK Domino Mosaic the 6th Graders Built |
Sixth Grade isn't the best context for a short course in Linear Analysis ;) So my challenge was to find smaller problems that are doable by 6th graders. The math unit is in a beta stage, and needs development. (More work to do!) What I have so far is enough to get across the idea that minimizing global error requires increasing local areas in some parts. Sixth grade students worked on measuring error, and on a task that required groups to cooperate to minimize global error, while accepting greater error in their section of the image. Even in math, we are better when we cooperate!
Finally there was the phase of cutting, gluing, and putting together the mosaic. While this activity isn't math or art per se, it's a nice activity that culminates in a tangible finished product. Sometimes it's important to do something to mark your achievement and appreciate something aesthetically pleasing. (And it only took one or two periods to assemble.) The tiles were connected and framed up by a local craftsman. I'll also note that there are extensions from this unit to other subjects (and more math). This unit could extend to MLK's biography, History, writing, Art (color and gray-scales), etc. I can see quite a long list of possibilities.
One of the best learning outcomes was something I heard from one of the Moms. She said that her son was so motivated by the project that he was really excited to go to school and do math that week! Moreover her son wanted her to take him to the public library so that he could learn more about Martin Luther King, Jr.
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| Working Together in Groups |
To me, inquiry-based learning or more generally teaching isn't fundamentally about teaching techniques or skills. These are important and necessary of course, and I'm not trying to minimize them. I focus and work on them daily! But teaching techniques, skills, and practices serve a larger vision for education, where students are deeply engaged as explorers and doers.
Real, meaningful education lies at the foundation of modern civilizations, and Martin Luther King, Jr. realized this. Here's one of his more famous quotes.
"The function of education is to teach one to think intensively and to think critically. Intelligence plus character - that is the goal of true education."
