## Tuesday, August 27, 2013

### "I Have a Dream": MLK Day, Math, Art, Inquiry

Here we are on the 50th anniversary of the "I have a dream speech."  I thought I'd share something related from the IBL Math world.

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.

 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.

 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."

## Friday, August 23, 2013

### The Sense-Making Continental Divide

Frequently I am involved in good discussions about what is IBL, whether it is the strict Moore Method or something else.   This is a good topic for discussion, and I'd like to share my thoughts on the issue.   I'm going to approach this topic with a simple, but useful model that highlights a major structural component of IBL instruction.

Math courses are taught in a variety of ways, and even within the IBL community there exists numerous differences.  This is something that should be expected, because environments and goals differ across institutions.  Just as we would not expect Michelin star restaurants to be identical across the world, we should expect that successful instruction will be different and varied to suit the needs across different institutions.  This honors the diversity of humanity, and allows a teacher to be true to her or his personality

Here's the model I have in mind:

In this model I use an idea I call the "Sense-Making Continental Divide."  A key feature that defines IBL instruction is that students regularly are encouraged to do the sense-making tasks, including validation of solutions or proofs, understanding statements of problems, and working from definitions and first principles.  IBL instructors set up courses to get students over the Sense-Making Continental Divide, where students are regularly doing activities that require students to think, decide, explain, evaluate, and reflect.  You have crossed over the SMCD if your students are (a) deeply engaged in rich mathematics and (b) have opportunities to collaborate and discuss ideas and solutions.  (This succinct characterization of IBL is from Sandra Laursen, University of Colorado Boulder.)

It's easy to see there are numerous options for implementing sense-making activities.  From the palette of teaching options is derived the multiple variations of IBL.  In my perspective, this explains why IBL comes in so many different forms.  We have more choices, and the "correct" teaching decision depends on real-time conditions in the specific class setting an instructor is in.

What typifies traditional instruction is that the instructor does the processing and sense-making through presentations.  "This is the proof of theorem 3.6..."  Students do not get many opportunities in class to do the structuring or validation.  In such classes, students might be unintentionally encouraged to memorize facts rather than make sense of the ideas.

The Hybrid IBL zone contains the different forms of IBL methods that are often a result of practical limitations instructors face.  There may be a required syllabus, or a course may be predominantly procedural in nature (e.g. calculus). A course may have large enrollment, or an instructor may not have the requisite skills or experience to comfortably run a full IBL course.

It could also be the case that the (full) IBL class instructor shares a solution on occasion in the event that students are floundering, and moving ahead would be more beneficial mathematically for the students.  Flexibility and adaptability are key traits of effective instruction.  An IBL course may change during the term to adapt to specific needs.

Is your course an IBL course of some kind?  One way to see is if your students are regularly over the Sense-Making continental Divide.