Henri and Kim Seashore (who's a math ed PhD student at Berkeley and would be awesome regardless of her current/future credentials) organized a conference last February at The Urban School in San Francisco where Henri teaches. I'm not sure if there will be a repeat this year as Henri and Kim now know how much work it is to organize a conference. Fortunately, a group of interested and available teachers (and pre-service teachers) have also been meeting a few times a year with the broad goal of talking and sharing things that typically can't be found in textbooks (or are seriously lacking).

We met this past Saturday morning. After an exiting dive into the uses of tangrams and Miri Math Geometry Tools fueled by the open ended question "how can these tools be used", we spent some time continuing the conversation from last year of how mathematical habits of mind can be explicitly taught in the classroom.

Tangrams: Picture taken from www.bcps.org |

Ideally you wouldn't even need the last sentence of the problem and students would go there on their own. Anyway, it's a really nice problem and I recommend spending some time with it if it's unfamiliar.

On Saturday, I had people work on this problem in groups, but asked people to reflect on places where students might get stuck and habits of mind that could be helpful in order to move forward. In the end we highlighted different habits of mind that people used themselves. Here's a link to what we compiled as a group. If you really want to get down and dirty, you can compare this to the list I made myself of potentially useful habits before the meeting.

Some big picture takeaways:

- The ability to
**collect lots of data**helps make this problem accessible. - The ability to collect lots of data in many different formats can lead to a rich discussion of
**different ways to organize this data**and how these different ways illuminate different patterns. I think too often this is where we scaffold (giving students the format to collect data), with the potential downside of eliminating this important mathematical thinking and the possibility for different representations that can be helpful in different ways - This problem is a naturally
**good group task**as cases could be divided among group members and there are a number of different solution paths that can be shared. - Many people had insights about the relationship between related rectangles such as 2x3, 4x6, etc,
**an intermediate place in the problem where students can feel success**. - It's important to develop the
**habit of listening to and understanding**other people's insights and methods, especially when they are different from your own. - Lots of opportunities to
**predict, test your prediction, create conjectures, and prove these conjectures** - There are good
**opportunity for variations and extensions**(one group that was familiar with the problem worked on the 3D equivalent). Someone else brought up the question of using other shapes or other paths. Students should never "be done." **Tying good problems to specific content is really hard**and society still values content more than habits of mind

"other shapes or other paths" ... that reminds me of the related problem of drawing your line at a 45 degree angle, bouncing off the sides repeatedly until you end in a corner. I wrote a short piece about it in the first Math Teachers' Circle newsletter: http://mathteacherscircle.org/newsletter/MTCircularSummer2011.pdf

ReplyDeleteWe thought about "other shapes" initially, but in the end, the problem is really not about the shape: it's about connecting two points. That's why the 3D version was a more interesting generalization.

ReplyDeleteBoth this problem and the one Joshua brings up start from a visual premise, and quickly take you into thinking about prime factorization. I first encountered the latter problem in Harold Jacobs' _Math A Human Endeavor_.

On the last point in your post: it's not just that *society* values content more. It is in fact not possible to solve problems effectively without a well-stocked toolbox of content.

The counterposing of those two sides of the coin is a classical mistake of both reformers and traditionalists. The art of teaching math is largely about navigating between those complementary poles.

My comments about the same meeting cover different ground, and can be found at blog.MathEducationPage.org.

Thanks for the kind words about my site!

--Henri

What's the point of the math mirror? I do any of my constructions now with a ruler and compass, rather than gridded paper, and while I don't always get measurable results, they're usually beautiful.

ReplyDeleteI don't understand what the mirror in the picture is supposed to do....

@Andrew: The mirror reflects anything you've drawn across the line formed by the mirror itself. Mathematically, the interesting question for me is, given specific tools, what can be built and what can't be built? For example, with a compass/ruler it is impossible to trisect an angle. Another common set of "tools" is the ability to fold paper. We were exploring what can and can't be made with our "tool" being this mirror.

ReplyDelete@Avery,

ReplyDeleteI've got a pretty good understanding of both why a ruler and compass can't trisect an angle, and how to approximate that.

Have you seen Whole Movement.com? I've seen some of the constructs that you can do with flimsy paper plates using this guy's directions... I have to say, it still wows me that I can make a tetrahedron and icosahedrons from paper plates...