cheesemonkey wonders

cheesemonkey wonders

Sunday, March 25, 2012

On Meeting Sam, or How Twitter and Blogging Refreshes My Teaching Practice... and My Life

So Sam Shah was in town for a visit, and a bunch of us got together Friday night for drinks and dinner in his honor.

Before anything, I should tell you that everything magical you've heard about Sam is true, including that elusive quality that Jason of Always Formative captured so well when he said that "hugging him is like being wrapped up in a freshly laundered rainbow."

If you teach, then you also understand that there is something both improbable and heroic about going out on a Friday night. Especially for a late dinner.

But these are my math teacher-blogger tweeps, the people who restore my faith in the power of teaching and learning, and nothing recharges my teaching batteries like connecting with them IRL (In Real Life). So I'm glad I got my butt off the couch and met up with them at Bar Tartine for an evening of conversation, laughter, and understanding.

There's something precious about having a circle of teacher-blogger tweeps that is hard to explain to teachers who don't use Twitter or blogs. Don't get me wrong, my colleagues at school are amazing and I love teaching with them. But my math teacher-blogger tweeps are the ones who really "get" me. They inspire me. They know me at a surprisingly deep level. They are the colleagues who are trying to improve as teachers in the same ways I am trying to improve as a teacher. They are the ones who respond to my Twitter distress calls with lesson ideas and foldables and encouragement and energy of their own. They share my love of office supplies and unicorns and my outrage at stupid copier breakdowns the suggestion that Khan Academy videos are the solution to all of American education's problems.

Sam, of course, is the grand wizard of intellectual generosity in the Twittersphere, the keeper of the Virtual Filing Cabinet and well as the frequently hilarious "Favorite Tweets," which is why I follow him around like a duckling.

And that is why it was such a gift for all of us to get to meet him IRL.

Sunday, March 18, 2012

The Big Ideas inside the 'big ideas'

As I've been organizing my job search materials, I've been reflecting on some of the Big Ideas I have learned are the most important among the too-many "big ideas" our textbook and state standards emphasize for Algebra 1.

One of the problems with the state standards is that they can't let go of anything as being less important than anything else. Which is why there are 26 overall standards, plus embedded sub-standards inside the standards, and the whole thing is a nasty ball of yarn to try and untangle.

One thing I've tried this year, which seems to be working well, is to choose my emphases based on what developmental psychologists have discovered about children's mathematical development. I am particularly grateful for the work of the British psychologists Terezinha Nunes and Peter Bryant, who also do a lot of work with Anne Watson of Oxford's math education program. Nunes and Bryant's book, Children Doing Mathematics, has really blown my mind open to what they call the "generative" quality of children's mathematical development and number sense — that is to say, kids develop their sense of number and of mathematics in layers, the way some inkjet printers work, with each pass of the printhead setting down another layer that completely transforms the image that is emerging on the paper.

In Nunes and Bryant's synthesis, as well as in their summary of others' research, kids' understanding of quantity is revealed to be an extremely fluid, dynamic, and multi-faceted set of tools. As they put it, "a successfully developed understanding of number comes from four distinct developmental threads" which they summarize as:
  1. the ability to COUNT discrete OBJECTS
  2. a deep familiarity with a wide range of QUANTITIES OF QUALITATIVELY DIFFERENT KINDS (such as both countable and uncountable quantities)
  3. the ability to COMPARE QUANTITIES OR COLLECTIONS of objects, assessing both similarities and differences regardless of their qualitative kind(s)
  4. the ability to use established notation for all of these (Nunes and Bryant, pp. 1-20)
Yet, as they have observed, our current curriculum tends to address only #1 and #4.

This helped me understand something I had been struggling with for a long time — namely, the fact that there are many incomplete understandings children develop that are sufficient for them in their context, BUT that are insufficient over the long term as a foundation for mature mathematical understanding. These are the mathematical versions of ideas like "the tooth fairy" or "Santa Claus." They are enabling fictions that are developmentally appropriate in their time and place, though they are not at all what we want our young adults to rely on by the time we release them out into the grown-up world of mathematics.

One of these incomplete understandings — one that drives university-level mathematicians like Keith Devlin head-banging mad — is MIRA, or the idea that Multiplication Is Repeated Addition. Their argument is that this is such a stunted understanding of multiplicative reasoning that it threatens to undermine the very foundations of civilization, dammit.

But in truth, like the idea of the tooth fairy or Santa Claus, MIRA does have a legitimate place in a child's generative mathematical development — as long as his or her teachers understand that, like the idea of the tooth fairy, it is an incomplete understanding that is meant to be expanded upon into a much richer and more scalar concept of multiplication.

The place where I am finding MIRA to be an extremely useful tool is with Algebra 1 students making their first forays into the abstractions of algebraic reasoning, which is to say, in dealing with polynomial arithmetic. I say this because young adolescents are such intensely concrete thinkers. When I ask them to consider combining like terms such as 5 elephants and 3 elephants, they can easily understand what I am asking for. But the moment we start investigating the idea of combining 5x^2 and 3x^2, their heads explode. Things only get worse when they are asked to combine 5x^2, 3x^2, and  6x^3. It seems like they forget everything they have ever known about the combining of like terms, and they start adding or multiplying exponents and or worse things than most of you can imagine.

This is the place where some teachers find algebra tiles to be helpful. But I find that algebra tiles have a grammar and a rhetoric of their own that is not easily extensible into polynomial arithmetic beyond quadratic thinking. Also with their color-coding, they also add in moving parts I find my students are not yet ready to think about.

But guide their attention away from abstraction for a moment, and ask them what happens if they combine 5 dogs and 6 apples. They understand the logic of ConcreteLand completely. In this case, I have found, getting them to think about 5x^2 the way they think about 5 dogs and about 6x^3 the way they think about 6 apples, and their conceptual understanding shoots through the roof. I can even use Brahmagupta's idea of "fortunes, debts, and ciphers" (positive numbers, negative numbers, and zero) to help students think about what happens in a trading economy where I might "owe them" 3 dogs (or 3 x^3) as we barter our algebraic quantities away for each other's lunch components.

Students still need a lot of practice and experience with this whole crazy abstract insanity to cement their understanding in place, and they can be expected to relapse several times into believing that they need to add exponents instead of thinking about coefficients as quantifiers. But eventually the idea of quantifying (and if need be, combining) x^2s the way they count and quantify dogs or apples gives them a surer footing as they begin to construct a new and deeper understanding of multiplying variables. And that is something I can eventually build a rich and scalar concept of multiplication on top of — on that would be appropriate to eventually deliver to Professor Devlin's lecture hall.

I do this knowing that even within a few months, this conceptual framework will be revised and replaced with other incomplete understandings many times over. But I do so knowing that I am teaching my students how to learn by giving them tools for understanding how to build tools that help them understand what the heck they are doing.

Thursday, March 15, 2012

Waiting for Gratitude: a reflection on pink slip day (or, Beware the Ides of March)

I got my pink slip early this year, and I'm finding that waiting for gratitude is a bit like waiting for Godot. But my dharma practice teaches me that waiting is just another word for trying to find a doctor's note that will excuse me from this human experience of groundlessness.

So as long as I keep waking up early anyway, I've been getting my butt out of bed and onto the couch to do writing practice on how this particular episode of groundlessness really feels -- trying to capture on the page what I am experiencing as I keep running out of runway.

Wile E. Coyote is my patron saint of groundlessness. I keep an enameled pendant depicting him hanging over my desk. He is nose-down, hanging by his left foot, having chased the Road Runner over the cliff yet again.

Like me, he really ought to know better, but he is a slow study. Like me, each time it happens, he looks out at the camera and blinks twice, before he crashes to the canyon bottom.

The hardest part of today was the fact that my students remained so bloody happy to see me and to spend time with me. My Algebra students wanted to wrestle with factoring nonmonic quadratic trinomials, while my English students wanted to brainstorm on their "Product of the Future" ideas for our science fiction unit. Being eighth-graders, most of their ideas for outstanding products of the future revolved around bathroom components, clothing/shoe/makeup accessories, or variations on teleporting devices.

My only product idea was for a Recess-Extender -- one that would stop time and allow me to take a nap during recess after I bolt my yogurt.

The best part of today was doing math with students -- finding patterns as we factored nonmonic quadratic trinomials, and saying "nonmonic quadratic trinomials." They love the words of mathematics, as much as the language of algebra. Anything they can use to stun their parents at the dinner table is a good day's work.

At the end of class in English (as we were cleaning up from the product of the future brainstorming), two of my 8th-graders who are in Geometry asked me about a problem they were struggling with. For about three minutes, I lost myself in the Pythagorean Theorem and in wondering how -- or whether -- we could prove that the area of the black region of a hexagon was equal to the white region of the hexagon.

This led to a quick discussion about equality, equivalence, and proof. And that made me feel sad as I remembered that I had just been laid off.