I think it all starts with word problems.  This is contrary to the model of American math education, and maybe beyond.  We teach you the skill and then teach you how to apply it.

You know what, though?  I'm not the one who is backwards on this.  Remember Psych 101?  When a person is exposed to a new piece of knowledge, she will attempt to incorporate that fact into her worldview.  If she can't find a schema for it or hasn't developed an advantageous strategy for assimilation, then she's barely better off than if she hadn't been exposed to the fact in the first place.  On the other hand, if she can effectively identify a schema and assimilate the fact into it, then her worldview is expanded and she is embiggened by the process.

Not everyone has had Psych 101, I suppose, so I'll illustrate.  Let's say that a young child sees a German Shepherd, which is the first time that she recalls seeing a dog that is as big as she is.  She doesn't know what to make of it; perhaps she is inclined to think of it as a "big animal" like a pony or a goat, except that she has developed a sense of "dogishness" (that adults would call "canine") that this animal satisfies in many ways except size.  That sense is her schema of "dog", and when an adult tells her that German Shepherd are also dogs, she is able to broaden her schema to understand that dogs can be big and perform tasks beyond companionship. 

However, this model is exactly what we don't do in math education.  The skill comes out of the textbook through the blackboard without giving any sense of how it works into your worldview.  This is tragically short-sighted, since mathematics is actually the formalization of the modelling our ancestors did to solve their real-world problems.  Sometimes we are so busy answering "how" that we forget to ever get around to "why".  Some students can get it anyways, of course, but we're far too comfortable with thinking that when other students can't get it that the problem is with the students and not with the math instruction.  Back when I was in high school, if 80% of students barely understood math and 10% of students mastered the concepts, we'd have enough understanding to sustain society's population of alphas and betas.  We don't live in that world any more; machines are currently doing much of the work that used to be done by Deltas and the Gammas are next in line for reassignment.

Let me give you an example I use at the beginning of one of my pre-algebra lessons.  I am leading a full-day field trip, so I made some packaged lunches for the adults and children.  Each adult lunchbox contains seven crackers and five pieces of sausage, and each child lunchbox contains three crackers and two pieces of sausage.  If there are ten children and four adults on the field trip, what is the total inventory of food in the lunchboxes?  Go ahead and solve that problem: note that it is a moderately difficult arithmetic problem that requires a few steps.  Still, in my experience, it is not beyond the intuition of the sufficiently motivated student to understand that they want to "dump out" all of the lunchboxes and gather all of the similar foodtypes together.  I've got a follow-up question too -- did you come up with fifty-eight crackers and forty pieces of sausage as your solution, or ninety-eight food pieces?  Whichever answer you came up with, why did you settle on that as the better answer?

So, once people have worked out a few problems like this in small groups, I show how you would solve this in "algebra language":

10 ( 3C + 2S ) + 4 ( 7C + 5S )
30C + 20S + 28C + 20S
58C + 40S

See, I don't need to drag students on a trip to appreciate how to simplify algebraic expressions using the Distributive Law and collecting like terms.  They made that trip themselves.  My job as a teacher is to point out some of the highlights they'd have spotted along whatever journey they made, and they're going to be able to retain and apply that knowledge effectively because they had already engaged their schema for solving pre-algebra problems.
I wrote a poem today and recited it in front of my classmates. Sorry, let me say that again. *I* wrote a POEM today and RECITED in IN FRONT OF MY CLASSMATES. A sonnet, no less.  It was an amusing thing that lead me to it. For the "final exam" of my literacy skills class, we were asked to reflect on everything that we had learned in the class and to synthesize it in any form we wanted ... except prose. We could make a poster or a diorama or a folk dance or a board game, anything at all except the one thing that I'm good at.

Thing is, this class was AMAZING. It's called "Teaching Language Skills in Middle and High School Content Areas I". Phew. Point is, reading comprehension isn't just the job of the English teacher. Who's going to teach you how to read a geometry proof or write a chemistry lab? But we should all be using the same vocabulary so when I ask a student "What connections are you drawing from the text?" the student can process this new sort of comprehension in the context of every other sort of comprehension they've done in school up to this point. So many other things -- engaging students in high level questions, effective small group learning, the process of inquiry, authentic assessment strategies, the role of technology in the classroom, holy cow. And I had to make a single artifact that captured such a complete transformation of my educational philosophy for about 10% of my class grade without access to my strengths.

So, given that another thing that I've learned in this course is that written language is the conscious product of thought, I wrote around five pages of crankypants rant impassioned analysis on how much I learned in this course and how one of those things is just how unjust this assignment was. I had started to form a risk-benefit analysis of turning that in when --

Well, maybe you linguistic people could tell me what happened, because it was new to me. I was reading the text aloud to myself and starting to sharpen it up, and I was thinking "Huh, that sentence has a very regular meter. Huh, those two sentences rhyme," and about twenty minutes later I had turned a page of ranty prose into a sonnet. It was a very weirdly powerful experience.  Perhaps the muse is not a mythical creature after all.

Reflection is an inauthentic task
Mere two dimensions capturing a space
Summing up is far too much to ask
A photo’s no replacement for a face
The walls cannot contain the things here earned
Nor closets hide my crafts ‘neath layers of dust
My room will overflow with all I’ve learned
To serve my students, praxis is a must
I’ll comprehend the things I ought to do
And teach my students skills to do the same
The value of my tutelage is true
Math in the world, not just a blackboard game
My purpose found, the strategies to aid
To serve me as I quest to make the grade
 

I was well outside my comfort zone in class when I found that we'd actually be presenting our projects to the entire class, but a fellow grad student with an English concentration read it over and told me that she thought it was really good and I was among friends.  It earned me a round of applause and my teacher was flattered to be the inspiration for, by my count, the fourth authentic poem I've ever written.  But it's a strange thing that so many things will paralyze me into inaction when the truth is that I am fearless in the moment.  I really must find the way to learn from this.

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Matthew Daly

December 2012

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