It’s Friday afternoon, and I’m sitting at work with a pile (no, it’s bit bigger than that) of things to plow through, that promise little satisfaction other than moving them off of my desk. I have a kid to pick up in a couple of hours, so I’m finding things to do to keep me otherwise busy. I will get to those papers.

Subscribers may have noticed a post that keeps refreshing. I’ve been trying to clean up a tiny problem (actually a couple of them) with that post, and I may have it done now. I was trying to get a graphic to display, and also to deal with some nit-picky spelling/formatting problems. I should have waited to press the ‘publish’ button for a day or so, I guess. Sorry. I might have put the graphic up on my flickr page, and linked from there, but the new community guidelines discourage us from doing that anymore. As far as that all goes, I’m wondering if they want for us to include a text link with any photos we blog, or will the linked image itself qualify as a link back. I’m not pleased - I wonder if anyone cares.

That last post - the one about math and vocabulary learning - wasn’t actually a blog post. It was a piece of my MEd project that I tried to scrub for blogging. No matter how much I edited it, I know it still smells like a research paper. I posted it because I’m having some ‘discussions’ with other teachers who see things a bit differently than I. Nobody wants to listen to another point of view - me included. So this was a bit of an effort to revisit an important connection that I made and make it public. I tripped over a little gem when I was looking back through the paper that’s relevant to this recent theme in my thinking about how we present math and reading to kids formulaically.

Alan Schoenfeld explained breakdowns in comprehension in classroom mathematics activities as he observed the function of both deduction and empiricism in the development of mathematical understanding. He saw a relationship between the emphasis on form as opposed to meaning, mathematical problems versus exercises, and passive versus active learning. With respect to the roles of deduction and empiricism Schoenfeld argued that students often failed to recognize the value of deductive reasoning and instead resorted to empirical methods of reaching a solution. In a case he cited involving a 10th grade geometry class, students ignored a previously developed proof in favor of a laborious procedure that involved straightedge and compass. I’ve observed a similar phenomenon in elementary school arithmetic when students will count fingers or objects to calculate 8 + 4 = 12, for example, and then use objects the next moment to calculate 8 + 5 = 13. Instead of reasoning that 8 + 5 must be one more than the known 8 + 4, they begin ‘from scratch’ each time. Schoenfeld referred to this phenomenon as “naive empiricism,” and characterized deduction as a mathematically useful “tool of discovery” (1985, p.173). He contrasted the “naive empirical” approach to mathematics with the approach taken by professional mathematicians. He accounted for the development of an appreciation for the value of argumentation as a matter of experience in the discipline.

This is all well and good, and as Schoenfeld noticed, when we are working outside of our comfort zone in deep and uncharted water, we depend on empircal methods. A dip of my toe into PHP and MySQL databases [Follow that link to a solid introductory resource that Chris has put together.] gave me a feel for what it’s like to be in over my head, and helped me to develop a fairly strong empathy for my kids who are a bit slow to pick some things up. Sometimes the empirical methods are needed to keep the ball rolling. The problem for teaching and learning occurs when we don’t move beyond that, and never bother to remove the props.

The reason I care about this at all, aside from the fact that I teach reading and math to little kids, is that when I went back to get the reading specialist ticket I decided to test out comprehension theory in the context of mathematics instruction. I gave myself a double dose in a lot of ways. Because my degree was in language and literacy, I looked at the NCTM Communication Standards and found my lit review reading list in the bibliography for that document. My whole project revolved around the way we talk about math with kids. Vocabulary was a big part of it, and so was classroom participation structure, communities of practice, shared cognition, epistemology, and God only knows anymore. I get a little bit worked up when it is suggested that I should just “show them how” and not confuse them with a bunch of reasoning. No more groaning on my part, I promise.

I think that as a profession, and as a nation, we are being driven in (at least) two directions simultaneously. It’s a conflict between conformity and creativity; between accommodation and rigor; between convergence and divergence; between authority and autonomy; between performance and understanding; between doubt and trust. It’s making me crazy, but I can’t quit because I sense that I’m sitting on top of a huge pile of junk that’s almost ready to implode. I want to be here when it happens.

Besides, I can’t afford to stay in Alaska if I quit my job now, and there’s nowhere in the world I’d rather be, even if I do still have to use dial-up at home and can’t watch the Olympics on television. Can you believe it?

Schoenfeld, A. H. (1985). Mathematical Problem Solving. Orlando, Florida: Academic Press, Inc.