Today Geoff Krall (@emergentmath) posed this on Twitter:
Thoughts/critiques of this framework? pic.twitter.com/vsTdBKbOhC
— Geoff Krall (@emergentmath) November 20, 2013
It's a cycle I've seen before, and you probably have, too. Students struggle on complex tasks (or, far worse, we just assume they'll struggle without giving them the opportunity) so we opt for the "back to basics" approach and change our mathematical practices to include a lot of repetition on lower-level, procedural tasks. We convince ourselves that this is the right thing until either the students tune out, or underperform, or both, and then we bark about "raising the bar" and the cycle begins anew.
Geoff's cycle reflects knowledge and practice, but I wanted to dig a little deeper into the idea of "student resistance." Jo Boaler (@joboaler) found herself in a similar position after she'd done some studies that linked reform-oriented practices to a more flexible and robust form of mathematical knowledge, but felt there were some stronger links to make between knowledge and a student's mathematical identity, or the way they see themselves as and becoming knowers and do-ers of mathematics.
Boaler described this process in an article titled The Development of Disciplinary Relationships: Knowledge, Practice and Identity in Mathematics Classrooms, published in 2002 in the journal For the Learning of Mathematics. You can find a preprint of the article on Boaler's faculty website, but I took the time tonight to add a summary of the article to the MathEd.net Wiki:
My takeaway from reading this particular Boaler article is that while both traditional and reform approaches can result in student learning, the attention to student identity and affect in the reform approach shapes student learning in a way that makes the knowledge more useful in more situations, or, in the relative absence of knowledge, gives students both the disposition and a set of practices to make mathematical progress. When students get caught in Geoff's "Cycle of Low Performance," it's not that they aren't learning. Instead, they just don't see their knowledge as particularly valuable, nor do they see themselves as active users of that knowledge. As teachers, we need to design our classrooms and activities in ways that give students opportunities to have some authority over their mathematical ideas if we expect them to use their knowledge productively.