A Quick-and-Dirty Guide to Fighting the Math Wars

I just posted this to a reply to a post by David Wees on Google+, but I thought it might be useful to some if it had some permanence here.

I've been in and out of "Math Wars" debates for 10+ years, and I find it's helpful to examine the issue at a more granular level. Here's a quick list of questions I jotted down:

What is your definition of mathematics? (Someone who answers, "It's a subject you learn in school" may have very different views from someone who answers, "It's a human activity we undertake to solve problems relating to number and shape.")

What is your philosophy of mathematics? (A Hardyist and a Mathematical Maoist have very different views, as do a Platonist and a Formalist. And for all the consistency in mathematics, this is not something with which we as individuals are necessarily consistent.)

What is our goal for students learning mathematics? (Is it to prepare them for work? For more school? To gain an appreciation of mathematics? For mental exercise?)

How should we assess mathematics? (Often when we claim that students do or do not perform well in mathematics, we are basing those claims on an assessment that may not embrace a balanced view of the issues above. Or, failing that, we make those claims without regard to the biases of the assessment.)

What learning theories do we use, and how do we use them? (A difficulty with learning theories is that in most all cases we can design curriculum and pedagogy around them that show they work -- at least to a degree. The workings of the human brain aren't easy to study, explain, or leverage in a classroom.)

How do we perceive "failure" or "success" of practices of the past? (I fear sometimes we stereotype certain historical movements, such as "New Math" and the "Back to Basics" movement, and we falsely assume that those movements were implemented in every classroom with high fidelity. We also sometimes forget that as time has passed, we are trying to teach higher and higher levels of mathematics to more and more students.)

How do we avoid false dichotomies? (False dichotomies were addressed in that article and Zwaagstra was wise to try to avoid them. But it's such an *easy* trap to fall into! [I've probably done it here without realizing it.] For example, he cited a paper by Alfieri, et al. (2011) that claimed through meta-analysis that "unassisted discovery does not benefit learners." But why would a well-trained constructivist teacher believe discovery should be unassisted? That's the same as assuming that a traditional teacher only has students listen to lectures and work problems in isolation. No teacher or student thrives exclusively on either. Interestingly, Zwaagstra in the next sentence says learners should be "scaffolded," an idea developed by Jerome Bruner in support of learning in a social constructivist environment.)

What skills, abilities, and philosophies do we believe teachers need to be successful? (I'm not sure we fully comprehend the effects on the received curriculum when it's taught by a teacher with skills, abilities, and philosophies that run counter to those supported by the curriculum. In such cases it's easy to misplace blame for poor outcomes.)

I'm sure there are more that I could add, but I strongly recommend that anyone who is serious about this debate to take on these issues one by one. Only if there is some agreement, or at least some sympathy and understanding, on these issues does it become truly productive to talk about "what works."