I'm lucky to have a wonderful advisor, Stacey Callier, in my grad school program. I work in a project-based school and, as she knows well, I often push back against the hidden assumption in PBL that math is a "tool" that helps us solve real-world problems (I think the terminology I used today was that math seems to always be the "servant of science"). We started talking a lot about the constructivist philosophy that is central to my work and we (mostly she) came up with this matrix:
1. The "HOLY GRAIL" label implies that this is where math education "should" be…which I'm not sure everyone would agree with (in fact, I'm not sure I can say that this is where I think it should be.
2. The top (applied vs. non) seems to get at "why teach math" while the right (constructive vs. non) seems to get at "how teach math." Is it ok to compare these two things in a matrix? Not sure.
Mostly, I labeled the top left "HOLY GRAIL" because I strongly believe that math should be taught in a constructivist fashion. If we can do this AND situate the math in a "relevant/applied" context shouldn't we do that?!?