This post is not about constructivism. Or inquiry. Or problem-based learning. At least, not directly, even though I like all of those things. I like them because (among other reasons), when done well, they give students agency, and a sense of ownership over their own learning, particularly as compared to the rote, passive, sponge-absorby way math (or science) is often taught. Students can learn that their minds have more value than as file cabinets or calculators, and feel that, as individuals, they're capable of generating knowledge, and knowledge that is thus personal to them is more likely to stick with them (I'm sure there's something I can cite here).
But most (certainly not all) math lessons designed in this way still have the same outcome: students figure out something for themselves that others have already figured out before them. If this is the desired outcome-- transmission and replication of a canon of knowledge-- then I'm all for these methods as the best way to accomplish that outcome. But I can't help but wonder: what would it take for students to learn that mathematics is not just something created by dead white men that they at best are coerced to re-discover and at worst have spoonfed to them, but rather a developing body of knowledge that they can and should co-create?
In science, it's easier for me to see how knowledge is a product of human creation and not just a set of unassailable facts, for two primary reasons: 1) the very language of science depends heavily on hypotheses/models and data, which constantly reinforces the idea that people-- not mythical stone tablets or textbooks-- are making predictions and drawing conclusions; and 2) the history of science, in my layperson's understanding, provides plenty of evidence that sometimes people get it wrong (the world is not flat, the Earth is not the center of the universe, just about everything in medicine, etc.), suggesting that what we "know" to be true is constantly evolving.
But what's the mathematical analogue? Are there similar "facts" or theories/concepts that we humans have gotten wrong and reversed course on over the years? I simply don't know my math history (and even less, my non-Western math history), so I'm hoping there are some good examples out there. Otherwise, it's currently feeling to me like what we know about mathematics has been more of an additive process over the centuries-- discoveries and new formulations based on need (like zero and negative numbers)-- rather than a revisionist one, which leads me to the rather depressing conclusion that students can construct bits of knowledge on a small scale but aren't going to learn that knowledge is subjective and biased from the perspective of the knowers (our version of "history is written by the winners").
I'm not saying high school students should be contributing totally new fodder to humanity's mathematical understanding, because I think most high schoolers are not going to be quite ready for that. But I do want them to recognize that they could, because a) I care about students' sense of self and the opportunity to realize that they can be a part of something-- it's not closed off to them-- and furthermore, they can actually be that something; b) I care about students' relationship to mathematics and their understanding that it isn't an instrument of torture created to make them feel shame and tedium; and c) I care about students' practical college/career opportunities and suspect that that excitement and sense of possibility and belief that what we accept to be true is all subject to change could inspire them to continue studying and learning, no matter what field they choose. Or in other words, that they matter, that mathematics matters, and that mathematics matters to them.