Failing on its own terms

Mathematics is unreasonably effective. In several ways. And that’s dangerous.

I am always surprised by the elegance of the Power Rule from Calculus. Humans worked very hard to make sense of instantaneous rates of change, but it surprises me that the slopes of tangent lines are computable, it surprises me that we can globalize these computations into a coherent new function that encodes these slopes at each point, and it shocks me that we can express these derivatives across the family of power functions in such a simple and elegant way: For any number n, (xn)’ = nxn-1.

I am also awed by how flexibly humans have applied mathematics to model systems and phenomena in order to understand ourselves and make predictions about the world, from the stars to microbes to the ways we behave in large groups.

But this elegance and flexibility is seductive. There are stronger metaphors, but one that should be uncontroversial is that with a hammer this powerful, all the world seems to be nails. If you are faculty, I am absolutely confident that you’ve had the frustrating experience of being at a department meeting with other professional mathematicians attempting to deduce a solution to a question for which deduction is absolutely not an appropriate approach, whether it’s about awarding your departmental undergraduate award or strategizing about the future of the program.

I think that lots of mathematics learning spaces select for people who approach tasks in this manner. I don’t remember this, but my mother says that she once told me to “get in the bath” as a euphemism for taking a bath that I didn’t want to take, so I went up and stood in the dry tub for a few seconds and went on with my day. This obnoxious and legalistic approach to language was useful and rewarded in math classes. And it’s clear that I was further trained to use these kinds of tools to prove theorems.

But this selection and training, paired with the elegance and flexibility, seems to convince lots of people that this approach is the only valid approach, that doing anything other than hammering nails is not only inferior but somehow also harming the hammer. In many ways, my work is focused on the fact that mathematics has a disciplinary worldview, an element of which is often the avowed belief that mathematics doesn’t have a disciplinary worldview. This element of (this version of) the disciplinary worldview is exclusionary and harmful, and I believe that other versions of a disciplinary worldview can exist that support the positive elements of mathematics while allowing us to redress the exclusion and harm. But this will take work, and and this work is subtle because of this pressure not to see pieces of the powerful worldview. Here are some ways this subtlety plays out.

People have said nasty, hateful things about mathematics education to me and in my presence. My personal hypothesis is that training that focuses us so much on deduction has made many of us impervious to data as evidence and dismissive of other ways of supporting claims, leading to people deciding in advance that education research cannot possibly have solid results. Unlike biologists, who can think about their students as biological creatures, psychologists who can think about students’ psychology, and historians who can use their historical tools to understand their classrooms in context (and essentially every other discipline) lots of mathematicians seem to believe that there is no possible overlap between our disciplinary research and our work as educators. This is false, but I’ll leave discussion of that for a longer, future essay. But I will point out that these other disciplines have explicit discussions of their methods and theoretical frameworks, which I think leads to stronger results than those possible without the conscious attention to worldview.

But this dismissive perspective on mathematics education also rests on an unsound vision of mathematics as a discipline. For example, I would challenge readers to define “proof”. I expect that almost none of us could give a definition of proof that is operationalizable in the way we demand of mathematical definitions. The few who can are likely really doing metamathematics, and these definitions don’t really match with the ways proof is used in the discipline. An education research colleague did some excellent work a few years ago to find multiple facets of a potential definition of proof, from an artifact itself to the argument to which that artifact refers to the conviction or communication it is supposed to support between humans, but found that accepted proofs regularly failed to have multiple of the supposedly definitional facets in different contexts. So the central object of our disciplinary methods does not meet our overt disciplinary standards. [I’m not trying to erase forms of applied mathematics that are not focused on proofs, but I am much better equipped to talk about this piece.]

This perspective leads many to use mathematics as the prototype for “objectivity”. There has been a lot of very angry yelling that “2+2=4” recently. First, this statement is not true without context, in much the same way that that (ex)’ is not equal to xex-1 with the familiar interpretation of the symbols. Sure, in context one could deduce the sum of two and two, but that context is a model, not a truth, and others are possible. For example, if fencing requires a support post every yard and two people each have the posts they need to support two yards of fencing, how much can they support together: five yards. Furthermore, 2 does not exist in the universe; it requires a conscious observer to collapse the stuff of the universe into objects that have some abstract property represented as 2. And moreover (running out of my favorite words here), work like that of Kurt Goedel shows that our attempts to formalize something as foundational as numbers will still include non-standard models! Beyond the mathematical analysis of this point, there’s the historical fact that trying to use “2+2=4” to invalidate anothers’ experiences comes from a particular, racist tradition of anthropologists using poorly formed linguistic questions to “decide” if Indigenous peoples were “civilized”. This tradition continues in the ways we treat mathematics tests as unproblematic measurements of something innate.

Some people are committed to maintaining this vision of a “pure” mathematics that rejects discussion of other aspects of our discipline. Setting aside the choice to use a word from eugenics and using it in the same way, this is still impossible. Lots of mathematicians seem to be Platonists, asserting that mathematics somehow exists a priori. I disagree, but even if we accept this point, it functions as an unsupported axiom, not a conclusion like this perspective tries to frame it (commonly known as assuming the conclusion). And even if this version of math were to “exist”, our discipline would be about the human work of connecting to the knowable pieces. Organizations devoted to mathematical research at the exclusion of dimensions of human identity are literally social clubs about not wanting clubs to be social, at least if taken at face value. And seen in context of the ways that power continues to work in our society, they seem more likely to be safe spaces for people who don’t want to be asked to feel responsible for making places safe.

As a final example, we mathematicians have a tendency to model the world as a zero-sum game. Content coverage vs active learning, rigor vs compassion, productivity vs inclusion, and many others. Mathematics is regularly used as a bogeyman that requires us to run classrooms that rush through ideas from an expert’s perspective but can’t allow for much learning, to treat students in ways that pretend to have high standards but block them from meeting them, and that organize our community around validating the current work of a select group while blocking the growth that could be. We are being used to claim that high school curricula that are not focused on selecting privileged students and concentrating resources on only them is destroying this country. Let me be clear about two points. First, these are not zero-sum situations. Active learning supports students in learning and in learning about learning all while we are able to focus on multiple ideas in parallel, and in my experience this means that my students get to explore more than would have been possible if my classrooms were focused on passive transmission; the research certainly makes it clear that we can set aside the fears about “what if they haven’t seen this thing for the next course”, though there is plenty of room to improve other aspects of these systems. Compassion supports students in excelling, and in my experience it is the compassion that supports the multidimensional excellence. And inclusion supports productivity, whether seen as more progress in the previous, narrow/extractive vision of productivity or as progress on the broader goals of a more diverse community. So if the conjecture is that it’s not possible to have both coverage and active learning, rigor and compassion, productivity and inclusion, then I am happy to provide a proof by counter-example to answer the conjecture firmly in the negative. But second, embedded in the first point, it’s clear to me that it’s the belief in the zero-sum that causes the undesirable outcomes, or somewhat conversely it’s specifically the focus on BOTH AND that leads to growth on either.

Stepping back from the examples above, here’s what I’m saying: mathematics as a discipline (the hammer) is regularly invoked in ways that fail to meet its own standards and in ways that fail to apply those standards in context, being distracted by the elegance and flexibility respectively of hammering nails. It’s failing on its own stated terms. This challenge is part of what happens when a group of humans share a disciplinary worldview, and we need to talk about it.

I’m disappointed with the AMS’s decision to terminate the space provided by these blogs because I think it blocks this conversation from spaces bound by a shared identity as doers of mathematics. The future of this particular blog is uncertain. Perhaps we will end up at the MAA. I certainly feel more at home in MAA spaces, but for me these are spaces bound by a shared identity as mathematics educators, where these kinds of conversations are already part of the fabric.

For those of you interested in resisting the erasure of our disciplinary worldview and the consequent harm, I would like to suggest a useful starting axiom: I have things to learn from other people about mathematics and mathematical spaces, especially from Black, Indigenous, and Latinx folks, and when I encounter something I don’t understand I should start from the assumption that I and those around me will be better served by working to understand it rather than working to explain it away.

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