In this week’s edition of Thinking Aloud, I’m thinking about thinking. More precisely, I am thinking about graph paper where my 5th graders draw doodles and (occasionally) show their computational work. And I am thinking about footnotes in academic works that introduce sources and tidy up arguments, and occasionally contain doodles. I’m currently reading a lovely book called The Weil Conjectures by Karen Olsson - a book that is part memoir, part essay, part intellectual history and that revolves around the philosopher Simone Weil and her mathematician brother Andre. And so my musings will be drawn from quotes from that book alongside my own reflections on teaching math to the 5th graders.
First, some brief updates on my writing, viewing, reading:
1): My review of Martha Nussbaum’s latest book The Cosmopolitan Tradition is now available online in ISI’s journal Modern Age: “while the Stoics would have us believe that even “the sage on the rack is happy,” I think we can conclude that he is not. And while we can agree with Socrates that it is “better to suffer injustice than to commit it,” we should recognize that neither is tolerable in the just republic. This is not to say that we aim simply for the absence of violence, as in some versions of libertarianism. A plant that isn’t watered will wither away. Often a person, particularly a child, who is abandoned to poverty will likewise wilt.”
2): I watched the golden-globe winning film, 1917, and it was extraordinary. There are maybe three or four scene cuts (fade to black, etc.) in the whole film: otherwise, it’s just one camera tracking the action, which makes for one of the most immersive films I’ve experienced. This is a film about war, yes, but it’s also about cows and cherry blossoms, postcards and poetry. It is well worth watching in theaters.
3): Before the New Year began, I finished reading Ordinary Girls - the breakout memoir by Jaquira Díaz that chronicles her years growing up brown, queer, and in poverty in Puerto Rico and Miami. Díaz writes with a carving knife, and some of the passages left me in gasping. But there is also beauty here, and insight.
4): Deaf Republic is a work of poetry that is both surreal parable and gripping protest art. Ilya Kaminsky presents us with an unnamed village being occupied by foreign troops. The villagers feign deafness, or maybe actually experience it (the poetry felt ambiguous), and Kaminsky explores how deafness and silence can be a revolt against tyranny but also maybe a complicity with it.
5): Alan Jacobs had an essay in Harper’s a while back lamenting the loss of Christian public intellectuals in the contemporary age. But I think Jacobs is as good of one as we are likely to get. In his book In The Year of Lord: 1943, Jacobs weaves together the thought of five luminary thinkers of the 20th century: C.S. Lewis, T.S. Eliot, Simone Weil, Jacques Maritain, and W.H. Auden. Jacobs believes these thinkers all faced the same question: how does Europe rebuild culturally after the War? And all these thinkers, in their own ways, turned to education, and advocated for a form of Christian humanism informing that education.
On Graph Paper and Scholarly Footnotes
As a classroom aide in 5th grade math class, one of my main duties is checking students’ homework, and helping them to find solutions to problems they got wrong. Often I will look at a student’s workbook and see no equations or computations, just an answer - and a wrong one at that. I try to remind students periodically that it is important for them to show their work, so that I can see what went wrong. But in the moment, this reminder doesn’t help me or the student to solve the problem.
The question in these situations is always whether the student simply made a computational error along the way (“oh, I forgot to carry the one”) or whether there is a deeper conceptual misunderstanding. And so when I am in this situation, of seeing a wrong answer, and of wanting to help a student rectify without giving them too much help, I use the following strategy. I say: “walk me through how you solved this problem.” This technique is really helpful for me, both because it typically allows me to spot the precise moment in which the student erred, but also because it helps me to enter into the mind of the student, to understand their misunderstanding, which better prepares me to walk through the concept with them.
More importantly, when I ask a student to walk me through the problem, I am giving the student an opportunity to narrate their own learning. And it is often the case that when my student begins to talk through the problem, they spot their own error, even in cases where it wasn’t simply a computational error. This process of narration reinforces mathematical thinking, yes, but it also makes visible and explicit the reasoning, allowing the student to think about their own thinking.
As the students puzzle through more complex problems, the graph paper becomes a place to write out their conjectures. Maybe if I plug in this number, or maybe if I shift this order, or maybe if. And perhaps when my student draws gold bars (“get it? 24K?”), he is making helpful connections, though I do pretend to frown, and say, “don’t let yourself get too distracted.”
I’m thinking about all of this as I read this extended passage in The Weil Conjectures:
The word conjecture derives from a root notion of throwing or casting things together, and over the centuries it has referred to prophecies as well as to reasoned judgments, tentative conclusions, whole-cloth inventions, and wild guesses.
Conjecture allows…[one] to press past the visible, to sacrifice the certainty of witnessing for the depth and predictive power of theory. There’s another old definition of conjecture that means something inferred from signs or omens.
Elsewhere, it’s hokum, claptrap, bull:
In contemporary mathematics, conjectures present blueprints for theorems, ideas that have take on weight but haven’t been proved. Couched in the conditional, they establish a provisional communication between what can be firmly established and what turn out to be the case. More than a guess, conjecture in this sense is a reasoned wager about what’s true.
I love this idea of conjecturing, and I think it perfectly captures what I am trying to do in this newsletter. At a political theory conference last year, someone asked a friend of mine about my political beliefs, and she said, “I went to college with him for four years and I still don’t really know.” And I am proud of that, because it means that most of my public speech is conjecturing, more than a guess, less than an ideology, sincere ideas with weight but no fixity. In other words, most of what I make available is my graph paper: some of it just doodles, some of it is half-baked speculations, and some of it is the rigorous reasoning that provides the framework for great conceptual feats.
I want to render public and in print, my efforts to think in real-time about this topic or that. I want to show you the sources shaping my thoughts, to let you see the dots that connect this theme with that one. It is easy to make assertions and claims, but I think the more valuable task is showing one’s work. So even if you disagree with my claim, you might still benefit from seeing the precise moment that my thinking veered into error. So think of this newsletter as being like footnotes in a thesis, or chicken scratch on graph paper, gesturing toward broader arguments and bolder claims, but couched in the conditional, wagered as maybe, and likely, and plausibly, and even almost assuredly.
Here’s another, shorter, excerpt from The Weil Conjectures:
How I would like to write something as clean and powerful as the best kind of mathematical proof. In pen, on quadrille paper: lines of black script conforming neatly, inevitably, to the faint blue squares. The sun would sly its way out from behind the clouds and beam right up your head, warm up some inner lobe of the walnut. A hint, a glimpse into the nature of things.
I hope that my writing does this, that it offers hints and glimpses. I do not presume to offer anything as totalizing as Euclid’s Elements, with axioms that withstand centuries of scrutiny and elegant proofs that still dazzle. Realistically, I have more humble aspirations, such as suggesting lines of reasoning that can be applied, or commending fruitful sources and texts filled with far greater insight than what I offer.
And anyway, even if my conjectures are totally off the mark, just like with my 5th graders, I still think it’s helpful to narrate the reasoning aloud, to think aloud, to think about our thinking, to reason together.
What I Am Reading Elsewhere:
Writing for The Dispatch, David French tells the surprising story of religious liberty in modern American history: a story in which Bill Clinton is a hero, and Scalia is an unfortunate villain. French explains why religious liberty has never been more secure judicially, even as Christianity becomes less and less of a dominant force in culture.
I think AOC’s comparison of our border camps to the Holocaust was imprecise and ill-advised: a much better comparison would have been to our internment camps in WWII. But amidst the backlash, arguments were made that it is inappropriate to draw analogies to historical events, particularly events involving great suffering. I think that argument is also misguided, but have struggled to articulate why. This essay in NY Books helps me to make my argument: “All historical analogies are interpretative acts, but interpretation is just what historians do. Those who say that we must forgo analogies and remain fixed on the facts alone are not defending history; they are condemning it to helpless silence."
I read this excellent essay in Jstor Daily that was about public speech, and relation to the Other, and power, and the nature of reasoning. And I got strong MacIntyre vibes (After Virtue, etc.) and when I tweeted it out, the author responded that the original draft interacted explicitly with MacIntyre. Anyway, you should for sure read it: “Reasons both arise from communities and are appeals to them. To offer a reason requires not only expressive work, but also engagement with an audience familiar with the terms we use, to whom we can explain our convictions, beliefs, and actions. It is always a reciprocal matter."
The New Yorker featured an interview with comedian John Mulaney, and here’s my favorite part: "And I started to feel like I was more myself when I was nine, ten, eleven than I was between the ages of fourteen and twenty-three."
In Public Discourse, a look at virtue in politics, and what it would look to cultivate a meritocratic class of leaders who take virtue seriously and who see themselves as public servants: "We should not be measuring merit in terms of potential and raw intelligence...but in terms of actual accomplishment and concrete knowledge. That knowledge should include humanistic subjects like history, philosophy, and literature."
Matt Stoller’s newsletter is typically granular analysis of anti-trust law and the various companies (Disney chief among them) that should be facing anti-monopoly enforcement but aren’t because of the Chicago School of law (cough, Robert Bork) dating back to the 70s. But in his most recent post, he took a step back to examine the ways in which democratically elected officials have shifted a lot of policy-making power to economists who wrap political assumptions and choices in neutral-looking technical jargon. His essay is provocatively titled: What is the Point of Economics?
Did ya know: in 2016, Kohl’s revenue on its credit card was 35 percent of profits. Macy’s was 40 percent... In the New Republic:"Macy’s is perhaps more appropriately described, not as a retail, but as a credit card and real estate company.” This is about the rise of "non-interest income" in the banking world.