One of the things that you face in college teaching is the question of when a student is entitled to say, “No mas!” and completely opt out of a major area of knowledge on the grounds that their mind simply doesn’t work that way, they’re no good at that sort of thing.
I think the official educational line prior to this point is often that anyone can learn anything if they try hard enough or if the right pedagogy is offered to them. Some professors will say the same thing in higher education: that the door is open to all, and that all can cross its threshold.
The moment I hit college myself, though, I decided that I was done with math. I kept at languages but by the time I was finished with one year of Latin and two years of Spanish, I decided I was done with them, too. Over time, I began to understand the nature of my issue with both subjects. My mind simply doesn’t like precision, it doesn’t like problems where there is a single right answer and a single wrong answer. In high school math, I invariably understood concepts really well and invariably made small computational errors when executing the concepts. In language, at least as it was taught at my undergraduate institution, it was the precision of the grammar that frustrated me.
I’m thinking about this again because of the course I’m auditing, in studio arts. My mind didn’t like perspective at all (though I have to say I now have an experiential understanding of the art-historical knowledge of Renaissance perspective that I’ve had for a while) but line and value are really attractive to me. Drawing plants and landscape with ink and a sumi brush was like a magnet to me.
The more I reflect on these habits of mind, the more aware I am of how much they influence what I do as a historian and cultural critic, and even the way I approach political questions. I don’t like binaries, ever. I’m not going to make grand theoretical claims about that. It’s just my cast of mind. Someone throws a stark right/wrong dichotomy at me, I’m going to look for a third way to see it, I’m going to try and shift the question or reframe it.
So partly I am wondering: if and when you understand that about how your mind works, can you go back to anything, any kind of knowledge, and do it your way? Abstractly, it seems to me that there is a mathematics suited to my cast of mind. There’s probably a way to learn language immersively that’s more suited to my intuitive approach. Partly I am wondering: when should you not accept your intuitive approach to learning as sufficient? When, if ever, should you force yourself to do something in a manner that your mind doesn’t like and is ill-disposed to accept?
Actually, you dropped math just at the level it would probably have gotten more congenial for you. Beyond a certain point (third-semester calculus), there are a lot fewer problems with one right answer, and a lot more with more or less elegant answers. It still might not have appealed, because the wrong answers are still clearly wrong, but the way of addressing proofs and such is rather different than things that exclusively value computation.
I found that my time as a math major (dropped, though, when the triple became impractical) helped my argument in political science immensely. If you’re a good writer, you can cover a great many weak arguments with good phrases. Math strips that away; if there’s a hole in your proof, there’s a hole in your proof. Practice in building mathematical arguments made my social science arguments sounder.
My short answer to the questions at the end would be “when you really need the result.” Maturity helps you to know when you really need the result, and when it’s not worth banging your head against the wall.
I had a similar but different reaction to math and computer science. (Foregin language isn’t the easiest thing in the world for me, but I never thought of it as precise the way I think of math and science as precise.) I was similarly made aware that my mind handles such precision badly–but at the same time I was enormously impressed by their rigor and their beauty. (A geometry class in ninth grade in particular impressed me with math’s beauty–Euclid alone has looked on beauty bare, and all that.) My practice as a historian, therefore, has perhaps been fuzzy, but always with an awareness that the rigor and strength of precision is intellectually formidable and aesthetically attractive. No allergies against binary dyads on my part–certainly no sense that they describe the world less well than fuzziness. My lack of mathematical rigor I see as a lack, and I am dubious that my fuzziness is a Philoctetian compensation.
The question is maybe not “when” but “for how long”? There’s something to be said for trying something you’re not particularly good at, and it’s too bad that most intellectual environments discourage this. Even undergraduate education, which is nominally about broadening yourself, usually forces you to hone in on the areas you’re good at petty quick, if only because doing poorly in a class is such a unpleasant experience.
So the answer to “when” is “whenever you can manage it”. But given there’s only so many hours in the day, you’ll probably have to ration out your periods of salutary ineptness. I’d say devoting yourself to something you know you’re going to be bad at is probably best tolerated a few months at a time max. About the duration of a single class.
“In high school math, I invariably understood concepts really well and invariably made small computational errors when executing the concepts.”
That was exactly the case for me, as well. I did tremendously well in math class discussions, and in answering questions, and in being one of the few, or sometimes only, hand in the air to explain a theorem; I was very good at — in these high school math classes, which were, to be sure, no more advanced than stopping just before pre-calculus — intuitively grasping the theory, and its implications and sketching it out.
But I consistently made small basicarithmetical mistakes in elementary addition, subtraction, multiplication, and division, when taking tests. Very annoying, and brought my grades significantly down, as well as my math SAT.
I’ve never had any talent for producing any art or music, though I greatly appreciate most forms of music, and a wide variety of visual arts.
Instead, I am and always have been highly verbal, generally analytic, and like to think I have a few other strengths. But I’m rather rigid in which categories my strengths and weaknesses fall in, it seems.
“I don’t like binaries, ever.”
That’s another trait I share with you.
“Someone throws a stark right/wrong dichotomy at me, I’m going to look for a third way to see it, I’m going to try and shift the question or reframe it.”
I find it to be completely involuntary; it’s relatively rare that I agree with someone that an issue has precisely those two sides, and only those two sides.
“Partly I am wondering: when should you not accept your intuitive approach to learning as sufficient?”
Probably not, but I’m fantastically undisciplined and lazy, and also blocked from a variety of things via depression and other mental hang-ups, so I’d like to hope that most other folks can do better than me at improving at that sort of thing.
Sorry, I missed the “when.” My only answer is “more often than I do.”
I used to be a bookkeeper, and found that congenial even as I hated the accounting class I took. One was theory/’shoulds’ and the other practically focused. So while I’m not allergic to numbers, I discovered a real aversion to theory that disregarded the practicalities of life/reality. Languages, like you, I persist at even though I find them as easy as walking through a concrete/stone wall (without a door). Immersion or classroom, makes no difference. Yet my mind is flexible and quick to find ways to make things work, pulling from knowledge from many disciplines.
So my answer to your final question (when should you not accept your intuitive approach to learning as sufficient?) – for me – is never. New and old ways change with every moment. I had to read Marx a dozen times before it clicked. And Serbian dative didn’t make sense for years; suddenly it did. Brains change.
When, if ever, should you force yourself to do something in a manner that your mind doesn’t like and is ill-disposed to accept?
_____________________
Well, I would prefer to brush my teeth side to side, but I know that to be effective, I have to do the up and down too. So I do the up and down, and also the circular. Also, I had to be dragged feet first to accept the soft-bristle theory, but I caved, and my gums are the better for it.
But if you’re talking about life directions and that, I couldn’t enable you more: I just quit being a lawyer because I figured out that I – surprise surprise – don’t like conflict. I’d rather work with than against. Lover not a fighter. Could I get over this, for the sake of my clients? There were lots of aspects of the job I was very good at, and getting better rapidly, but my babyish true self just held its breath and turned blue, so I quit. Life is short. I don’t know what I want to do now. I invested a lot in that lawyer thing. But hey.
I’m curious as to what textbook was used for your Latin class. I teach from Wheelock, and I often find that students have exactly the opposite problem – they dislike the rather broad range of possible translations that Wheelock offers for any given item. They’d like to have a single correct answer they can hang on to.
I worry about the way in which the system is set up to punish students for persisting with something that doesn’t play to their strengths. Pretty much every Latin class I teach has at least one student who doesn’t find learning the language at all easy, but wants to learn it anyway, and manages, by putting in an extraordinary amount of effort, to survive. This student will probably get what doesn’t look like a very good grade – but which, for that student, is actually an impressive achievement.
There’s pretty much no way to reflect that (outside of letters of recommendation), and it’s not only personally admirable – I’m fairly sure that this kind of capacity to stick with the difficult will pay off in other areas of life.
You probably would have liked the application of math to economics. There you can calculate your heart out and never have to produce a single result. In fact, it seems that the more answers you can offer to an economic problem the better you are. I think it was G.B. Shaw who said something to the effect that “thousands of economists will never reach a conclusion.”
But seriously folks, I taught high school math for a while decades ago and I always gave the students who didn’t like math a pass. It was easy to find mathematical things that would get them to think logically (more or less) without insisting on process and precise answers. In fact teaching students about thinking would have done much more good than memorizing the Quadratic Equation.
I wish I remembered, Gavin. It’s probably in a box in the basement, actually. Brown cover–would have been used in the mid-1980s.
Actually, I liked the precision of Latin as far as the puzzle-solving quality of it. I think at some point, my mind is just lazy: I want some magical way of knowing the boring stuff so I can get to the good stuff.
Your first commenter said it all, but as a former math/english double major, I saw a lot of similarities between formalist and structuralist literary criticism and the kinds of pattern recognition the upper-level math courses required of you. Eventually I chose English for grad school b/c I felt like writing proofs was like jumping out of an airplane and hoping your chute would deploy, I couldn’t imagine how to come up with something new in math, and I loved to read and write about literature.
My ironic take on your divide is that I managed to go through college and grad school without taking a single history course, yet since I started the dissertation, historical approaches to lit have been my main research focus. Talk about playing catch-up!
Gavin, do you have an opportunity to say to those students who persist what you said here?
I think that kind of personal attention, and honest admiration, from a teacher will mean much more than a grade. If they need letters of recommendation later, they’ll also know they can turn to someone who can offer more than the usual boilerplate.
Doug: very true. But it’s not so much that the student himself or herself doesn’t have an emotional sense of just how much they’ve achieved. Generally they do.
It’s that right now Tim’s question is IMO a little optimistic. It’s not just “when is it OK for a student to say that something just isn’t their thing, and give up.” It’s also”when is it – pragmatically – OK for them not to do that.” Grades send signals about what the institution wants from students. (Obviously, this is part of a larger question about what college education is supposed to achieve in the first place.)
I have a lot of high school students tell me: “I’m a math science student, I can’t do history.” I tell them, “If you can write a proof, you can write an essay. It’s the same skill.” And the ones that buy into that do quite well because they can find the similarities and build from there.
“I don’t like binaries, ever.” This assumes you divide the world into the binary and the non-binary. Binarier and binarier!
Tim: the thing that struck me about this post is that your description of your experience with math matches mine precisely. And I’m in the process of applying to math grad school right now.
More generally, I find that the really good math work is about reconceptualizing a problem. As a fairly famous and easy example, a lot of cryptography requires the generation of large primes. One of the important developments came when one mathematician realized that you didn’t actually need large primes; you just needed numbers that had a high probability of being large primes.
More generally, good math work generally involves either finding a way to see one problem as an example of an easier problem, or looking at a bunch of different problems and seeing how they’re all really the same. So I’m not convinced that you wouldn’t have enjoyed real math if you’d hung in until you got there (usually starts after multivariable calculus). Sadly, we spend all of grade school and high school training people in the boring stuff, so by the time they get to the cool stuff they’re sick of it all.
I think at some point, my mind is just lazy: I want some magical way of knowing the boring stuff so I can get to the good stuff.
Yes. And it’s even worse with stuff you’re trying to teach yourself, outside of a classroom setting. If the task is difficult, and you’re on your own, it’s hard to telling the difference between simple laziness (work hard and you’ll be rewarded eventually) and being frustrated because you’re trying to do something that’s never going to work well, no matter how much effort you put into it.
It’s easier when the reward comes at the same time as the hard work, rather than being deferred.
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As an art major in college, I was TERRIBLE at math and stayed far away from the “math option” in college. But there’s nothing like reality to change things. When my husband and I started our own graphic design firm some 20 years after my college graduation I had a great epiphany when I discovered the concepts of “income” and “expenditures”! Suddenly math made practical sense and my little grey cells–the part of my brain dedicated to math skills and until then little used–clicked into action. I’m sure that several of my algebra teachers are rolling in their graves!
As Mary Carter notes, life circumstances can make skills easier to learn a little later. (In my case, it was returning to viola after 20 years of not playing it.) So I’d encourage you to go back to math and foreign languages in your copious free time.
For what it’s worth, I agree with those here on the squidginess of math. I’ve written about math standards earlier this fall, from a non-math standpoint (http://www.shermandorn.com/mt/archives/001035.html), but the general point is that if your primary weakness was making errors in calculations after getting the concepts right, that’s only a small part of math. ‘nough said about that, I think, especially since I’m probably one of the few history Ph.D.s who have also taken real analysis in college.
On foreign languages, I’m trying to tackle that myself, a little bit when I have time, by starting the easy way for someone with my skill level (and staleness of such): find a news/current-affairs podcast that has both English and Spanish versions. Listen to the Spanish one. Listen to the English one. Listen again to the Spanish one. Take two aspirin tablets. The one dual-language broadcast I’ve found (Democracy Now) allows me to (re)learn a bit of Spanish and avoid the more bombastic interviews. I’ll head to the Spanish version of Chinesepod and Coffee Break Spanish after I’ve brushed up on my Latinate current-events vocabulary. I don’t think there are any shortcuts, though, at least not for me.