I am posting the following as a comment on this post on Reddit’s Wicked_Edge. I’m pretty sure Reddit doesn’t allow responses that long, but once I had the bit in my teeth, I couldn’t let up. And it was interesting to look back on a time when I was just making it up as I went along and to see that my instincts were right. But it was all ad lib. Then and, of course, now. Life is pretty much all ad lib: the art of drawing without an eraser, as John Gardner said.
Here begins my response in the thread:
Extremely good point. Sometimes—and in fact, we see it often—a person wants to postpone starting a skill (e.g., playing the piano) until he has mastered the skill. He dreads (and hates) the inevitable clunkers made in learning: the wrong notes, the wrong beat, the lack of expression, the crudity of phrasing… his adult (and, note, trained) ear detects it all, and it becomes intolerable. “Much better,” he decides, “that I wait until I can play without all those mistakes before I start learning.”
God, I’ve been there. Drawing. Still can’t do it, for exactly the reasons described above.
You’re totally right that people should jump in and start, and I see now not only the error I made, but why I made it—and many similar errors of the same genus (e.g., drawing, as mentioned above). Now that you’ve opened my eyes, I certainly will change my argument on this point.
I’m reminded of a freshman math tutorial I took over at the start of the second semester at my alma mater, St. John’s College, Annapolis, because of the untimely departure of the tutor. By the start of the second semester, all freshman tutorials are deep into Euclid’s Elements (no electives there: all students follow the same course of study), and the theorems are getting more challenging.
Even before this point in the Elements, some tutors will have adopted a kind of kindly Mr. Chips persona and will alert certain students—going through the entire tutorial, no favorites (though some tutors do play favorites and seldom call on certain students). No favorites in general, but the hardest theorems go to the students the tutor has decided are brightest, an odd strategy, since it is in opposition to the very idea of teaching, which is to help the ignorant, not the educated. The tutor merely suggests to the students whose turn it is that it might be a good idea if they glanced through theorem such-and-such before the next tutorial meeting.
That was pretty much their experience, waiting (as it were) for their piece of kibble, tutorial by tutorial. There were 11 students, as I recall, so I removed from a card deck the Joker, King, and Queen, and assigned each student a specific card value. We went around the class and each called his/her card: “Ace,” “deuce,” “three,” and on through “Jack.”
Then I explained: I’ll shuffle the cards before each class, and for each theorem, the student to demonstrate it at the board would be determined by the value of the top card in the (diminished) deck. So, I explained, “I am not selecting you, and I’m not asking for volunteers (the idea of “volunteer” becomes irrelevant). The gods will pick who goes to the board.
“And if your card comes up, you will go to the board, immediately. Whether you’ve even read the theorem is irrelevant. If the gods pick you, you must go. We’ll certainly help you through the demonstration—we’ll provide the words and arguments if you fail at any point. The idea is only to see the theorem demonstrated at the board, and the gods will pick the demonstrator. Feel honored, not frightened.”
And so I picked the first one, whom it’s kinder not to identify. This youth at first said, “Mr. Ham, I really didn’t prepare, and I’m sorry…” and I interrupted. “That’s not a problem, truly. Whether you’re prepared on not, the job is very simple: go to the board when your card turns up. The gods want you to.”
So the student went to the board. “Painful” ain’t in it. It was quickly apparent that the student had not so much as glanced at the theorem, and help was needed from the very statement of the theorem.
You can imagine: we went line by line through the entire theorem, help needed at every step. I think it took the full hour.
It was an unusual tutorial meeting, to say the least. But I thought it had been a learning experience.
So the next day, they troop in and take their seats (habitual, not assigned, but adhered to very closely; changing one’s seat at the table was not something most would do lightly).
I felt we had made progress the previous day. I took out the deck, shuffled, and turned up the first card. It was the same student.
As soon as the card turned, there was a very peculiar silence in the room. But—and they already accepted this—the gods had spoken. The student, without protest but with slumping shoulders, moved to the board and picked up the chalk listlessly. And then we reprised the previous meeting: again the student had not even looked at the problem, having unconsciously assumed that the gods would not pick the same student two days in a row, wasting a chance for someone else to demonstrate.
But the gods are hard to read and have their own ideas. So once again we stumbled, line by line, through the demonstration, in a kind of Euclidean call and response. (A student must utter aloud all the words of the theorem and proof; saying “what you said” is simply beyond the pale—not so much “not allowed” as “unthinkable.”)
It was easier this time. The student at the board was more accustomed to how to do that sort of demonstration efficiently, having had plenty of practice the previous day, and the chorus in their seats were more on their toes because of their own previous practice. But, easier or not, there was still the embarrassment.
And that completed the lesson. From that day forth every student was prepared on every theorem for every tutorial meeting (and, indeed, perhaps this carried through to the other tutorials and the seminar).
Naturally, learning the theorems became easier and easier. Patterns emerged and were recognized. In learning one theorem, a student would see that this was just a twist on a previous theorem. “Gimme a 47B without the motor,” said in a cartoon about a shop selling custom memorial wreaths for Graceland. “I just do the previous theorem, except do this differently,” the student thinks, having begun to build from what’s been learned the net that easily catches new ideas and incorporates those into the net.
So soon the whole class was comfortable with the theorems, and being selected by the gods did, indeed, become somewhat an honor—at least I think so—even though it was admittedly sometimes a bit scary. But if the gods picked you, you went to the board.
To round out the memory: the next time I had to be away from the tutorial on a recruiting trip (I was also director of admissions), I asked Miss Leonard to take my place. She had been a Euclid tutor often, so needed no prep, and she agreed.
The day arrived, and she entered the room. By this time they were in Book XIII, the construction of the (only) five regular solids that are possible in 3-space: the Platonic solids. These are daunting theorems, and at this point the universal practice was to assign a theorem explicitly to some chosen student, aloud, in class, the day before the demonstration, so the student could memorize the theorem as though memorizing a speech in a play—that is, word-perfect, but sometimes with little (sometimes no) grasp of the meaning of what they were saying. It could be like a fifth grader reading aloud a speech in Macbeth about the drive and demands of ambition, and they of course have not the remotest idea of what ambition can be and can do to people. They totally miss the point.*
So Miss Leonard naturally asked, “Okay, who’s prepared for the next theorem,” expecting the student chosen the previous day to volunteer, and things would move smoothly along with no hitches: perfection unmarred, just as the adult beginning piano player wants to play from the start. But, of course, it is through mistakes that we learn. Avoiding mistakes means avoiding learning.
The students cried, “The cards! The cards! Didn’t Mr. Ham give you the cards?!”
Miss Leonard was totally baffled. “Cards? What cards?”
“A number!” someone said, and they all chimed in. “A number! Pick a number! Between 1 and 11.”
So Miss Leonard said, “Eight,” and one student groaned and stood up and trudged to the board, and she absolutely nailed the theorem: stated the general theorem, drew and labeled the diagram (the “setting out”), restated the theorem in terms specific to the diagram (the “statement”), did any construction (and at this point, all the theorems required construction), and then gave the logic of the proof, with understanding, able to restate things, able to answer questions, able to point out any nice touches.
Miss Leonard’s jaw dropped. Later, she told me, “A random studen—did they all know it?”
“Sure,” I said. For one thing, at that point learning the next theorem was not so difficult. It was like learning a new verb in a foreign language: it has its unique points, but it works pretty much like any other verb, is conjugated in its family, either the obvious family or, if not that one, then it’s memorable. Learning each new theorem, in the light of all the previous theorems, is not a biggie.
In sum, they learned Euclid well, learned how to prepare and to present, and learned not only the value of good preparation but also how it feels, the feeling of actual learning and understanding—what you’re going for. Once they learn that, they can apply it in many fields and contexts.
The path of learning goes through mistakes, and if you take that path, either at the direction of the gods or voluntarily because you now understand how it works, then you learn. What helps is to focus your attention more on your progress than on your current level of accomplishment. Progress will be stunning when you start, though the level of accomplishment will doubtless leave much to be desired. Eventually you grasp the pattern of effort/mistakes/effort followed by learning. You find the rhythm. The rate of progress will inevitably slow, but by then the performance is rewarding. Demonstrating, playing the piano, or shaving with a straight razor.
*Stringfellow Barr, one of the two founders of the St. John’s Program, saw Macbeth when he was quite about 7 or 8 (I can’t recall, but he was precocious), and for days afterward went around in a cape (dishtowel tied around neck), pretending to be Macduff. He told us, “To read Macbeth and think that Macduff was the main character is about as bad a reading of the play as is possible.”
On seeing how much he liked the play, his father told him, “Did you like that? He wrote a lot more.” And the young Stringfellow started reading through the plays. (I mentioned he was precocious.) When, a year or so later, he read Macbeth, he thought, “You idiot! You dolt! You totally missed the entire point of the play!”
“And,” he told us, “Ever since, every time I reread the play, I feel the same way about my previous reading.’
Post Script: I did have to modify the rules a bit. Each time you were hit, one of your cards was taken from the deck, to decrease the odds you got dinged twice in a row. But you always had to have more than one card. And after everyone had demonstrated, the deck was restored and reshuffled.
Also, I learned not to deal the cards out one by one during the class. Until the last card was shown, everyone was quickly reviewing the next theorem in case his number came up. Instead, I’d deal out as many cards as I expected theorems to be demonstrated, plus one for safety. Then I’d turn them all over at once. That way, although those four might be reading ahead, everyone else would follow what was going on, since they knew they would not be presenting.
Filed under: Books, Education, Shaving
