How to Survive in Your Native Land II

The theme of the book, if we get down to it, is honesty in teaching. No question why it’s aggravating sometimes and inspiring others, why this guy Herndon grates on your nerves with his pompousness and his insistence that he’s got some way to do it, even when he’s more than forthcoming about his failures, failure after failure, and how he seems to cling to a vision that doesn’t work again and again, but then also sees, rightly, that the schools are doing the same, even worse, really, and when a colleague in the book says of the teachers, all teachers, “We’re the dumb class,” you really start to get what he means, because the teachers fail over and over in the same way, and never learn the thing they’re there to learn, which is how to teach, and Herndon is included here, so, ok, he’s honest, but still, Herndon, what do I do with this?

Here’s a quote, coming at the end of three pages on how now that he told the kids what they’re supposed to do, according to the school, but doesn’t actually force them to do it, he has time…

Time to talk about all that, without worry, since the official part of the school work is going on, or not going on, without your total involvement in it. Time to read your book in there too, look at the want ads in the paper if you feel like it, telling everyone to leave you alone, time to cut out of the class and go visit the shop or the art room or some other class to see what’s going on, knowing everyone will get along while you’re gone…

Time to live there in your classroom like a human being instead of playing some idiot role which everyone knows is an idiot role, time to see that teaching (if that is your job in America) is connected with your life and with you as a human being, citizen, person, that you don’t have to become something different like a Martian or an idiot for eight hours a day.

So what we’re talking about here is honesty, doing things because you want to do them and not because some bureaucracy (Noman, Herndon calls them) said that’s how it’s done.

And so Herndon bugs me sometimes, but still, I’m all for honesty in teaching, and in particular, I’m for honesty in mathematics, which we’re most dishonest about of all subjects (or else honest without knowing it, like the first grade teacher who tells a bunch of bright eyed first graders who love solving puzzles that it’s time for math even though none of them want to do it and it’s such an awful subject, and those kids learn her honest feelings and learn that they’re supposed to dislike the subject before they’ve even met it properly, so maybe this is a kind of accidental honesty, but really, why does the teacher do something she hates so much anyway, except she feels that she has to? And unfortunately, in the doing, she breaks the Hippocratic Oath teachers should take but don’t do first do no harm). And when I think about working with kids, I get excited because we’ll be able to work together, and I want to be surprised by their ideas and hear questions I haven’t heard before, and I want to really do math with them, because I’m a mathematician and so are they if they have half a chance, and if you do work in schools, you’ve got to be honest, I agree with you there, Herndon, and there will be all sorts of pressures telling you not to be, and I hope that if you’ve gotten the taste of the real joy of honest teaching then you’ll be inoculated against the dangers, just like when I saw real math (not in school) I had something in me that abided and which I could never lose, no matter how much of the fake stuff they threw at me. Cuz beauty is truth and truth is beauty and math is one place where you get both all wrapped up together.

A Sort of Maze

Link: A Sort of Maze

When I was a child, I went through a period of maze drawing. There was something deeply compelling to me in the question of how anyone can tell the good direction from the bad. They were a stand in for all kinds of creative activities, where the choice seemed, well, random. Try writing a melody. You have some finite number of choices for what the next note will be. Choose right, and you end up arriving at the end, and your prize is the thing you thought you heard in your head, but which wasn’t really all there yet, even though you were reaching for it, and it felt like it was there altogether.

So mazes were a help in understanding both creativity and randomness, which themselves, to me, seem inextricably, tensely connected to each other. John Cage (“The first question I ask myself when something doesn’t seem to be beautiful is why do I think it’s not beautiful. And very shortly you discover that there is no reason”) and Jackson Pollock (“I don’t use the accident. I deny the accident.”) are both artists, though one is all chance and the other claims to be none.

That’s what I think of when I look at a puzzle game like this one, late at night, thinking of myself as a child, who wondered if it could be possible to play a perfect game of chess just by accident, just because you were that lucky.

(I wonder too whether playing games like these are good for students. I think the answer is yes, though I imagine a physical version, like you might find at ThinkFun Games, would be even better. I had a student once who couldn’t do algebra to save her life. I gave her a Rush Hour-like puzzle. To solve it, she kept having to look beyond where she was at the moment and see the whole puzzle. After she solved it, she could, magically, find x. It was a surprising correlation. I don’t know where the inspiration to give her that puzzle came from…)

There was a blithe certainty that came from first comprehending the full Einstein field equations, arabesques of Greek letters clinging tenuously to the page, a gossamer web. They seemed insubstantial when you first saw them, a string of squiggles. Yet to follow the delicate tensors as they contracted, as the superscripts paired with subscripts, collapsing mathematically into concrete classical entities – potential; mass; forces vectoring in a curved geometry – that was a sublime experience. The iron fist of the real, inside the velvet glove of airy mathematics.

Gregory Benford, Timescape

Subversive Suggestions

I inherited from my dad a bookshelf of books on teaching, many of which were written in the sixties and seventies and feel as anachronistically radical as, say, the Declaration of Independence (…whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it[!!!]).

I’m reading How to Survive in Your Native Land now, but I just discovered this little list of suggestions from another that graces my bookshelf, called Teaching as a Subversive Activity. It’s a list of things that will never happen (some rightly, because they would be terrible suggestions from a practical viewpoint). Why? Because they’re all about shaking up institutional behavior, and putting people into situations where they have no choreographed, bureaucratically-administered response. It’s like when Kasparov played Deep Blue. To have a chance, he had to get it out of it’s pre-programmed library of openings as quickly as possible. Throw a couple of monkey wrenches in the machine, and people have to start acting like people again, instead of teachers, students, administrators, or whatever other role they’re supposed to fill. So we can safely predict that the institution will resist; a bureaucracy never accepts changes that threaten itself, and is hostile to change in general.

I like some of these suggestions a lot (11 could be quite interesting; 13 is probably a great idea, and I think they’ve adopted it here); others are deeply problematic (like 6. And 10, practically speaking). Consider the list for yourself, and, as a thought experiment, imagine what a school that subscribed to these rules (or better, the philosophy behind them), would look like. Would you want to go there? Would you predict disaster? There are private and charter schools that have adopted some of these suggestions.

Institutionalizing these suggestions, most likely, would be catastrophic. But holding the ideas in your mind makes space for the questions: what are we trying to do with these schools in the first place? and: is there any other way they could be?


By Postman & Weingartner

1. Declare a five-year moratorium on the use of all textbooks
2. Have “English” teachers “teach” Math, Math teachers English, Social Studies teachers science, Science teachers Art, and so on.
3. Transfer all elementary teachers to high school and vice versa.
4. Require every teacher who thinks he knows his “subject” well to write a book on it.
5. Dissolve all “subjects”, “courses”, and “course requirements”.
6. Limit each teacher to three declarative sentences per class, and 15 interrogatives.
7. Prohibit teachers from asking any questions they already know the answers to.
8. Declare a moratorium on all tests and grades.
9. Require all teachers to undergo some form of psychotherapy as part of their in-service training
10. Classify teachers according to their ability and make the lists public.
11. Require all teachers to take a test prepared by students on what the students know.
12. Make every class an elective and withhold a teacher’s monthly check if his students do not show any interest in going to next month’s classes.
13. Require every teacher to take a one-year leave of absence every fourth year to work in some other “field” other than education.
14. Require each teacher to provide some sort of evidence that he or she has had a loving relationship with at least one other human being.
15. Require that all the graffiti accumulated in the school toilets be reproduced on large paper and be hung in the school halls.
16. There should be a general prohibition against the use of the following words and phrases:
Teach, syllabus, covering ground, I.Q., makeup, test, disadvantaged, gifted, accelerated, enhancement, course, grade, score, human nature, dumb, college material, and administrative necessity.

From Problem to Question to Proof to Problem

I just had an absolutely wonderful meeting with a student I’m working with, a second grader by the name of Millan. The kid is a natural mathematician, and a joy to work with. Allow me to describe what happened today.

Every time we meet, he brings me a question; this is his central duty in between our meetings. In the past he’s asked me questions about cutting slices of spheres (inspired by an exhibit at the Pacific Science Center) and how multiplication by 9 works, and today, thinking of our previous work on projections, he asked:

Is there a shape you can shine a light on such that no matter how you turn it, you always get a square?

He was inspired by a particular property of spheres that we discussed: they always cast circular shadows. (This isn’t immediately obvious, and the implications are actually serious: Aristotle used this fact, along with the observations of the Earth’s shadow on the moon during lunar eclipses, to argue that the Earth was a sphere!) It’s a beautiful question. Probably too hard for a second grader to answer rigorously, but worth thinking about. If Millan wanted to think about it.

In fact, he did, and together, we managed to solve it. First, he noticed that the shape must have all square faces (and he clarified that square shadow should not change as the object rotates, so all the square faces must be identical). Why? Because you could always have your light source close enough to the face that the shadow is just a projection of the face. So now we had a new question:

What 3D shapes have all identical square faces?

This is still tricky, but more tractable than the other question. The cube is the obvious one (it unfolds in the drawing below) I observed that there must be a band of squares all in a row, like the four down the middle in the picture.

An unfolded cube. Glue same colored sides together to rebuild it.

So you get some kind of shape on the sides after you fold that band together, and then you need to fill in the empty parts of it with squares. After Millan did some careful analysis of the possible ways to do it, he realized that if the shape had any angle besides a right angle (or a straight angle), it wouldn’t work. We were able to dispose of all the options. The cube was the only choice (implicit here was a desire for convexity. Something to return to later).

The whole thing was elaborate enough that I thought we should write it up. Millan opened his notebook and wrote:

The only 3D shape with equal square faces is a cube.

To which I added, at the beginning:


And then, underneath:


We spent the next twenty minutes recollecting the argument and writing it down. It was an amazing accomplishment for a second grader, in my opinion. After it was done, I told him about the phrase “quod erat demonstratum,” and how mathematicians write QED at the end of their proofs. Since he’s now a working mathematician, I told him he deserved to write it at the bottom of his proof. And so he did.

But we weren’t done yet! He returned to his original question and polished it off. The only option for the shape that projects squares must be the cube, by his theorem, and the cube doesn’t work (shine a light from the corner and you get a hexagon). So after the followup question was answered, the original question was answered!

Does a mathematician stop there? Never! A mathematician sees in every solved problem an opportunity for a more general question or extension. I thought of asking Millan about what shapes could be built from triangles, but before I could, he suggested hexagons. And can you build a shape from hexagons?

Almost... there's those pesky pentagons in there. Why don't they just use all hexagons?

Almost… there’s just a few pesky pentagons in there. Why don’t they just use hexagons?

He’ll think about that for next time. What a pleasure, the young mathematician.

For a limited time, you can listen to this series on the history of mathematics on BBC radio. I’m hearing about Fourier right now, who, by the way, was apparently an excellent math teacher, who encouraged questions from his students and peppered his lectures with historical anecdotes.

Telling stories of mathematicians and from math history is actually one of the best reasons to lecture. The story of Galois (which you have 2 days left to listen to!) is one of my favorite stories, since it involves a duel, a death, and a letter. The Fourier one is quite good, though, especially because they give such a good description of the music, and how complicated sounds can be broken down into simple ones.

I’ve linked to Cantor, since that will be live longest. I haven’t listened to it yet, but Cantors work on infinity is one of the most incredible feats of human mental daring ever, and his life, well, full of pathos.

Thank you, BBC.

You can learn more about a person in an hour of play than you can from a lifetime of conversation


Speaking of How to Survive in Your Native Land, here’s a beautiful remark from that book, where the author is describing his colleague, Frank, teaching his students to diagram sentences.

He was the only man I’ve ever heard give a good answer to that old kid question, Why do we have to make these diagrams?

Why? Because they are beautiful, said Frank.

Thesis and conference

Last Thursday, I defended my thesis. The process was challenging, in that I have a tendency to be casual with certain details, and in this context I was called to task over each one of these. Most unexpected was being caught about a misplaced minus sign (not what you expect to be caught on in this context). Basically, I only cared if there was an arrow, but a member of my committee wanted to know whether it was pointing to the right or to the left.

It was a surprisingly difficult point.

In any case, I’m finished now, and I’m heading out of town on a brief vacation. Then I’m off to an annual conference in honor of R. L. Moore. Moore pioneered a technique of teaching math sometimes known as the Moore method, which is essentially a classroom setup that involves the students doing all the work: the professor provides a frame (some axioms and statements of theorems) and then the students have to prove virtually everything. It’s a great way to learn math. My math seminars at Swarthmore were usually taught in some variation of the Moore method, and when I got to graduate school I was shocked that we were back to boring old lecture.

Of course, the really hard work of math is always in the actual doing of math: the problem solving, and question asking. The question for the teacher is, how much class time should you spend on telling and showing students how to solve problems, and how much time letting them solve problems. It’s more time consuming to let them solve the problems, so you can “cover” much more material if you just lecture to your students. However, it’s very common to have students in this environment who feel like they understand but can’t really remember how to do stuff on their own. The gap between “knowing it in class” and “knowing it on the test” is vast, alas. Part of this is because we tell the students what to do but don’t devote class time and energy to helping them with the hardest part: doing it on their own. So a different approach which has the students working at the center may cover less material, but the students tend to understand it a lot better.

One variation on the Moore method that I’ve used a lot is the conceptual workshop (discussed in chapter 4 of Teaching with Your Mouth Shut, by my father, Don Finkel), where student groups work through a series of questions, which lead them to bigger ideas. I wrote a workshop for my differential equations class to lead them to invent a method to solve the simplest kind of 2nd order differential equations. In a purer view of the Moore method, you could just let students do all the work, and not break up the ideas into steps at all. Unfortunately, not every student is ready to attack a big problem right away. So the teacher must decide where to step in with a hint. Ideally, the students will be in that perfect blend of success and frustration where they stay engaged with the struggle. Too easy and there’s no point; too hard and students will give up.

It’ll be good to see what others are up to in the world of math teaching.

Why so few posts of late?… a thesis synopsis

Perhaps you’ve noticed the dearth of blogging lately here at mathforlove. Here’s the story: I’m defending my thesis—“On the Number of FM Partners of a K3 Surface”—this Thursday afternoon, so at the moment, I’m ensconced in preparation. Or I would be, had I not also gotten sick this past week, and been pretty much knocked out. 

But not to worry! I’m almost all better now, and getting to work again. Fortunately, most of what I need to do is done. I’ve written the thesis, my committee has more or less signed off on it. The final okay comes Thursday, hopefully, but no one has raised any red flags thus far. I’ve also written my talk. And I’ve even starting picking out the juiciest parts from my thesis for a paper, to be submitted to a journal later this summer. I’ve also been reading through the literature, and trying to anticipate what my thesis committee might ask. 

So what is left to do? Strangely, not too much. The defense will be an hour talk, followed by an hour of questions. I just need to be ready to answer whatever they might ask. As my advisor noted, what I’ll be facing is some extremely smart people who don’t know much about what I’m doing, so they’ll ask me about whatever it reminds them of that they’re interested in.

Let me take one opportunity to describe what my thesis is about here. (When I defend, I’m going to consciously try to remember not to be overly, metaphorically simple—mathematicians in this context want to see me be as dense as necessary, and handle the heavy language and the power tools. Whereas I’ve been trying for so long to explain mathematically deep ideas in simple language and stress the accessibility that I sometimes forget that I’m talking to an audience of geniuses who’ve been thinking about mathematical ideas for a living for years.)

A K3 surface… well, it’s a little technical to define. It’s sort of doughnut-like. But, weirdly, it’s actually not too important. So don’t worry about it. (Okay, an example of one is: graph x^4+y^4+z^4+w^4=0 in projective 3 dimensional space.)

An FM partner… well, the cool thing about K3 surfaces is that there’s a bunch of different ways of looking at them. There’s a way to keep track of the types of geometric objects that can live on them. There’s also a rather abstract way to keep track of a type of curves that might be on them. There’s also a completely different way to keep track of the curves that are hard to track algebraically. The details aren’t important, but what’s amazing is that all of these processes involve another K3 surface, which does all of this at the same time: it keeps track of the geometric objects on the first, its curves have the same structure in the abstract algebraic way, and the curves that are hard to describe algebraically have the exact same structure too. The new K3 surface is called an “FM Partner” of the first. 

The point is, we’ve got these K3 surfaces that are partners of each other. The question I was trying to explore is: can you figure out how many partners a K3 surface has?

Well, it’s a hard question. People knew a bunch of stuff before I started, and I’ve added a little bit to that. I looked into a specific example of K3 surfaces and gave a pretty decent count for those. I also showed that they could have a lot of FM partners—as many as any number you pick (not infinitely many though). I also showed how to study partners of partners. Turns out there’s a pretty nice structure to them.

There are a few things I would have liked to get that I haven’t yet. But they’ll have to wait till later.

Anyway, that’s my first and last attempt to explain what I’ve been up to for the past two years in a public place. I like problems that are accessible and understandable; it’s kind of a shame that the work I’ve been involved in is so opaque it’s difficult to share even with other mathematicians. But, alas, that’s how the field tends to be. It’ll be nice to get to devote more energy in the future to math that I can share with others.