Zen in the Art of Underlining

My father, were he alive, would have turned 67 this week. He died in 1999 from non-Hodgkin’s lymphoma. One of the treasures he left behind, or rather, collection of treasures, were his books. I just finished one of these books, called Zen in the Art of Archery

There’s a pleasure in reading books my father read, not least because he inked up his books; there are copious underlined passages and the occasional cryptic reference. (My favorite in this book is the underlined “The Master, long accustomed to my tiresome questions, shook his head” which is accompanied by a note in the margins: cf. Don Juan and Carlos, which I can only assume points here.) To track his notes is to follow his unfolding thoughts on this book and on whatever he was thinking at the time; it is to look through time to a moment when a man who would become a master teacher was my age, and when his ideas about teaching, the career to which he would devote his life, began to take shape.

Before he died, my dad wrote the excellent Teaching With Your Mouth Shut. What’s astonishing about the book is that it is written to be experienced rather than read: in exploring forms of teaching besides lecture, it actually invites the reader to become more than a passive “listener” to the thesis of the book. It’s a phenomenally cunning trick, actually, and I’m always impressed when I return to the book.

And here I am with a book he read, reading now to myself the underlined words: “You are under an illusion… if you imagine that even a rough understanding of these dark connections would help you.” My father believed in a vision of teaching that was less about transmitting facts and more about transformation, a kind of change that can be invited but not forced. You lead someone to a contradiction and let them struggle with it; you discover a paradox together and search for the extra dimension that resolves it. 

I see my dad now as someone who was just like me: a human being who loved struggling with great questions. I love to follow the signs of his struggle in the books he read, as if I were tracking someone’s path through the wilderness. 

“He is a living example of the inner work, and he convinces by his mere presence.”

Only the last clause is underlined here. I imagine my dad found the danger of the convincing presence, whether he was a living example of the inner work or not, the central point in his own reckoning. A teacher can inspire in positive and negative directions, and in a democracy we need students who are skeptical, even of the most convincing presences. On the other hand, he may have been thinking something else entirely. This is followed by:

How far the pupil will go is not the concern of the teacher and Master. Hardly has he shown him the right way when he must let him go on alone.”

Apt indeed. I’ll never know my father’s thoughts completely, but tracing their wake helps me better know my own. 

Happy birthday to you, Dad, the master teacher and perpetual student. May we all learn to play as seriously as you did. 

Murder on Gilligan’s Island

I went to a linguistics lecture in college to hear a speaker talk about how they had written Dr. Seuss type books for kids. One gem from that talk that always stayed with me is that there are three topics that are immediately interesting to everyone, across all cultures and age groups:

1. Death & Danger

2. Power and Status

3. Relationships between the sexes

After the speaker elucidated these, he pointed out that these are precisely the topics we scrub out of school curricula. In other words, the three things that everyone is guaranteed to find interesting are mostly absent from schools.

While I was teaching elementary school, I started embedding math problems in fairy tales (which I would write), and I tried to make the whole thing as exciting as I could. I found that it bought even the rowdiest classes an extra 10 minutes or so of focused work after I read it (dramatically). For example, I think I had an ogre in one that opened it’s terrible mouth, and had 7 beetles apiece on each of its 12 teeth. At the end, there was one question: “how many beetles are in the ogre’s mouth?” The kids had a printout of the story, and would need to look through it to find the setup, and then they could figure it out. A simple problem (and a relatively quick lesson), but the difference in focus was palpable. For a while I was thinking of writing dozens of them and publishing a little book of “back pocket math story/lesson,” which a teacher could pull out when they needed something fun and stimulating that didn’t require too much prep. Up till now, though, I haven’t pursued it.

How nice, then, to see this cool lesson on logarithms on Math Mama Writes, a really cool blog I’ve been reading lately. The setup is like a game of Clue on an island—someone’s been killed, and the kids have to do imaginary forensic work to try to figure out who did it (involving Newton’s law of cooling—it’s a lesson on logarithms).

I’m curious how well the lesson worked, but I have to say, I like the concept a lot. There’s a story that allows the kids to get involved right away, and it’s a matter of (pretend) life and death. Rousseau suggested a similar tactic in Emile, when he had the “test” for the application of the things he and his student had been working on in trigonometry come when he arranges for them to get lost at night and have to use the stars to find the way home. Now that’s motivation. Teachers today have to settle more often for games and stories, but if the execution is right, it can have a powerful effect. 

On the other hand, there are times when the math, unadorned, is the best motivation. Another problem I heard recently: how many squares are there on a chess board? (hint: not 64… consider the bigger squares). I think most kids (and adults) find something that motivates them to think about that right away. Still, nice to have both adorned and unadorned mathematics at your disposal. 

What skills do you need in order to teach well?

I often stress the perspective that the first thing you need do be a great teacher is love: love of your subject, love of teaching, love of learning. The skills you need to actually teach come naturally from the fundamental interest, care, and honesty.

But is this all? In the comments of the last post, Katie referred me to this recent article in the New York Times magazine, and since I read it, I’ve been trying to work out what I think of it. Briefly, there are a lot of small technical tricks to becoming a good teacher(check out the videos they provide. I particularly like the last two, about hand signals and joy). I’ve spent time working with different ages and different backgrounds, and the truth is that knowing some of the technical tricks for holding kids’ attention can be enormously helpful. The best way to learn these techniques is by teaching, reflecting on what works and what doesn’t, and observing other teachers and trying out what you see work for them. I’m a little dubious of any attempt to work these things out in an abstract setting. And even though these techniques can be effective, you have to teach in a manner that is honest for you. The idea that we can train teachers (or students) to learn a bunch of techniques that they can apply quasi-mechanically is a dangerous one, and it has a way of cropping up in the background of our national discussions of education.

The point is that if you are a teacher, you have to own your education as a teacher. It’s great to see these techniques. Some will work for you and some won’t. It’s up for you to decide what you want your classroom to look like. Knowing the possibilities is great, but you don’t have to be exactly like anyone else. 

On page 6 of the article, we see another important point:

ANOTHER QUESTION IS THIS: Is good classroom management enough to ensure good instruction? Heather Hill, an associate professor at Harvard University, showed me a video of a teacher called by the pseudonym Wilma. Wilma has charisma; every eye in the classroom is on her as she moves back and forth across the blackboard. But Hill saw something else. “If you look at it from a pedagogical lens, Wilma is actually a good teacher,” Hill told me. “But when you look at the math, things begin to fall apart.”

And here’s the rub: if you don’t know the math (or whatever subject you’re teaching), or don’t know how to start thinking your way into it, classroom techniques aren’t going to help you so much. It’s like the technical pieces of teaching are the cup, and the content is what goes inside. If the cup is cracked, not much is going to stay inside it, but you’ve got to have something to put in there too.

The other thing that gets me about the videos I linked to earlier is that they all highlight classes where run in a very similar way: teacher at the front, kids responding to them in something like a one-on-one way. In my experience, getting kids to work in groups is often the most effective tool a teacher has. I’d like to see more examples of this when it comes to effective techniques. My best classes I’ve ever had are the ones where I’ve been redundant in the classroom: the kids are working together, they’ve got some problem that completely absorbs their attention, and I, the teacher, am just hanging around, seeing what they’re up to.

Then I start thinking about the problem too, and there’s no teacher in the class at all; we’re all just people, seriously playing.

On Not Teaching Mathematics

A huge thank you to Jade, who referred me to this compelling argument for not teaching mathematics (or teaching less mathematics, or teaching it in a different way). 

Should we, as Gray argues in the link, dispose with the teaching of mathematics in elementary schools? Sadly, it would probably be a positive step. What I’d prefer to see, of course, is teachers who enjoyed and understood mathematics and could pass it on to their students. But at the moment, as Paul Lockhart so eloquently argues (and congratulations, Paul!), the situation is pretty bleak, and the desiccated body that passes for mathematics in some (most? virtually all?) schools needs to be scrapped. It’s like there’s a hack doctor claiming everyone needs more medical intervention, and we get sicker after each unnecessary surgery. The first rule of education should be Hippocratic: first, do no harm. There are a lot of things we need to stop doing in schools, and especially in math education, and probably the harm that we’re doing is greater than the good.

As Rousseau put it:

The training of children is a profession, where we must know how to waste time in order to save it

But that’s not really the end for me. I’ve been struggling with this for a long time, and I guarantee we’ll never see wide scale setting aside of math, and that’s not really the answer anyway. The answer is to teach relevant, appropriate, interesting math with kids and not worry so much about the rote mechanics till later. A real mathematics foundation is about courage, curiosity, finding and owning the discoveries and connections. It’s about love (thus, the name of this blog).   

One of my hats, as a matter of fact, is teacher of future teachers, and starting Monday I’ll be co-teaching a math course for future elementary teachers. And I will tell you that most elementary teachers do not much care for math. However, they want to do right by their students, and once they start getting to have a real relationship with math, they take to it. My goal is to have all of these teachers realize that they don’t need to cram stuff down their students throats (as it was once crammed for them), and that there’s a way to play, have fun, and discover naturally… and that everyone ends up learning more this way anyway.

Were the students under Benezet really not doing any math? They were just avoiding the hateful stuff of math. They were thinking, which is where the real stuff of math is. It’s a huge lesson.

Incidentally, I’m reading What’s Math Got to do with It, by Jo Boaler at the moment. I’ll have a proper review here eventually, but so far I’m very impressed with it as a primer for parents and teachers. We’ll be reading it for class. Updates to follow!

I wish I was special

There’s a great story in the beautiful, tragic, triumphant story of Ramanujan. Ramanujan emerged from India in the early 1900s (out of nowhere as far as Western mathematicians were concerned), and quickly emerged as the greatest natural mathematician of all time. He teemed up with Hardy and produced incredibly novel results. I have heard it said that he had a personal relationship with every number less than 10,000. Think about that. A personal relationship. These were his friends. And he needed them… he was out of place in England, sick in its winters, stranger to its customs. He died young, at the age of 32.

The classic story is that Hardy visited Ramanujan in the hospital during one of his bouts of illness, and remarked, making conversation, that he had taken a taxicab with a very boring number to get there. The number was 1729. Ramanujan responded, “No, Hardy! No, Hardy! 1729 is the first number that can be written as the sum of cubes in two different ways.”

What’s truly impressive about this, when you stop to think about it, is not that he knew that 1729 had this property… it’s that little word ‘first.’ No number below it has that property, and Ramanujan knew that. He had already checked them all. Or he just knew, the way we know which friend we need to go see when we need cheering up.

I was thinking about this because someone just asked this question: what’s the smallest positive integer that isn’t special? And that took me to this very interesting webpage, which shows that most integers have something special about them. There’s a paradox here, actually: the smallest number that isn’t special… that’s pretty special, isn’t it?

So right now I’m wondering: what’s special about the number 424?

Meanwhile, my little brother was playing this very song on the ukulele this morning.

"I am a mathematician, and I would like to stand on your roof."

Link: “I am a mathematician, and I would like to stand on your roof.”

Speaking of excellent talks, you should definitely watch this one. This is a look at African fractals, which are surprisingly ubiquitous throughout Africa, and which has some great conclusions, including:

1. The computer owes its creation to African fractal making (by way of Leibniz and Boole).

2. The answer to capitalist excess may be found in the self-organizing algorithms of African trade.

Also, there’s a clear story of how natural mathematics is to our story, as humans. This kind of pattern making springs out of the woodwork. This is a particularly cool example of it, though.

Confessions of a Converted Lecturer

Link: Confessions of a Converted Lecturer

This is an absolutely excellent talk, which I can’t recommend highly enough.

It’s not just that he talks about the problems with lecturing and the benefits of using other methods. What’s phenomenal about this is that he actually collects solid data, and scientifically argues his point. Anyone who’s interested in teaching, and especially those who are interested in teaching math or science should watch this talk.

It does make me consider my own teaching as well. I’ve long been an advocate of using other methods besides lecture, and have found ways to have my students really work stuff out for themselves in my differential equations class, for example. And yet, I haven’t really done as much as I could be doing in this context. I suppose the reason for this is that I feel like there’s no way to go over all the material that I have a curricular mandate to cover without lecturing. Also, the book we use is pretty weak. But now I wonder—surely there’s a better way.

I’m looking forward to the day when I can teach outside the auspices of any curriculum, and let the students’ learning take precedence over every other consideration.

In mathematics, the art of asking questions is more valuable than solving problems.

Georg Cantor