Avery Pickford is a teacher I used to work with. He regularly beat me at scrabble (and I’m pretty formidable in most crowds), and he taught me ultimate tic-tac-toe, where you add a box after every turn, and need to get four in a row to win. Now, he’s just started a blog. The first entries include a description of what makes great problems great and a call for questions. Welcome to the online conversation, Avery!

## The End of Strogatz’s Series in the Times

Link: The End of Strogatz’s Series in the Times

It’s been a nice run, and tremendously well received. Here’s the last installment of Steve Strogatz’s New York Times math articles. This one actually tries to prove that there are different sizes of infinity. It’s a tricky argument, but if you stick with it, I think it can be followed.

## Addendum to post on a free education

Link: Addendum to post on a free education

I just found this effort to use TED talks as high school curriculum. Nice idea.

## Why do we keep paying for what’s free?

Matt Damon said it first, in Good Will Hunting: one day, you’re going to realize that

You dropped 150 grand on a $%*#ing education you could have got for $1.50 in late charges at the public library.

Here’s the paradox of college in the age of the internet: students are paying more and more for what’s available for free. Why does this happen?

Today, it’s possible to go to, say, MIT, for free. Except you don’t have to go. Let’s say you want to learn calculus. From home you can attend lectures, download your textbook, and even do all the homework and take the exams. If you compare this textbook by Strang to a standard, written-by-committee monstrosity, it’s actually written by a human, for humans. Here’s page 2, talking about comparing the odometer and the speedometer on a car [v stands for the velocity, the reading on the speedometer; f stands for distance travelled, the reading on the odometer]:

If we know the velocity over the whole history of the car, we should be able to compute the total distance traveled. In other words, if the speedometer record is complete but the odometer is missing, its information could be recovered. One way to do it (without calculus) is to put in a new odometer and drive the car all over again at the right speeds. That seems like a hard way; calculus may be easier. But the point is that the information is there. If we know everything about v, there must be a method to find f. What happens in the opposite direction, when f is known? If you have a complete record of distance, could you recover the complete velocity? In principle you could drive the car, repeat the history, and read off the speed. Again there must be a better way. The whole subject of calculus is built on the relation between v and f.

The question we are raising here is not some kind of joke, after which the book will get serious and the mathematics will get started. On the contrary,

I am serious now-and the mathematics has already started. We need to know how to find the velocity from a record of the distance.(That is called differentiation, and it is the central idea of differential calculus.) We also want to compute the distance from a history of the velocity. (That is integration, and it is the goal of integral calculus.)

Much nicer than your average math text, wouldn’t you say? What was the last textbook you saw that used first person pronouns? And it’s free.

Economically, something doesn’t click in all this for me. The content of college, more now than ever, is free and easily available. What are students paying for?

One answer, of course, is the pedigree (this is what the longhaired guy answers Matt Damon in the clip above). But it’s hard for me to believe that it will last forever: actual skills are starting to matter more than ever. The more important impediment is that it’s just hard to motivate yourself to watch twenty hours of lecture, do sixty hours of homework, and check yourself to make sure you’re really understanding without other students around you doing the same thing, and without a teacher or TA to turn things in to, and to check out how you’re doing (though given how large so many lecture classes are, many students barely check in with anyone anyway).

Here’s the fix then: I should offer a calculus class run in the following way: a group of 10 kids meets with me twice a week for an hour each. In between classes, they watch the best teacher in the country (someone at Harvard or MIT, say) lecture on calculus (free); they read the materials from that part of the (free) calculus text and do the assigned homework problems from the syllabus (free). When they meet with me, I don’t lecture at all (why would I? They get that for free already); rather, they bring their questions, their difficulties, and we discuss together to correct misconceptions and strengthen real understanding. If everyone’s on top of everything, I’ll assign a particularly challenging problem to stretch their thinking further. They all grade each other’s homework, ensuring that they actually write for people, instead of a nebulous grader, and if they’re homework is poorly done or arranged, they’ll hear about it from each other. There will be quizzes/tests every three to four weeks that I grade to make sure they’re actually learning and absorbing all the material. These could be adapted easily from the MIT syllabus. And I’ll charge $1000 per student.

I’ve just designed a calculus class that is demonstrably better than MIT’s (I give the students everything MIT offers, plus 3 extra hours of dedicated attention a week in small group) for a fraction of the cost. Any high schoolers interesting in taking calculus and actually understanding it—not just having to repeat it when they get to college—should come flocking to me, right?

If the economics are right, and people actually care about what they’re getting, then I should have no trouble filling up my calculus class. If you know any interested students in Seattle, refer them to this post. If I get 10 students on board, I’ll actually do this next September. This began as a thought experiment, but if students respond, let’s make it a real experiment.

Will it happen? We continue to buy bottled water, even though it’s often “glorified tap water,” actually from the same source as tap water, but sold at as much as a 1000 times the price. Why are we irrational in this way? Why do we keep paying for what’s free?

## Question, Questions

I don’t know why it’s taken me so long, but I’ve decided to start teaching the art of inquiry to everyone I tutor. Step 1: all my private students will have the homework of bringing in a question to every tutoring session. I’ve run classes on that principle, but for some reason I never applied it to my tutees. I’ve finally started to catch up to myself.

I’ve been reflecting on the difference between teaching high school students and elementary school age kids. There’s a way that I have an affinity for the older kids sometimes: they’re able to connect dots that are further apart, and there’s a very clear arc to the subjects that they’re working on. The younger kids are great too, though. One thing I’m considering is how best to structure my time with them, since they often don’t have curricular need from their student.

One parent offered me some worksheet books. I haven’t looked at these yet, but I think I will. Often, math workbooks are stilted or problematic in different ways, but there are usually great things in them, and younger kids often need more structure. Totally open questions don’t always occupy them in the same way, or for as long. (The first time I taught fourth grade, coming from high school, the difference in attention span was a huge challenge. You’ve got to break things up into smaller bites.) And because there isn’t as much mathematical infrastructure built up in their minds already, there are fewer places to latch things on. And while I have great curricular arcs, it might be time for me to start adapting some preexisting work to my (and my students’) needs, so I can give them worksheets (as opposed to open questions) to take home with them. Part of it depends on whether they need/want homework from me at home anyway.

Questions, of course, are the best medicine for kids of all ages, and I developing the habit of articulating questions will help them prep the ground for the more major architecture to latch on. The math on it’s own, of course, is always so compelling. Yesterday evening a student ended by saying, “It’s really great to actually understand this stuff rather than just memorize it.” For so many kids, learning math is like memorizing monologues in foreign languages; the passion that originally motivated the work is thoroughly obscured. We (the student and I) had started with the law of sines, but quickly worked backwards to definitions of trig functions via circles and then the Pythagorean Theorem. And some remarks about the square root of 2, and how it upset people. How they called it “Alogon, the unutterable.” How they said:

It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantly destroyed and shall remain forever exposed to the play of the eternal waves. —Proclos

For just a breath of that passion today. (Not too much… I don’t need to see people killing each other over these things either. As they have in the past.)

The good news is that once you reveal to the student that all this work was motivated by questions (how long is the diagonal of a square?) in the first place, the field opens to them, and in most cases becomes captivating pretty darn quickly.

What about the younger kids? I met with a second grader today and told him he’d need to bring me a question each time we met from now on, and asked him if he had any for me today. He asked how kaleidoscopes make triangles (and is that all they make?). So we talked about projective geometry, as in the shadow cast by different shapes. Then I pointed out that if you want to know what types of shadows a circle cast, it should be the same as looking at sections of a cone. And then we talked about the reflective properties of those shapes, and how you can build an elliptical pool table with a hole at one foci and the ball at the other, and any direction you it it, it will go in the hole. And then we talked about lasers bouncing around inside circles (which you could imagine as circular billiards), and, as an immediate connection, star polygons, and finally, as a direct consequence, prime numbers. This gives me about eight directions to go for next week. If I brought in a curriculum, I might be able to guide it in the way I thought best. If he brings in a question (or several) and we let that question guide us, I think it will be best of all.

## Accolades and Puzzles

Math for Love just made the list of 50 best blogs for math majors!

In honor of that, I’m going to point to one of the other blogs listed there, which I expected not to like, but actually found pretty fun: MAA MinuteMath.

When I see the words “minute” and “math” together, a shadow reaches into my internal landscape, as I’m made aware yet again of the cloud, the blight, that is the misguided “skill building” effort. Not that there’s anything wrong with skill building; there’s nothing wrong with locusts either—it’s just that when you see them, it’s usually the sign that there’s too many of them coming, and there won’t be a lot of other sustenance around for a while.

But the MinuteMath blog is another thing entirely: a collection of the kind of “cute,” harmless, occasionally maddeningly hard puzzles that, if you’re in the right kind of mood, give way to satisfying answers very quickly. It’s the mathematician’s equivalent of the Tuesday crossword puzzle in the Times. (These problems actually come from the AMC math contests for 8th-12th graders).

Here’s an example: What’s the probability that a randomly picked factor of 60 is less than 7?

You could do this by writing down all the factors of 60, and see how many are less than 7. But that’s boring and slow. Or, you might imagine that if you have a*b=60, then one of the numbers (say a) has to be less than the other, and in particular, it has to be less than the square root of 60 (or else a*b would be bigger than 60). And the square root of 60 is somewhere between 7 and 8 (since 7*7=49 and 8*8=64), so half of every pair is 7 or less. But 7 isn’t a factor of 60, so exactly half of its factors must be less than 7. Done!

Of course, it happens much more smoothly and satisfyingly in your own head. The writing is a little clumsy, and is mostly meant to point the right direction for you to have the idea. Still, I hope it follows, for those interested in thinking about it.

I’m not sure why puzzles are so attractive. In Ancient Puzzles, Olivastro refers to them as the detritus of intellectual exercise. Somehow, they’re the unimportant but amusing leftovers of discovery, but we get to relive the sense of discovery through them. And they seem unimportant. Even though no one will conceivably use my work on K3 surfaces in any practical way, it feels like I’m involved in an important pursuit, at least by the standard of mathematical endeavor. On the other hand, I avoided combinatorics and graph theory, fields where I’m arguably more intuitive, because they didn’t feel as “important.”

(Though probabilistic graph theory turns out to be one of the most applicable, important, and hot areas of math going right now. And arguably, I only became an algebraic geometer because my advisor played basketball with me and asked me to work with him. So perhaps I’m wrong in every respect.)

In any case, the appeal of puzzles is undeniable. If you’re a math major (my new audience), or someone with a taste for tricky, little puzzles that don’t go anywhere in particular but take some insight to solve (which is exactly what I’m in the mood for at this hour), check out the MathMinute blog.

If you feel like more accessible problems that occasionally point in interesting directions, you might take a look at Saint Ann’s problems of the week. They’re great for kids and adults, and some are great lesson builders: beginning problems leading to deeper principles and questions. Consider this problem for example. If you ask yourself one additional question—what is area of each square—and stick with it for a while, you’ll almost have to end up with the Pythagorean theorem, or at the very least it’s motivation. There’s a great, deep idea embedded in that one little problem.

## My Advisor and His Wife

Link: My Advisor and His Wife

They always seemed like a great couple to me, and I can’t resist referring to this article on the two of them. The artist and the mathematician: both tapped in to the creative process.

They also believe that both mathematics and contemporary art are widely misunderstood.

“For him, math is not about numbers,” Tihanyi said. “For me, art is not about pretty pictures.”

## 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.