Month

“The area of a circle, we know, is

A = (pi)*r*r,

where r is the radius of the circle. Most of us first learned this formula in school with the justification that teacher said so, take it or leave it, but you better take it and learn it by heart; the formula is, in fact, an example of the brutality with which mathematics is often taught to the innocent.”—Petr Beckmann,*A History of Pi*

A = (pi)*r*r,

where r is the radius of the circle. Most of us first learned this formula in school with the justification that teacher said so, take it or leave it, but you better take it and learn it by heart; the formula is, in fact, an example of the brutality with which mathematics is often taught to the innocent.”—Petr Beckmann,

On Vacation

I’m on vacation in Yellowstone this week, visiting my amazing girlfriend Katherine. She’s also a mathematician, but she’s much more than that. Since I won’t be updating with any long posts for a few more days, I’d like to refer you to her blog, which records her adventures on her bike trip from Banff, Canada to the U.S.-Mexico border.

If anyone is still holding on to the stereotype that math people are straight laced and not well rounded, her stories of racing wildfires, confronting bears, and biking through lightning storms should break those stereotypes wide open.

The Valedictory Speech

Thanks to my sister, Zoe, for blogging about Erica Goldson’s incredible, devastating valedictory speech, available here, and also copied to the bottom of this blog post. It’s always powerful for me to see my father’s words; they’re so often surprising, challenging, and inspiring.

I don’t think I have much to add at the moment. This speaks for itself.

Here I stand

There is a story of a young, but earnest Zen student who approached his teacher, and asked the Master, “If I work very hard and diligently, how long will it take for me to find Zen? The Master thought about this, then replied, “Ten years . .” The student then said, “But what if I work very, very hard and really apply myself to learn fast — How long then?” Replied the Master, “Well, twenty years.” “But, if I really, really work at it, how long then?” asked the student. “Thirty years,” replied the Master. “But, I do not understand,” said the disappointed student. “At each time that I say I will work harder, you say it will take me longer. Why do you say that?” Replied the Master, “When you have one eye on the goal, you only have one eye on the path.”

This is the dilemma I’ve faced within the American education system. We are so focused on a goal, whether it be passing a test, or graduating as first in the class. However, in this way, we do not really learn. We do whatever it takes to achieve our original objective.

Some of you may be thinking, “Well, if you pass a test, or become valedictorian, didn’t you learn something? Well, yes, you learned something, but not all that you could have. Perhaps, you only learned how to memorize names, places, and dates to later on forget in order to clear your mind for the next test.School is not all that it can be. Right now, it is a place for most people to determine that their goal is to get out as soon as possible.

I am now accomplishing that goal. I am graduating. I should look at this as a positive experience, especially being at the top of my class. However, in retrospect, I cannot say that I am any more intelligent than my peers. I can attest that I am only the best at doing what I am told and working the system. Yet, here I stand, and I am supposed to be proud that I have completed this period of indoctrination. I will leave in the fall to go on to the next phase expected of me, in order to receive a paper document that certifies that I am capable of work. But I contest that I am a human being, a thinker, an adventurer - not a worker. A worker is someone who is trapped within repetition - a slave of the system set up before him. But now, I have successfully shown that I was the best slave. I did what I was told to the extreme. While others sat in class and doodled to later become great artists, I sat in class to take notes and become a great test-taker. While others would come to class without their homework done because they were reading about an interest of theirs, I never missed an assignment. While others were creating music and writing lyrics, I decided to do extra credit, even though I never needed it. So, I wonder, why did I even want this position? Sure, I earned it, but what will come of it? When I leave educational institutionalism, will I be successful or forever lost? I have no clue about what I want to do with my life; I have no interests because I saw every subject of study as work, andI excelled at every subject just for the purpose of excelling, not learning. And quite frankly, now I’m scared.

John Taylor Gatto, a retired school teacher and activist critical of compulsory schooling, asserts, “We could encourage the best qualities of youthfulness - curiosity, adventure, resilience, the capacity for surprising insight simply by being more flexible about time, texts, and tests, by introducing kids into truly competent adults, and by giving each student what autonomy he or she needs in order to take a risk every now and then. But we don’t do that.” Between these cinderblock walls, we are all expected to be the same. We are trained to ace every standardized test, and those who deviate and see light through a different lens are worthless to the scheme of public education, and therefore viewed with contempt.

H. L. Mencken wrote inThe American Mercuryfor April 1924 that the aim of public education is not “to fill the young of the species with knowledge and awaken their intelligence. … Nothing could be further from the truth. The aim … is simply to reduce as many individuals as possible to the same safe level, to breed and train a standardized citizenry, to put down dissent and originality. That is its aim in the United States.”This was happening to me, and if it wasn’t for the rare occurrence of an avant-garde tenth grade English teacher, Donna Bryan, who allowed me to open my mind and ask questions before accepting textbook doctrine, I would have been doomed. I am now enlightened, but my mind still feels disabled. I must retrain myself and constantly remember how insane this ostensibly sane place really is.

And now here I am in a world guided by fear, a world suppressing the uniqueness that lies inside each of us, a world where we can either acquiesce to the inhuman nonsense of corporatism and materialism or insist on change. We are not enlivened by an educational system that clandestinely sets us up for jobs that could be automated, for work that need not be done, for enslavement without fervency for meaningful achievement. We have no choices in life when money is our motivational force. Our motivational force ought to be passion, but this is lost from the moment we step into a system that trains us, rather than inspires us.

We are more than robotic bookshelves, conditioned to blurt out facts we were taught in school. We are all very special, every human on this planet is so special, so aren’t we all deserving of something better, of using our minds for innovation, rather than memorization, for creativity, rather than futile activity, for rumination rather than stagnation? We are not here to get a degree, to then get a job, so we can consume industry-approved placation after placation. There is more, and more still.

The saddest part is that the majority of students don’t have the opportunity to reflect as I did. The majority of students are put through the same brainwashing techniques in order to create a complacent labor force working in the interests of large corporations and secretive government, and worst of all, they are completely unaware of it. I will never be able to turn back these 18 years. I can’t run away to another country with an education system meant to enlighten rather than condition. This part of my life is over, and I want to make sure that no other child will have his or her potential suppressed by powers meant to exploit and control. We are human beings. We are thinkers, dreamers, explorers, artists, writers, engineers. We are anything we want to be - but only if we have an educational system that supports us rather than holds us down. A tree can grow, but only if its roots are given a healthy foundation.

For those of you out there that must continue to sit in desks and yield to the authoritarian ideologies of instructors, do not be disheartened. You still have the opportunity to stand up, ask questions, be critical, andcreate your own perspective. Demand a setting that will provide you with intellectual capabilities that allow you to expand your mind instead of directing it. Demand that you be interested in class. Demand that the excuse, “You have to learn this for the test” is not good enough for you.Education is an excellent tool, if used properly, but focus more on learning rather than getting good grades.

For those of you that work within the system that I am condemning, I do not mean to insult; I intend to motivate. You have the power to change the incompetencies of this system. I know that you did not become a teacher or administrator to see your students bored. You cannot accept the authority of the governing bodies that tell you what to teach, how to teach it, and that you will be punished if you do not comply. Our potential is at stake.

For those of you that are now leaving this establishment, I say, do not forget what went on in these classrooms. Do not abandon those that come after you. We are the new future and we are not going to let tradition stand. We will break down the walls of corruption to let a garden of knowledge grow throughout America. Once educated properly, we will have the power to do anything, and best of all, we will only use that power for good, for we will be cultivated and wise. We will not accept anything at face value. We will ask questions, and we will demand truth.

So, here I stand. I am not standing here as valedictorian by myself. I was molded by my environment, by all of my peers who are sitting here watching me. I couldn’t have accomplished this without all of you. It was all of you who truly made me the person I am today. It was all of you who were my competition, yet my backbone. In that way, we are all valedictorians.

I am now supposed to say farewell to this institution, those who maintain it, and those who stand with me and behind me, but I hope this farewell is more of a “see you later” when we are all working together to rear a pedagogic movement. But first, let’s go get those pieces of paper that tell us that we’re smart enough to do so!To illustrate this idea, doesn’t it perturb you to learn about the idea of “critical thinking.” Is there really such a thing as “uncritically thinking?” To think is to process information in order to form an opinion. But if we are not critical when processing this information, are we really thinking? Or are we mindlessly accepting other opinions as truth?

2 Probability Puzzles

Here’s a really neat puzzle that Paulos mentions in *Innumeracy*, which I’m now in the middle of, answered by von Neumann. The second, since we’re on the topic of probability, I can’t resist putting in since it’s my favorite coin problem of all time:

- What’s a way to make a fair game with an unfair coin? Say the coin is skewed so that it lands on heads 60% of the time and tails 40% of the time (for example). How can you make a game you have a 50% chance of winning, as you would by flipping a fair coin once?
- How do you make a game you have a 1 in 3 chance of winning with a fair coin?

Answers will be in the comments later, from you or from me.

On Danger

Of the stacks of math and pop math and math & culture books now cluttering my table, I just opened one and laughed out loud. The book is *Innumeracy*, by John Allen Paulos, and and the discussion was about the relative danger of different threats. Which should we be more worried about?

- Choking to death on a piece of food.
- Getting killed by terrorists on a trip abroad
- Dying in a car accident

Well, circa 1985, the answers are that you have a 1 in 68,000 chance of choking to death; a 1 in 1,600,000 chance of getting killed by terrorists when traveling, and a stunning 1 in 5300 chance of dying in a car accident. Given these figures, my response is to say, “I’m never going to worry about getting killed by terrorists, I’m going to be an extremely conscientious driver, and I’ll try to remember to chew my food well.” Which is actually how I live my life, roughly (I know it’s 25 years later, but I doubt the statistics have changed much). If you’re familiar with the mathematics, it helps you assess risks, and not pay too much attention to those (like sharks attacks, terrorists) that aren’t likely to affect you.

Here’s what made me laugh:

Confronted with these large numbers and with the correspondingly small probabilities associated with them, the innumerate will inevitably respond with the non sequitur, “Yes, but what if you’re that one,” and then nod knowingly, as if they’ve demolished your argument with their penetrating insight.

Funny because it’s true!

Really big and really small numbers aren’t intuitive for lots of us. I love visualizations of them if you know any. A few of my favorites are:

and of course, the amazing classic Powers of 10, which I didn’t realize was available online! (I think it may be the older of two versions.)

There’s much more to the innumeracy that Paulos is talking about than not understanding orders of magnitude: not understanding chance is huge as well. These things are counterintuitive, of course, because we didn’t need them to live until quite recently in our species development. Now, cultural numeracy is vital.

My doppelganger heard the starting gun go off...

You’re going to see a shallow, petty side of me in this blog.

I know that there is a tremendous need to bring the beauty of math to the culture at large, and that I can’t do it alone.I know that a lot of intelligent, passionate people are working on the issue, and that there’s room for all of us.

Still, I feel a little possessive when people horn in too closely on what I think of as *my territory*. Like that pop math book I’m planning to write this year. Or the radio program/podcast on math I was going to launch with my girlfriend.

So, it’s with mixed feelings that I share with you Tom Henderson (interviewed here), a man whom I’m destined either to work with or hunt for sport. Why? He beat me to the punch on starting a math podcast. Worse, he’s writing a book, called Punk Mathematics, which will no doubt cover a lot of the same ground as my future book, and for a similar audience (In fairness, I’m not too hooked in with the punk scene, and whatever book I write will probably be substantially different. And it probably won’t have swearing). And like me, he does improv comedy. It’s like racing your f-bomb dropping doppelganger, and he has a head start. (Doppelganger is strong, I suppose; I’m sure there are substantial differences in our writing styles, presentations, philosophies, etc. But we do also both have beards.)

Most impressively, Tom used kickstarter to raise an astonishing amount of money to support his book writing process. If you would like to donate to his cause, do, by all means… but if you do you also have to promise to donate to me as well when I use kickstarter to raise money for my book. He appears to be the first person to use the site for a math related idea, and, in keeping with my contention that we, as a culture, are hungry for mathematics, he has one of the all time most popular pitches on the site.

My hat is off to you sir.

This post is an addendum to the last post

Great comments in the last post! In the spirit of self referential comedy, I have to include one of the suggestions mentioned there:

If you go to the original comic, you get an extra self referential joke in the rollover.

The courage to dream

One reason it’s exciting to be in education right now is that business (well, some business) is on the side of great education.

A great team, and tons of meaty problems to solve. … It’s open, collaborative. … We’re facing problems that are pretty unusual. … We take the smartest and most passionate team-oriented people we can find and put them in an environment where they can thrive. We value innovation, teamwork, and good clean fun. … We’re still a small company, so one person can make a big impact.

Now if you worked at this company (it’s Meebo), and were talking to schools about how you wanted them to teach, what would you recommend?

In fact, business leaders have been asked, and here’s what they say they value most highly. Here’s the they want, according to the study *Tough Choices or Tough Times*, from the National Center on Education and the Economy:

Strong skills in English, mathematics, technology, and science, as well as literature, history, and the arts will be essential for many; beyond this, candidates will have to be comfortable with

ideas and abstractions, good at bothanalysis and synthesis,creative and innovative, self-disciplined and well organized,able to learn very quickly and work well as a member of a team and have the flexibility to adapt quicklyto frequent changes in the labor market as the shifts in the economy become ever faster and more dramatic. [my emphasis]…

That kind of leadership does not depend on technology alone. It depends on a deep vein of creativity that is constantly renewing itself.

None of this seems too surprising. Aren’t these the skills you’d expect to be in high demand in the information age?

My contention, of course, is that mathematics is a phenomenal way to learn creativity and flexibility (and even writing!). I hope I’ve managed to give examples of what learning mathematics can look like in earlier blog posts. Does that picture resemble what you learned in school? It doesn’t for me. If I hadn’t gotten exposed to the beautiful ideas of math outside of school, I don’t think I would have kept doing it.

We are at a moment of possibility. Gone are the robber barons of last century, wanting mindless factory drones. We can allow ourselves to question what is possible in a classroom, and at a school. Some phenomenal schools have already sprouted up, local answers to this question.

Consider the essay here, which ends like this:

Current educational “reform” is a smoke and mirrors distraction. For decades reform has been a series of piecemeal attempts to do the same thing we have always done, just differently. The real question, “Should we even be doing what we have always done?” is not being asked. What we need, is to reimagine school from the ground up, drawing on the truths we have learned about how humans learn. We need to take what we know about the power of environments that encourage and nurture creativity and innovation and not “reform” school, but finally begin to create what can honestly be called “school.”

If we have the courage and the vision, there are allies in industry to support us.

Free At Last

If you want to see someone really committed to freedom in education, check out *Free At Last: The Sudbury Valley School*, which you can read online at the link, if you want. I went there for the first chapter, on teaching arithmetic, but stayed for the later chapters.

But read that first chapter, with the surprise at the end. Consider how amazing it is that students could learn six years of arithmetic in twenty contact hours, and then consider how it’s not so surprising after all. In fact, consider the research (both these quotes from a larger article, available here):

… early childhood may simply be an inefficient period in which to try to teach skills that can be relatively quickly learned in adolescence.

—

Prime Time for Education: Early Childhood or Adolescence? by William D. Rohwer, Jr., Harvard Educational Review, Vol. 41, No. 3, August 1971, page 316, from the summary.

Several groups of important investigations on the teaching of arithmetic have contributed findings that have led schools to make changes in the organization of the curriculum. One group of studies dealt with the effect of postponing or deferring the teaching of arithmetic in the primary grades. Included in this group are the studies by Ballard in 1912, Taylor in 1916, Wilson in 1930, and Benezet in 1935-36. In these studies formal arithmetic instruction was withheld in one group and administered as usual in another group. At the end of the experimental period, the comparative achievements of the two groups were measured. In each case the experimenter recommended the postponement of “formal” arithmetic – Ballard for two years or the age of seven, Taylor for one year, Wilson for two years, and Benezet until grade 5.

On the basis of these and other studies the plan of eliminating formal arithmetic instruction from grades one and two, sometimes also grade three, has been adopted by a considerable number of school systems. In some systems there is not even an approved plan of informal or incidental arithmetic. Such a procedure fails to recognize certain very important facts about the studies referred to above. A careful reading of the reports of these four experiments shows that while formal practice on computational processes was postponed in the experimental groups, there was a great deal of use made in these classes of various kinds of activities, games, projects, and social situations through which the child was brought into contact with numbers and given the opportunity to use them informally in meaningful ways. It is especially clear in the studies by Wilson and Benezet that arithmetic was not in fact postponed at all. It is evident that what happened in these two studies was that computational arithmetic was replaced by what I called earlier in this paper, social arithmetic. In each study the plan was to emphasize number meanings, to develop an understanding of the ways in which number functions in the daily lives of children both in and out of school, and to develop what is called number “readiness” for the more formal work to follow … .

—

Mathematics Teacher, Volume 31, October, 1938, pages 287-292, article “Deferred Arithmetic” by Leo J. Brueckner, from a paper read at the annual meeting of the National Council of Teachers of Mathematics in Atlantic City, J. J., Feb. 26, 1938.

[1938! We’ve been working on this problem of how to teach math for so long, and there have been so many examples of places that have solved it (check out UCDS or PSCS if you live in Seattle. They have solved the problem, as far as I’m concerned), and we can’t as a whole, get it together!]

Notice that those kids who learned arithmetic at Sudbury Valley probably already had a wide experience with non-formal arithmetic. When they wanted it, they learned the formal part in a flash. And indeed, that’s the way it should go: exposure, play, experience, “number readiness,” with formality following when the child is ready. According to most research I’ve seen, [for example], this is usually around 9-12 years of age, though it can come a few years earlier or later.

And that, as I read chapters following, is the amazing, courageous thing about Sudbury Valley: they let the motivation come from the students, and when it comes, they take it seriously. It’s an amazing thing to really trust in people like this. And there are ways to mess it up. If you go in naively and say, “let’s just let kids do their thing,” and do nothing else, it can fail pretty dramatically. There’s more happening: the deal-making, the environment. This is artfully artless teaching. They make it look easy, when really it’s simple. Not the same thing.

“It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”—Laplace

Big news on P vs NP?

Perhaps you’ve heard of the Millenium Problems, the seven $1,000,000 prize math problems. They’re pretty hard. In fact, I kind of assumed we’d see one solved every forty or fifty years. Well, an eccentric Russian named Grigori Perelman solved one already (the Poincare Conjecture). And now someone (his name is Deolalikar) has taken a serious shot at the P vs NP problem. We’ll see if he is vindicated and claims the prize.

P vs NP is the most important problem in theoretical computer science right now. Wikipedia’s explanation of it is so good, I’m going to quote at length:

Informally, it asks whether every problem whose solution can be efficiently checked by a computer can also be efficiently

solvedby a computer.

More thoroughly [note: I’m not hooking up the links for the quoted parts. Just search for them on wikipedia if you want to know more]:

In essence, the question

P=NP? asks:Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be computed quickly?

The theoretical notion of

quickused here is that of an algorithm that runs in polynomial time. The general class of questions for which some algorithm can provide an answer in polynomial time is called “class P” or just “P”.For some questions, there is no known way to find an answer quickly, but if one is provided with information showing what the answer is, it may be possible to verify the answer quickly. The class of questions for which an answer can be verified in polynomial time is called

NP.Consider the subset sum problem, an example of a problem which is easy to verify but whose answer is suspected to be theoretically difficult to compute. Given a set of integers, does some nonempty subset of them sum to 0? For instance, does a subset of the set {−2, −3, 15, 14, 7, −10} add up to 0? The answer “yes, because {−2, −3, −10, 15} add up to zero” can be quickly verified with three additions. However, finding such a subset in the first place could take more time; hence this problem is in

NP(quickly checkable) but not necessarily inP(quickly solvable).An answer to the

P=NPquestion would determine whether problems like the subset-sum problem that can be verified in polynomial time can also be solved in polynomial time. If it turned out thatPdoes not equalNP, it would mean that someNPproblems are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time.

Roughly, polynomial time means that as the complexity of the problem grows, the difficulty in solving it doesn’t grow too fast.

The weird thing is, most experts (though not all!) agree that P≠NP. So why is the proof important?

To answer, I refer you to the beautiful discussion here, from which I cherry-pick the following quote from the mathematician Atiyah:

I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions—they are not just repetitions of each other.

The lesson for teachers and students of the subject? Your study doesn’t end with a right answer… it shifts into high gear.

There's no sport in blowing minds anymore

You can use math to blow people’s mind so easily and so casually that it almost feels unfair. I had the pleasure of doing so tonight at a volunteer orientation meeting at the Puget Sound Community School, a pretty awesome Seattle school in the tradition of the Sudbury Valley School (see my earlier post). I’ve been chatting with them about coming in some Friday and teaching some cool math to the students there.

PSCS Teacher: What are you thinking of doing?

Me: I was thinking of “Which infinity is bigger?” For example, there are an infinite number of square numbers, 1, 4, 9, 16, and so on. But they fit inside the whole numbers. So does that mean there are more whole numbers than square numbers? Is that infinity bigger?

Other Volunteer, listening in: You just blew my mind.

It’s becoming a crutch. Maybe I should practice blowing people’s minds with my own ideas for a while, instead of raiding the wealth of little-known, bewilderingly gorgeous reality benders that mathematics provides. It’s so much harder to do it that way, though.

On that note, I found this on another tumblr blog (though it was already reblogged, so I can’t credit the original source):

Is it normal to burst into hysterical tears multiple times a day over how utterly, indescribably, ethereally, inconceivably beautiful mathematics is?

Hitchhiker's Guide

I made it part way through the newest of the *Hitchhiker’s Guide to the Galaxy* movies last night. I stopped right after the pivotal moment where the supercomputer gives the answer to life, the universe, and everything: 42. When they ask what the story is, the supercomputer points out that they never really gave it the question. Forty-two is the right answer, no doubt, but finding the question will take some *real* work. An absolutely classic moment.

Today I started thinking about how perfectly that scene fits the state of mathematics in mainstream education. We’re all answers, no questions.

I’ll give you my favorite example. What’s the area of a circle? If you remember your high school geometry, you’ll recall that there’s a formula for this, namely:

"The area of the circle is pi times the radius squared."

Pause here. Does this answer mean any more to you than 42? Probably it means less. What’s pi, after all, and how does it all work? We need to slow down, and remember our question.

What is the area of a circle?

No tricks. No formulas. You’re just a human being, looking at one of the simplest and most fundamental shapes there is, just like you’re ancestors once did before anyone had any idea about formulas. What do you do?

In fact, let’s say you knew *everything else you wanted to know about the circle*: the diameter (i.e., the distance across is, or d in the picture above); the circumference (or distance around, c in the picture above); the radius isn’t pictured, but it’s just half the diameter. Anything you want, let’s say you know it. How can you figure it out.

Maybe not everyone can solve this problem. But it’s worth starting with the question. When you get to the answer, it has a chance of actually meaning something. Forty-two, after all, was never much help.

"They lost interest because they stopped asking questions."

Perhaps you saw the recent Newsweek article on creativity. It’s worth reading, and while I’ll leave the statistics and arguments in the article (creativity can be measured kind of reasonably; creativity scores in the U.S. are declining, but scores in the rest of the world are rising; most schools are educating in a way that squashes creativity, etc.), but I want to include the anecdote, because it’s so lovely:

Consider the National Inventors Hall of Fame School, a new public middle school in Akron, Ohio. Mindful of Ohio’s curriculum requirements, the school’s teachers came up with a project for the fifth graders: figure out how to reduce the noise in the library. Its windows faced a public space and, even when closed, let through too much noise. The students had four weeks to design proposals.

Working in small teams, the fifth graders first engaged in what creativity theorist Donald Treffinger describes as fact-finding. How does sound travel through materials? What materials reduce noise the most? Then, problem-finding—anticipating all potential pitfalls so their designs are more likely to work. Next, idea-finding: generate as many ideas as possible. Drapes, plants, or large kites hung from the ceiling would all baffle sound. Or, instead of reducing the sound, maybe mask it by playing the sound of a gentle waterfall? A proposal for double-paned glass evolved into an idea to fill the space between panes with water. Next, solution-finding: which ideas were the most effective, cheapest, and aesthetically pleasing? Fiberglass absorbed sound the best but wouldn’t be safe. Would an aquarium with fish be easier than water-filled panes?

Then teams developed a plan of action. They built scale models and chose fabric samples. They realized they’d need to persuade a janitor to care for the plants and fish during vacation. Teams persuaded others to support them—sometimes so well, teams decided to combine projects. Finally, they presented designs to teachers, parents, and Jim West, inventor of the electric microphone.

Along the way, kids demonstrated the very definition of creativity: alternating between divergent and convergent thinking, they arrived at original and useful ideas. And they’d unwittingly mastered Ohio’s required fifth-grade curriculum—from understanding sound waves to per-unit cost calculations to the art of persuasive writing. “You never see our kids saying, ‘I’ll never use this so I don’t need to learn it,’ ” says school administrator Maryann Wolowiec. “Instead, kids ask, ‘Do we have to leave school now?’ ” Two weeks ago, when the school received its results on the state’s achievement test, principal Traci Buckner was moved to tears. The raw scores indicate that, in its first year, the school has already become one of the top three schools in Akron, despite having open enrollment by lottery and 42 percent of its students living in poverty.

Reflecting on this story, and then about how much wrong we do in education, I start to get furious.

Problems on the Board II: Simplify!

I asked a pair of girls (age 8 and 9) I tutor to ask questions about the chessboard, and got another really great lesson out of it, this one highlighting the importance of making things simple.

With a little prompting, we came up with a nice list of questions:

- How many squares are there on the chess board?
- How many squares of any size?
- How many rectangles?
- How many chessboards would it take to fill the space needle? To reach the moon? To cover the Earth 10 times?
- Can you bend a chessboard into a sphere?
- How many corners, or vertices are there on the board?
- How many edges?

When they get interested in a problem (and these kids, like most, were interested in their own problems right away) the tendency is to try to solve them right away. I thought it might be nice to work on the problem of how many edges there are, but one of the girls suggested working on how many rectangles there are. Take a look at a chessboard.

There are a lot of rectangles on this thing. So many, I’d say, that if you try to start counting them, you’re setting yourself up for a frustrated end, especially if you’re a kid. Here’s the dialogue between us:

Me: Do you know what a mathematician does when they face a problem that’s too hard to solve right away? They make it simple. (I fold the board in half.)

Them: Oh, we could start on a four by four board! (They immediately start to count rectangles.)

Me: Mathematicians really prefer to start with really easy problems. (I fold the board in half again)

Them: We could start on a two by two board!

Me: Sure. Though a mathematician would really like to start with something even easier. Like a one by one board. In fact, a mathematician would probably start with a zero by zero board. How many rectangles are there on that board?

Them: (laughing) Zero!

Me: See? We’ve already solved a problem! How many rectangles are there on the one by one board?

Them: One.

Me: All right! We’ve solved two problems; let’s write down what we’ve got so far (we do). What about the two by two board?

Them: (counting): 1, 2, 3, 4, … 5. No. There are 1, 2, 3, 4, 5, 6, 7, 8, 9. Nine.

By this point, we’re all involved, and we’ve got real momentum. One of the girls starts to think of possible patterns in the numbers 0, 1, 9, but can’t think of what might come next (another great reason to start on the zero by zero board: we get an extra piece of data, and can look for patterns before we’ve even started the hard work). So, inspired, they dive in to the three by three board.

It’s complicated enough to take some doing, and after they had tried to count them all, missed some, tried again, got closer, tried again, and got the right answer, I suggested writing down their process instead of doing everything in their heads. We ended up with the following table:

The numbers in each box represent the numbers of each type of rectangle. For example, there are 4 two by two rectangles (i.e., squares) on the 3 by 3 chessboard, and there are 6 two by one rectangles, and 6 one by two rectangles (which look the same, but one is vertical, one horizontal). Adding up all the kinds of rectangles, this confirmed the next number in our pattern.

0, 1, 9, 36…

Do you see what should come next? We didn’t yet, but our momentum was better than ever. One of the girls jumped in to counting all the rectangles on the four by four; the other, daunted by the magnitude of that effort, started looking for patterns in the table we’d made. By the end, both had solved the four by four board, and one, having found a pattern in the table, had conjectured the number of rectangles in the five by five board. Our pattern now looks like this:

0, 1, 9, 36, 100, 225,…

Do you see a pattern now?

One of the nice things about this lesson was that, in addition to the deeper skills of analyzing patterns, simplifying, organizing, etc., there was a need to employ some basic technical skills: both girls ended up motivated to add up long columns of numbers. So this ended up giving us motivation to practice complicated addition problems that, without this framework, would have been boring. One of the girls, faced with the task of adding up the numbers on the table of all rectangles representing rectangles on the four by four board, found that adding the numbers in columns first makes it easier.

For anyone who wants to dive in at home, here’s a little extension: if you start with the number of *squares* on the chessboard, there are two natural generalizations: find the number of rectangles on the chessboard, or the three dimensional variation, find the number of cubes in a bigger cube (i.e., how many cubes are there in a rubik’s cube?). These questions seem totally different, but I if you start to write your answers down for each, you’ll notice a similarity between them that bespeaks a connection between these questions and the structures.

Questions on the Board

The place: the reading room at Elliott Bay Books. Large but with no natural light, and imperfect lighting.

The time: this afternoon at 1:30.

The crew: 7 kids, in the 2-4th grade range.

If art requires inspiration, and math is an art, then my job is, in part, to provide inspiration. I brought in a chessboard.

We warmed up with the handshake problem, and then I presented the board. What are some questions we could ask about it? Here’s what the kids came up with:

- How many squares are there of any size?
- What pictures of letters could you make out of squares (the same color)?
- How many edges on all the squares on the chessboard?
- How many diagonals?
- How many corners?
- Is the perimeter of a square different from the inside of a square?
- How many sides if the squares are cut out?
- How many sets of chess pieces could we fit on the board?
- How many letters would fit inside the chessboard?
- How many squares are there if non of the same size are allowed to overlap?

It’s actually a very rich list of questions, and this group had no trouble digging in.

Spoilers follow.

The first problem is a very natural and interesting question to ask, and one I would have brought up if the kids hadn’t. Two kids, worked on it together, oblivious to the outside world, until — breakthrough! — they noticed a very cool symmetry:

the number of 1 by 1 squares is 64 (or 8 x 8);

the number of 2 by 2 squares is 49 (or 7 x 7);

the number of 3 by 3 squares is 36 (or 6 x 6);

and so on till

the number of 8 by 8 squares is 1 (or 1 x 1).

So the answer, interestingly, is just a sum of squares numbers!

Another student found the number of diagonals on the board, and then went back to find a pattern, starting with smaller board sizes and working his way up to 8 by 8. A very tidy pattern, in the end. The last group (of 4) tackled the last problem, which is more technical and involves extra cases. What’s the pattern, though? Two of them went home determined to figure it out.

Here are some pros and cons of letting the kids ask their questions and choose what to work on:

Pros:

- Time is set aside specifically ask questions, one of the most important parts of doing math (or anything).
- Every kid owns their work, from start to finish.
- Every kid gets to choose what to work on, and every single person in the group is engaged pretty much the entire class (something that is essentially impossible if you lecture, no matter how compelling an orator you are).
- Kids get to present their own findings to the class at the end.
- This is a math class where the kids do what real mathematicians do.

Cons:

- It’s much harder to create an arc for the class, since I don’t choose the problems ahead of time, and I can’t guarantee as many “wow” moments. It takes a subtler hand on the tiller.
- Since not everyone is working on the same problem, there can be less overall class cohesion. Sometimes students aren’t that interested or tuned in to what other kids have figured out.

In my opinion, the pros outweigh the cons. However, I think there’s even a way to eliminate the cons altogether. The first meeting was great; as we keep meeting I think it will get even smoother.

What’s next? Math games, I hope. Infinity, as well, possibly via Zeno and infinite divisibility. I’ll be thinking about it.

Announcement: A Math Circle for 2-4 graders.

When: Tuesday, July 13, 1:30pm-3pm

Where: The reading room at Elliott Bay Books, 10th and Pine in Capitol Hill.

What: An opportunity for kids to explore some of the best stuff in mathematics with a working mathematician.

Cost: $25.

Please contact me at finkelitis@gmail.com if you’re interested in joining us. If you’d like to know more about me, check out my tutoring website, or my blog.

The Hankering (Knots)

It was only a matter of time, I suppose, before I felt the need, the yen, the hankering for some mathematical activity again. To that end, I borrowed my girlfriend’s copy of The Knot Book by Colin C. Adams.

It’s about knots.

More specifically, it’s about knot theory, which was a pretty hip subfield of topology last I checked. In other words, studying knots is a serious mathematical pursuit if you do it seriously enough.

I like Adams’ perspective on math as an activity. Here’s how the book starts:

Mathematics is an incredibly exciting and creative field of endeavor. Yet most people never see it that way. Nonmathematicians too often assume that we mathematicians sit around talking about what Newton did three hundred years ago or calculating a couple of extra million digits of pi. They do not realize that more new mathematics is being created now than at any other time in the history of humankind.

He basically has me at hello. And I like his idea to use knots as a quick way in. We’re all used to them and have some intuition about them, but even the problem of telling knots apart is still open.

His exposition on knots themselves is promising thus far (I’ve just begun the book). And he includes some great history of where the study came from. In the 1880s, Lord Kelvin hypothesized that atoms might be knotted ether, and his colleague Peter Guthrie Tait started tabulating all possible knots, in the hopes that this would help him understand chemistry in some fundamental way. Of course, the whole notion of ether was disproved before the end of that decade, and scientists left knots behind. Mathematicians picked up the thread, though, and knot theory became a serious field. In the 1980s, it turned out that there are applications in studying knotted DNA molecules. So knots are important in the real world after all.

Which must be nice for Tait, who I imagined felt pretty crummy about having spent years of work fleshing out a totally defunct theory. I wish he could have lived to see the usefulness of his work.

Here’s a fun tidbit from the math: you can add knots together, and even more, you can break knots down into sums of the simplest kinds of knots. Taking our cue from how we do the same when we factor numbers, we call these simplest knots (which can’t be broken down further) *prime* knots. For your consideration, here are some of the first prime knots. See if you can guess what the numbers below each knot indicate.

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.

“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?

**TEACHING AS A SUBVERSIVE ACTIVITY**

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.

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:

Theorem.

And then, underneath:

Proof:

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 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.

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

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.

New Blog From an Old Colleague

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!

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