Math for Love

I'm a mathematician. This is my blog.
It's about math, creativity, culture, education, and beauty.
It's for everyone.

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Sep 14

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Sep 12
“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

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Sep 11

I’m back from vacation! Yellowstone was great.

Here’s a little challenge (or point of interest) for anyone who likes magic. This video is an honest portrayal of a card trick (with somewhat overdramatic music). It works like this: a deck is shuffled perfectly, i.e., from wikipedia:

In a perfect shuffle or perfect faro shuffle, the deck is split into equal halves of 26 cards which are then pushed together in a certain way so as to make them perfectly interweave.

In the video, the card handler does 8 perfect faro shuffles in a row. The result is surprising. The mathematical challenge: why does this trick work?

(I had to solve this problem as a homework problem in a group theory class in college, but I remember hearing the question much earlier, and having a good time thinking about it.)

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Sep 9

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.

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Sep 5

I guess I shouldn’t be surprised they made a movie out of it.

Speaking of 2-D vs 3-D, designing two dimensional worlds and getting kids to imagine explaining three dimensions to two dimensional people can make for an amazing curriculum. Here’s some ideas to get you started:

  1. 2d people can’t have digestive tracts like us, since a tube that runs through a 2d person cuts them in half. How can they take in and digest food?
  2. What about lungs, circulatory system, and other bodily systems?
  3. Can you build a 2d car? How do you get into it? How does the wheel turn?
  4. If you know any physics (like dissipation of gases, propagation of waves, etc.) what does the 2d version of these look like?
  5. How can you explain what a cube is to a 2d person?
  6. How would a 4d person explain what a 4d cube is like to us?

Any questions you can think of to add?

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Sep 4

I find Jim Tanton's videos so charming that I can't understand why they're so rarely viewed. Look at this one, on the relatively well known two pancake problem. It's clear to anyone, relevant to a student of calculus—it's a lovely application of the intermediate value theorem, and makes the intermediate value theorem meaningful and obvious (if you slowly move your knife across the pancake, at some point half of it will be on the left and half on the right)—and, most impressively, novel: I’d never seen the pizza problem or the quarter pancake problem before, and they’re great additions to the lesson. And it’s just a fun problem.

There’s a three dimensional analogue called the Ham Sandwich Problem: can you cut, in a single cut, a ham sandwich so that both slices of bread and the ham in between are cut precisely in half? The link gives an abstruse discussion.

Having finished two math books recently, I’m very interested in looking at Jim Tanton’s Solve This! The art of coming up with great problems is quite different from knowing math. From what I’ve seen (I referred to Tanton earlier in this post), Tanton is a master of thinking up clever problems.

Here are a few additional observations and questions for his video:

  1. Can there be a 3 pancake problem? Can you show with a simple example that it’s impossible to cut 3 pancakes in half with a single cut?
  2. Notice that when he’s cutting a pancake at different angles of rotation, all the cuts go through a single “center” point. Is he doing that on purpose, and is there some mathematical reason for it? [Hint: think about easy shapes, like squares or equilateral triangles… does every cut that cuts them in half go through some special point?]
  3. With 2 cuts, could you cut 2 pancakes into thirds?

Other thoughts/questions/observations/answers?

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Sep 3

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, and I 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 in The American Mercury for 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, and create 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?

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Sep 2

Here’s his full act, if you want more.

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So many rhetorical questions in this video that it gets hard to watch, but skip ahead to the middle, where Art Benjamin starts doing his human calculator tricks (and beating people on calculators). The question is: is he a genius? He answers, honestly, no. He’s a computer, in the ancient sense.

He’s a total kick to watch. At the same time, this is exactly the kind of math that isn’t important, from a mathematical perspective. It’s the kind we let the computers do.

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Aug 31

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:

  1. 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?
  2. 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.

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