Links on Monday

• The day after I added Puzzling Queen to my blogroll, she linked to this amazing collection titled Geometry and Perspective. Looks to me like Escher may have been influenced by this stuff from the 1500's.

• Review of a movie about Godel, Cantor, Turing and Boltzmann, at Reasonable Deviations.

• Diandra, at Cocktail Party Physics, asks: "Which is better: going from a car that gets 34 miles per gallon (mpg) to one that gets 50 mpg, or changing from a car that gets 18 mpg to one that gets 28 mpg?" It's not as straightforward as you'd first think.

• What if you have a coin that lands heads more often than tails? Can you devise a way to use it for a fair toss? Yep. Bill the Lizard explains how.

• Gwen Dewar blogs at Parenting Science, where she passes along research of interest to parents. From an article on helping young children develop their math skills, here's a fascinating bit on the connection between math, language, and an innate counting ability:

Recent research [was] conducted by cognitive neuroscientists on kids who speak only Walpiri or Anindilyakwa, two native Australian languages (Butterworth et al 2008). These languages include number words for only three numerosities--“one,” “two,” and any imprecise quantity that is "more than two." Yet 4- to 7-year old speakers of these languages performed as well or better than English speakers when they were asked to

• briefly examine a small set of tokens and then assemble an identical set of tokens from memory

• listen to a series of up to 7 taps and then place the corresponding number of tokens on a mat

• spontaneously subdivide a set of 6 or 9 items into three equal sets when they were told to “share” these items among three toy bears

Who's Lying With Statistics?*

In my wanderings, I came across this (at O'Reilly Radar):

How the UK Government Spun 136 People into 7 Million -- a radio show looked into the government's claim of 7 million illegal filesharers and discovered it came down to 136 people in a survey admitting they'd used it.

But when I read the original article at PC Pro, I was moved to reply:

Your headline is more misleading than what the government originally reported. The 136 was 11.6% of the responses to a "survey of 1,176 net-connected households". That's a good sample size for a survey.

[pinero50 says "Extrapolating 3.9 million from a sample of ~1000 odd still seems pretty suspect to me." Pinero, I know it seems strange, but that's a basic thing you learn when studying statistics. A sample size on the order of 1,000 gives very accurate results, if picked randomly.]

There were problems, including having an interested party be involved in the research, but a more sensible comparison would be 3.9 million (mentioned at the end of the article) vs 7 million.

Commenters are right to question the wording used in the survey. If it just mentioned file-sharing, all bets are off as to how many people share illegally.

Why is it possible (if you ask a well-worded question and pick people randomly) to accurately gauge what's happened in a large population by asking only 1,000 people to respond to a survey?

It's counter-intuitive at first. But try this thought experiment: Imagine a silo full of grains of corn. You want to pull some out and examine it to get a sense of the quality of the corn. Imagine that this is a high-tech silo, and the corn gets thoroughly mixed, so if you reach in and pull out a scoop, it will be a good random sample of the corn. Can you see that it doesn't really matter whether the silo is 10 feet tall or a hundred? The cup of corn you pull out should give you a sense of what's in there, as long as there's not too much variability (i.e., as long as the kernels as all close to the same size, water content, etc).

The people surveyed are like the kernels in the scoop. In statistics courses, you learn to put together something called a confidence interval, so you can come up with a precise way of talking about how accurately the sample reflects the population. You end up with something along the lines of: "We're 95% sure that 3.9 million people are doing illegal file-sharing, with an error margin of .05 million."

A more accurate (and less attention-grabbing) title? Perhaps How the UK Government Spun 3.9 Million People into 7 Million. But who'd buy your news if it weren't inflammatory?




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*My title takes off from a charming little book called How to Lie with Statistics, by Darrel Huff, written in 1954, an enjoyable and educational read.

Two Puzzles from Down Under

Pat Ballew posted these at Pat'sBlog, and I worked on them last night when I had trouble sleeping. They come from the puzzle corner of the Gazette of the Australian Mathematical Society.

Problem 1. Digital deduction.
The numbers 2^2009 and 5^2009 are written out on a piece of paper in the usual decimal notation. How many digits are on this piece of paper?


Problem 2. Piles of stones.
There are 25 stones sitting in a pile next to a blackboard. You are allowed to take a pile and divide it into two smaller piles of size a and b, but then you must write the number a×b on the blackboard. You continue to do this until you are left with 25 piles, each with one stone. What is the maximum possible sum of the numbers written on the blackboard?

A few of us worked them out in the comments over there. I won't spoil your fun if you're seeing it first here. What I liked about these problems was that from each of them I learned something new that extends well beyond the problem at hand.

Obama Speaks to the Nation's Schoolchildren

President Barak Obama will speak today, in about an hour, to the nation's schoolchildren. (Well, the ones who are allowed to listen, anyway.) My comments at the end...

Here's what he plans to say (From the White House website):

Hello everyone – how’s everybody doing today? I’m here with students at Wakefield High School in Arlington, Virginia. And we’ve got students tuning in from all across America, kindergarten through twelfth grade. I’m glad you all could join us today.

I know that for many of you, today is the first day of school. And for those of you in kindergarten, or starting middle or high school, it’s your first day in a new school, so it’s understandable if you’re a little nervous. I imagine there are some seniors out there who are feeling pretty good right now, with just one more year to go. And no matter what grade you’re in, some of you are probably wishing it were still summer, and you could’ve stayed in bed just a little longer this morning.

I know that feeling. When I was young, my family lived in Indonesia for a few years, and my mother didn’t have the money to send me where all the American kids went to school. So she decided to teach me extra lessons herself, Monday through Friday – at 4:30 in the morning.

Now I wasn’t too happy about getting up that early. A lot of times, I’d fall asleep right there at the kitchen table. But whenever I’d complain, my mother would just give me one of those looks and say, “This is no picnic for me either, buster.”

So I know some of you are still adjusting to being back at school. But I’m here today because I have something important to discuss with you. I’m here because I want to talk with you about your education and what’s expected of all of you in this new school year.

Now I’ve given a lot of speeches about education. And I’ve talked a lot about responsibility.

I’ve talked about your teachers’ responsibility for inspiring you, and pushing you to learn.

I’ve talked about your parents’ responsibility for making sure you stay on track, and get your homework done, and don’t spend every waking hour in front of the TV or with that Xbox.

I’ve talked a lot about your government’s responsibility for setting high standards, supporting teachers and principals, and turning around schools that aren’t working where students aren’t getting the opportunities they deserve.

But at the end of the day, we can have the most dedicated teachers, the most supportive parents, and the best schools in the world – and none of it will matter unless all of you fulfill your responsibilities. Unless you show up to those schools; pay attention to those teachers; listen to your parents, grandparents and other adults; and put in the hard work it takes to succeed.

And that’s what I want to focus on today: the responsibility each of you has for your education. I want to start with the responsibility you have to yourself.

Every single one of you has something you’re good at. Every single one of you has something to offer. And you have a responsibility to yourself to discover what that is. That’s the opportunity an education can provide.

Maybe you could be a good writer – maybe even good enough to write a book or articles in a newspaper – but you might not know it until you write a paper for your English class. Maybe you could be an innovator or an inventor – maybe even good enough to come up with the next iPhone or a new medicine or vaccine – but you might not know it until you do a project for your science class. Maybe you could be a mayor or a Senator or a Supreme Court Justice, but you might not know that until you join student government or the debate team.

And no matter what you want to do with your life – I guarantee that you’ll need an education to do it. You want to be a doctor, or a teacher, or a police officer? You want to be a nurse or an architect, a lawyer or a member of our military? You’re going to need a good education for every single one of those careers. You can’t drop out of school and just drop into a good job. You’ve got to work for it and train for it and learn for it.

And this isn’t just important for your own life and your own future. What you make of your education will decide nothing less than the future of this country. What you’re learning in school today will determine whether we as a nation can meet our greatest challenges in the future.

You’ll need the knowledge and problem-solving skills you learn in science and math to cure diseases like cancer and AIDS, and to develop new energy technologies and protect our environment. You’ll need the insights and critical thinking skills you gain in history and social studies to fight poverty and homelessness, crime and discrimination, and make our nation more fair and more free. You’ll need the creativity and ingenuity you develop in all your classes to build new companies that will create new jobs and boost our economy.

We need every single one of you to develop your talents, skills and intellect so you can help solve our most difficult problems. If you don’t do that – if you quit on school – you’re not just quitting on yourself, you’re quitting on your country.

Now I know it’s not always easy to do well in school. I know a lot of you have challenges in your lives right now that can make it hard to focus on your schoolwork.

I get it. I know what that’s like. My father left my family when I was two years old, and I was raised by a single mother who struggled at times to pay the bills and wasn’t always able to give us things the other kids had. There were times when I missed having a father in my life. There were times when I was lonely and felt like I didn’t fit in.

So I wasn’t always as focused as I should have been. I did some things I’m not proud of, and got in more trouble than I should have. And my life could have easily taken a turn for the worse.

But I was fortunate. I got a lot of second chances and had the opportunity to go to college, and law school, and follow my dreams. My wife, our First Lady Michelle Obama, has a similar story. Neither of her parents had gone to college, and they didn’t have much. But they worked hard, and she worked hard, so that she could go to the best schools in this country.

Some of you might not have those advantages. Maybe you don’t have adults in your life who give you the support that you need. Maybe someone in your family has lost their job, and there’s not enough money to go around. Maybe you live in a neighborhood where you don’t feel safe, or have friends who are pressuring you to do things you know aren’t right.

But at the end of the day, the circumstances of your life – what you look like, where you come from, how much money you have, what you’ve got going on at home – that’s no excuse for neglecting your homework or having a bad attitude. That’s no excuse for talking back to your teacher, or cutting class, or dropping out of school. That’s no excuse for not trying.

Where you are right now doesn’t have to determine where you’ll end up. No one’s written your destiny for you. Here in America, you write your own destiny. You make your own future.

That’s what young people like you are doing every day, all across America.

Young people like Jazmin Perez, from Roma, Texas. Jazmin didn’t speak English when she first started school. Hardly anyone in her hometown went to college, and neither of her parents had gone either. But she worked hard, earned good grades, got a scholarship to Brown University, and is now in graduate school, studying public health, on her way to being Dr. Jazmin Perez.

I’m thinking about Andoni Schultz, from Los Altos, California, who’s fought brain cancer since he was three. He’s endured all sorts of treatments and surgeries, one of which affected his memory, so it took him much longer – hundreds of extra hours – to do his schoolwork. But he never fell behind, and he’s headed to college this fall.

And then there’s Shantell Steve, from my hometown of Chicago, Illinois. Even when bouncing from foster home to foster home in the toughest neighborhoods, she managed to get a job at a local health center; start a program to keep young people out of gangs; and she’s on track to graduate high school with honors and go on to college.

Jazmin, Andoni and Shantell aren’t any different from any of you. They faced challenges in their lives just like you do. But they refused to give up. They chose to take responsibility for their education and set goals for themselves. And I expect all of you to do the same.

That’s why today, I’m calling on each of you to set your own goals for your education – and to do everything you can to meet them. Your goal can be something as simple as doing all your homework, paying attention in class, or spending time each day reading a book. Maybe you’ll decide to get involved in an extracurricular activity, or volunteer in your community. Maybe you’ll decide to stand up for kids who are being teased or bullied because of who they are or how they look, because you believe, like I do, that all kids deserve a safe environment to study and learn. Maybe you’ll decide to take better care of yourself so you can be more ready to learn. And along those lines, I hope you’ll all wash your hands a lot, and stay home from school when you don’t feel well, so we can keep people from getting the flu this fall and winter.

Whatever you resolve to do, I want you to commit to it. I want you to really work at it.

I know that sometimes, you get the sense from TV that you can be rich and successful without any hard work — that your ticket to success is through rapping or basketball or being a reality TV star, when chances are, you’re not going to be any of those things.

But the truth is, being successful is hard. You won’t love every subject you study. You won’t click with every teacher. Not every homework assignment will seem completely relevant to your life right this minute. And you won’t necessarily succeed at everything the first time you try.

That’s OK. Some of the most successful people in the world are the ones who’ve had the most failures. JK Rowling’s first Harry Potter book was rejected twelve times before it was finally published. Michael Jordan was cut from his high school basketball team, and he lost hundreds of games and missed thousands of shots during his career. But he once said, “I have failed over and over and over again in my life. And that is why I succeed.”

These people succeeded because they understand that you can’t let your failures define you – you have to let them teach you. You have to let them show you what to do differently next time. If you get in trouble, that doesn’t mean you’re a troublemaker, it means you need to try harder to behave. If you get a bad grade, that doesn’t mean you’re stupid, it just means you need to spend more time studying.

No one’s born being good at things, you become good at things through hard work. You’re not a varsity athlete the first time you play a new sport. You don’t hit every note the first time you sing a song. You’ve got to practice. It’s the same with your schoolwork. You might have to do a math problem a few times before you get it right, or read something a few times before you understand it, or do a few drafts of a paper before it’s good enough to hand in.

Don’t be afraid to ask questions. Don’t be afraid to ask for help when you need it. I do that every day. Asking for help isn’t a sign of weakness, it’s a sign of strength. It shows you have the courage to admit when you don’t know something, and to learn something new. So find an adult you trust – a parent, grandparent or teacher; a coach or counselor – and ask them to help you stay on track to meet your goals.

And even when you’re struggling, even when you’re discouraged, and you feel like other people have given up on you – don’t ever give up on yourself. Because when you give up on yourself, you give up on your country.

The story of America isn’t about people who quit when things got tough. It’s about people who kept going, who tried harder, who loved their country too much to do anything less than their best.

It’s the story of students who sat where you sit 250 years ago, and went on to wage a revolution and found this nation. Students who sat where you sit 75 years ago who overcame a Depression and won a world war; who fought for civil rights and put a man on the moon. Students who sat where you sit 20 years ago who founded Google, Twitter and Facebook and changed the way we communicate with each other.

So today, I want to ask you, what’s your contribution going to be? What problems are you going to solve? What discoveries will you make? What will a president who comes here in twenty or fifty or one hundred years say about what all of you did for this country?

Your families, your teachers, and I are doing everything we can to make sure you have the education you need to answer these questions. I’m working hard to fix up your classrooms and get you the books, equipment and computers you need to learn. But you’ve got to do your part too. So I expect you to get serious this year. I expect you to put your best effort into everything you do. I expect great things from each of you. So don’t let us down – don’t let your family or your country or yourself down. Make us all proud. I know you can do it.

Thank you, God bless you, and God bless America.

Mostly, I love it. And it seems to me that this is a speech conservatives would love too, for exactly the same reason I have a bit of trouble with it.

What I don't love is that it is almost completely a Puritan work ethic sort of thing. I think we learn best when our learning is fun and playful, and a good learning challenge sucks you in so that you want to work hard (really hard) to 'get it'. I wish he would have addressed that. I also wish he would have talked about how good it feels to accomplish something you've worked hard at. And my third wish is that he would have addressed what it means to think for yourself. Learning should not be (only) about learning facts, figures, and procedures. It should be about learning how to think about issues deeply.

But the truth is, being successful is hard. You won’t love every subject you study. You won’t click with every teacher. Not every homework assignment will seem completely relevant to your life right this minute. And you won’t necessarily succeed at everything the first time you try.

My success has come through following my heart. I do think we should be able to love everything we study. I do think homework should be inspiring. (Doesn't have to be relevant, if the kid wants to do it anyway.) I like the message that we need to work hard. But I want kids to work hard because they love it, not just for some future gain.

Math Teachers at Play #15...

...is up at Homeschool Math.

Next one will be hosted by Maria D, at the mathfuture wikispace. I see there aren't any hosts listed after her at the blog carnival site. Is anyone ready to be the next one to take this on? Denise (whose marvelous blog, Let's Play Math!, has been quiet lately) sends you great suggestions about how to do it. My first time, I had trouble with extra formatting junk, but I got that figured out for my second time around. It's a great learning experience.

Freedom to Learn

Peter Gray has been writing a series of articles called Freedom to Learn, on the Psychology Today blog. He covers this topic from every conceivable angle - how we evolved to learn, our innate craving for freedom, the young child's rage to learn, the contradiction posed by a coercive 'learning' environment like school, ... In today's post, he repeatedly calls school prison.

At some level of their consciousness, everyone who has ever been to school knows that it is prison. How could they not know? But people rationalize it by saying (not usually in these words) that children need this particular kind of prison and may even like it if the prison is run well. If children don't like school, according to this rationalization, it's not because school is prison, but is because the wardens are not kind enough, or amusing enough, or smart enough to keep the children's minds occupied appropriately.

But anyone who knows anything about children and who allows himself or herself to think honestly should be able to see through this rationalization. Children, like all human beings, crave freedom. They hate to have their freedom restricted. To a large extent they use their freedom precisely to educate themselves. They are biologically prepared to do that. That's what many of my previous posts have been about (for an overview, see my July 16, 2008, post). Children explore and play, freely, in ways designed to learn about the physical and social world in which they are developing. In school they are told they must stop following their interests and, instead, do just what the teacher is telling them they must do. That is why they don't like school.

I agree with just about everything he writes. And yet ...

Here's my reply to his article:

Like Mark [who wrote the one reply before mine], I didn't personally experience school in such a negative way. I liked going to school. I especially liked learning math, which was not something anyone I knew did outside of school. I also liked being in an environment that was predictable. My parents were loving and alcoholic (still are) - there was the gift of lots of freedom at our house, along with the trauma of being screamed at sometimes for little or no reason. School was calm, and had its routines.

I objected to some of those routines. I stood up for the pledge, so I wouldn't be punished, but I thought about every word, and decided which parts I would choose not to recite. I got out whatever book we were currently working from, but I put another book inside, to read at my own pace. I was terribly embarrassed when I was called on while my mind was far away, but that didn't stop me. My addiction to reading got me through school still thinking. (I also recognize that I was damaged by school. I love to sing, but for years I thought I couldn't sing. I knew I sang badly when I was young, and music class didn't teach me to sing better, it just made me embarrassed. Math class probably functions that way for many kids.)

Like Mark I used to think, "Just give me harder classes, and get me away from bozos who don't want to learn." But as a teacher, I've seen that many of the students who act up in class are very smart, and do want to learn, but feel the same inner demand for freedom that made me read my own books. That particular strategy just doesn't work for them.

If school weren't required for every child, rigorous classes could kick out anyone not playing by the rules. (In karate class, for example, which most kids join freely, disrupters will be asked to leave.) But because school is a requirement for all children, that's not possible.

Peter, I love what you write, and I want to agree with all of it, but I think you're oversimplifying some parts of this. School is not the only thing wrong with our modern society, and we can't throw out schooling without changing other things first (or maybe at the same time). Parents living in difficult urban areas will tell you their kids are much better off in school than on the streets. Many people who come from low socio-economic situations will tell you that education is their hope for their children's escape. Many children, already damaged by coercive parenting and homes where thinking for yourself is considered dangerous, would use the freedom they crave in self-destructive ways.

I love math and I want to share that love. So I teach. Mainly I teach at college level, so the students are not coerced in the same ways you describe.* But they feel coerced still, and bring much of their baggage from their K-12 years with them.

If you've read Deborah Meier's book, The Power of Their Ideas, I'd like to know what you think of it. She created a school in Harlem that was (and is) part of the public schools, but works in a way that is much more respectful of each child. But it's not at all like the Sudbury schools. When I heard her speak (years ago), she was quite clear that kids were not in charge. She compared her school to a family. I think there must be contradictions even at her school, but it at least addresses the issue of class. I'd love to read a conversation between the two of you.

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* The Kaplan's have performed some real magic. In their math circles, they've created a non-coercive environment where they get to work together with the children, doing some deep mathematics.

First Day of Class

I had the kids use 12 square tiles to make as many different rectangles as they could. I showed them that the shape is written like this: 3x4 ("we say 3 by 4"). Then we talked about the area being 12 square inches, and that the x usually means times. By golly, 3x4=12, too. Our newest student, who is 7, thought that was totally funny! She laughed for each one. The 6 by 2 rectangle has an area of 12, and, yep, 6x2=12. Hee hee hee.

The size of the rectangle turns out to be the same as a multiplication problem! Imagine that... She made my day.

WCYDWT: Loop-the-loop, big possibilities

I'm having trouble following the physics in these, but it seems like they'd make some great problems for a more extended investigation. Check out a bike and a car, both going through a loop-the-loop where they are completely upside-down in the middle.

I wonder if I could design a calculus unit around these...

On Using Technology for Doing Math

Right now I'm not thinking about how we use technology in the classroom. This post is about how we use technology as a problem-solving tool (or crutch?), when we're working on math problems ourselves.

This post was inspired by a conversation with Rick, at Exploring Binary.

Rick: I have to learn to go to Wolfram Alpha for mathematical queries such as the one I mention in the article. Typing 4, 24, 124, 624, 3124 into Wolfram Alpha gives the answer I sought directly: a_n = 5ⁿ – 1.

Sue: But then you wouldn’t have had as much fun! And I’d say that’s the problem with Wolphram Alpha. Having fun with math is often hard work. It’s so much easier to click your way to an answer. But just not the same.

Rick: For me, it goes beyond Google and Wolfram Alpha — it’s the availability of computers in general. I often find myself slapping together a small program or script or spreadsheet before I sit down and think about a calculation first. Sometimes this leads to serendipitous discoveries; other times it leads to more “Well, Duh!” moments .

Maybe it’s good I started my serious study of math before the computer era really got underway. When I started high school ('70), calculators weren’t yet in common use. When I started college ('74), we didn’t have graphing calculators. I remember drawing hundreds of graphs while I was in calculus. For a long time, I felt like I wasn't a real mathematician because I never learned how to use a slide rule. All the math people I knew could use one, because they'd been doing difficult calculations for a few years more than I had, and hadn't had calculators. There was no internet (in my personal life, anyway) until well after I was done with my formal education (’89).

I saw a game for learning factoring, called Divisor Miser, at the Colorado NCTM conference in ‘82, I think. Someone had programmed it on a Vic-20 or TRS-80, or something like that. But I don’t remember computers being used to solve math problems in the computer course I taught in the mid-8o’s.

I taught junior high for a bit, and one of the math teachers wanted me to help him write a program to solve a probability problem. But he had the calculations all wrong, and the right calculations were simple enough that a computer program would have been silly. I was shocked at his bad understanding of math. He was our department chair. I knew almost no probability at the time, but I learned enough from the student materials (extra credit stuff) to figure out that problem. Yikes!

And yet, I love how technology can help us see. Here are some examples that come to mind:

1. Dan wanted to simulate an amusement park ride he calls a scrambler, and pulled up Geometer's Sketchpad. Nice results.
2. I used the matrix solution capabilities of my TI-83 when I was solving the regions in a circle problem. (Put n points on a circle, connect each to each with a straight line, how many regions in the circle?) If you already understand the solution to that, you may think this was a totally unnecessary use of technology. But it helped me to see. I'd like to post about that problem soon.
3. At last year's Math Circle Teacher Training Institute, this problem was posed: "Can you tile a rectangle completely, if you require that only squares are used and each square must be a different size from all the others? How?" We used Geometer's Sketchpad to play with our sketches of how it might work. I dropped out of the group thinking about this. When they presented their solution the next day, they mentioned having used Mathematica to solve some equations.
4. Back to my conversations with Rick. The one above is on his blog. We've been having another conversation at the same time in the comments to my post about the Carnival of mathematics. I wondered about the pattern in decimal fractions, .99 in particular, when they're converted to binary. He gave me .99 in binary, and pointed out that it repeats. I started playing with the conversion by hand, got frustrated, and turned to Wolfram Alpha for assistance. (My first time using it for a serious purpose.) Without any more tedious calculations, I could think about patterns. Turns out .9 is a repeating decimal in base 2 with a four-digit repetition, but .99's representation in binary takes 20 digits. Weird. I haven't seen the light yet on that one.
   

What do you think? Tool, crutch, or both?

Links on Sunday

How to Read Mathematics, by Shai Simonson and Fernando Gouvea. Excellent. I'll definitely be using this with my students. (There's one little problem. They say 'term', about half-way down, when they mean factor.)

Check this out, by Alison Blank: Math is not Linear. Her blog is great, too.

Cool graphic of a wave, made using Mathematica.

Detexify lets you draw online. You draw one symbol, and it'll tell you what you drew. If you'd like to start using LaTeX online, this will find symbols for you. You can also use a site called sitmo. Maria explains that here. A few people have raved about Detexify. I'm not sure why. Is it just too cool that it does character recognition? Or is it more powerful than sitmo?

Video: Bobby McFerrin on the Pentatonic scale. I looked up pentatonic on Wikipedia, and the business about hemitonic or anhemitonic was beyond me. But I liked seeing the connection with the Orff method in kid’s music education.

Off the math topic, but on the I'm-writing-a-book topic: Uncertain Principles pointed me to this post about settling in to write.