Links on Thursday

I've been saving these up, I guess ...

I keep hearing about thought experiments in physics lately. Here's a good post on that idea.

This post is mostly about John Conway, the game of life, and how that relates to segregation (watch the video).

Here's a post on the n-queens problem. (For n=8, put 8 queens on an 8x8 chessboard so none are attacking any others.) On Sunday evening I played around, trying to find a solution, and couldn't. On Monday morning I showed the problem to Artemis (the boy I tutor), who said, "That has 92 solutions, unless you don't count rotations and reflections. Then it has 12 solutions." I wanted to get past his memory and work with thinking about it. He put 4 queens on a graphpaper board, and said "now go from there". I told him I'd done that much and gotten stuck. He and I crossed out places queens couldn't go, and I suddenly saw a solution. I told him he was a good teacher. [He also showed my his attempt to multiply out (A+B)^5. He isn't yet grounded in why you do the steps you do; he did all the right steps plus a bunch more. We'll get there... :^) It sure doesn't feel like work, tutoring him.]

This one is not really math. NASA is planning to crash something into the moon tomorrow, and it should be visible with garden-variety telescopes. Lesson plans available, and I might do something with the kids I teach, if I can pull it together in time.

Tanya Khovanova doesn't like IQ tests, and has made a funny question up that she's pretty sure won't appear on anyone's IQ test.

This woman knows how to take learning into her own hands! She had a concussion, and decided to create a multi-player role-playing game called SuperBetter to help herself recover. Wow! (Thanks, Dan, for pointing me there.)

Sweeney Math has a nice systems of equations project for an algebra II class, or a stat class. "Students find data online that they are interested in comparing. (Sales of video games v sales of movies, Wins of their favorite sports team v wins of their friend's favorite sports team, Women's race times v Men's race times, Success of movie with many sequels v another, Sales of Abercrombie v sales of American Eagle, etc) They graph and find best fit lines for each set of data, then answer some thought provoking questions about the results." One of the questions is the point of intersection and what it means. I like it.

At God Plays Dice, there's an interesting (to me) book review post titled Counterexamples in X, where X is a field in mathematics.

I don't have an interactive whiteboard available where I teach, but I do have a 'smart classroom' (computer hookup and internet projection). Most of the examples in this Interactive White Boards interview can be used in my classroom, I think.

I can't remember now why I did this, but I'm still intrigued... I went to Wolfram Alpha and typed: factor 1782^12+1841^12. It's just a bunch of big numbers. Why do I like it?

Oops! There was one more. I remember from when I was young, a letter to Ann Landers, claiming that you get wetter in the rain when you run than when you walk. Here's a good physics analysis debunking that.

Why Math? Why School?

Part I. Why Math?

Deborah Meier (author of The Power of Their Ideas) co-writes the Bridging Differences blog with Diane Ravitch. Meier's topic today is 'Why School?' She is discussing what she hopes students will learn in school. Deborah Meier has done amazing work, and I usually like what I read at her blog. But her conception of math seems terribly shallow to me:

Sufficient mathematics to make sense of what they find in the media—statistics, probabilities, forms of graphing, percentages, et al to a high degree of sophistication by the time they are 16. Basic arithmetic computation by 13.

I agree that one reason to learn some basic math is to be able to have intelligent opinions about national issues: Is it more costly to have single payer health care or what we have now? Is social security doomed because more and more of our population is the elderly? I would never have expected to be interested in a blog on the tax code, but Mary O'Keefe (who runs the Albany Area Math Circle) writes great posts about issues like this on her tax blog. How can you understand national budget questions if you get nervous about numbers?

So yeah, these are reasonable goals, but dreadfully insufficient.

Here's what I wrote in response:

Deborah, I have a deep respect for you and the work you've done. So I was distressed to see your opinion of what math students should know - mostly arithmetic and statistics. Well, that's a fine start, but it is not enough.

Shouldn't they know enough math to understand science? Shouldn't they see the beauty of math? Two books, accessible to anyone, that I'd highly recommend, are The Cat in Numberland, about infinity, and Powers of Ten. (I've blogged about a number of fun math books at my blog, Math Mama Writes.)

What Diana said above about literature and history applies to mathematics as well: We will teach mathematics because it is important and beautiful. We will teach it not because it will save our society, not because we "must" know particular techniques, but because we simply do not have it in our hearts to do otherwise.

I don't feel like I was particularly eloquent. If you think math is important for more than these basic uses, please go on over there and say your piece.

And I'd love to hear from you here. Why is math important? And what math is vital for schoolchildren to learn? I love math, but I don't feel clear on why people who don't love it should learn it (beyond the basics discussed above).


Part II. Why School?

Meier labeled her post 'Why School?' in response to Mike Rose's new book with the same title. I've enjoyed his previous books, Lives on the Boundary and Possible Lives, so I expect to like this one too. Whether it will answer some of the questions I find most difficult remains to be seen.

My personal vision of the ideal school is more like a kids' community center, where the children decide how to spend their time, and are surrounded by resources and adults who want to share in the learning adventure. Deborah and her respondents talk about what should be 'required'. I don't think it's possible to require students to learn anything. The best we can do in a system based on requirements is to show the students (who are usually still eager to learn, if it appeals to their own values and priorities) why our subject is vital. That's why I love the work Dan Meyer and Kate Nowak are doing making high school math topics relevant for their students.

These two questions go deep for me: Why Math? Why school?

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.

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.

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.

Finding More Good Math Talk

My blog list in Google Reader is long. I'm making it longer today. samjshah wrote about making resolutions for the school year, and Kate mentioned "...blogs we all read". I wanted to have a list of all the math teacher blogs. Sam pointed to a list at Moving Forward. They both said looking at people's blogrolls is better than any attempt at a comprehensive list. But the list at Moving Forward is great. Maybe Scott McLeod is weeding out the not-so-great ones.

I'm not even halfway through it and I've already found:
• Solving Problems at Kiss My Asymptotes
• A cool calculator trick that goes well with the 2 payment plans (linear and exponential increase) problem, at Math Teacher Mambo. (I can't get a link for just that post. It's in July.)
• Second Time Around, a fun experiment I'll use with my stat classes, at Continuities

Enough for one day! I've added the blogroll widget so you can see my list.

Physics, Hangmath, and Math Trails

Just a few cool links, really, before I run off for a week at family camp, with no internet.

Brent, at The Math Less Travelled, wrote a great review of The Mathematical Mechanic, by Mark Levi. If I weren't on a budget, I'd buy it right now, after reading this. (I see that UC Berkeley's library doesn't have it yet, either. Bummer.)

In my morning explorations today I also rediscovered the blog of the Albany Area Math Circle, which looks like it's full of intriguing posts. First I found Hangmath, which looks like a very fun game - I'll look forward to trying it with students in September. And then, in the next post down, she described something I heard about earlier this summer, and have been meaning to explore further.

Math Trails, also known as Math Walks
While I was attending the Math Circle Institute, Amanda Serenevy gave us a leaflet about a project she'd developed for her Riverbend Community Math Center, about taking a math walk in South Bend. I can't find it now, but it looked intriguing and is another activity I'd like to try out in September. So I was excited to see Mary O'Keefe's post on Math Walks.

Most math walks involve arriving at various destinations and solving some puzzle (or math problem) presented by what you see at the location. O'Keefe's is more focused on the history of her campus, and a local mathematician. Googling 'math walks' found me this post on the Futures Channel, about Ron Lancaster's work. I noticed that he uses the term math trail, so started searching on that. Here's a site I'll come back to when I'm ready to set up my own math trail.

I also found this on Wikipedia, about Kay Tolliver, who's won a Presidential Award for her teaching. She started the National Math Trail project, where students and teachers can post information about math trails they've developed.

Anyone find any exciting math trails near them, or have any experience creating or using one?

Monday Math Madness #37: Spiral Numbers, Part 2

I really like this contest, held at Wild About Math! every other Monday. Maybe because I won a prize the first time I entered. Or maybe because the first person I encouraged to try one of these got it, after just enough effort to make his victory sweet.

Background from #36 (part 1):

Imagine arranging the positive integers in a spiral pattern.
The numbers from 1 to 16 look like this in the spiral pattern.

10  9  8  7
11  2  1  6
12  3  4  5
13 14 15 16

The location of each number corresponds to an X,Y Cartesian coordinate where the number 1 is at the origin: (0,0). 2 is at (-1,0). 3 is at (-1,-1). 4 is at (0,-1). 5 is at (1,-1). 6 is at (1,0). 7 is at (1,1) and so on.

Here's this week's contest problem:

1. Come up with an algorithm that tells what number is at an arbitrary X, Y coordinate.
2. Come up with an algorithm that tells the X, Y coordinates for an arbitrary positive integer.

Check out all the details at Wild About Math, and maybe you'll win a prize.

5 Finds

1. Mega-math. The site is way old (and clearly states it's not maintained), but I like it... Yesterday at WIldcat I tried out the story-game of The Land of Many Ponds (playing with a graph theory sort of graph), where each of the 3 players goes different distances, and one is trying to escape while the other two are trying to meet up with it (all with good intentions). Some of them wanted to play a few more times. But we didn't get into much analysis... Today I'm going to try out the Usual Day at Unusual School, a play where some characters always tell the truth and some always lie.

2. Why Is Her Paycheck Smaller? Here's a great graph from an article in the NY Times. I'm wondering how to put together a good social justice math lesson from it.

3. Here are some gorgeous sculptures. A lot of the kids at Wildcat like art lots more than they like math. I'm going to collect all the mathy art sites (and artsy math sites) I find, and see what I can do with them next fall.

4. I thought about my much-loved game of Nim while reading back posts on Denise's Let's Play Math blog.

5. A comment on dy/dan's blog led me to this article on Habits of Mind.

Math, Crochet, Coral Reefs, Global Warming, Oh My!

I was following a reference in the book What's Math Got to Do with It? (Jo Boaler), in which she quoted Margaret Wertheim as saying she was enchanted by pi. I loved the quote and thought I should read it in its original source.

I googled and found The Institute for Figuring, whose photo gallery shows hyperbolic geometry illustrated through crocheted pieces. Why might you care about hyperbolic geometry, if you're not particularly into math? (from the press page:)

For Ms. Wertheim ... the project embodies the “beauty and creativity that comes out of scientific thinking,” what she refers to as “conceptual enchantment.” As it turns out, the gorgeously crenellated, warped and undulating corals, anemones, kelps, sponges, nudibranchs, flatworms and slugs that live in the reef have what are known as hyperbolic geometric structures: shapes that mathematicians, until recently, thought did not exist outside of the human imagination.

... the Hyperbolic Crochet Coral Reef is much more than a warning about global warming. It marks the intersection of the Wertheims’ [Margaret's and her twin sister, Christine's] various passions: science, mathematics, art, feminism, handicrafts and social activism.

Now those are some renaissance women! They know that it's all connected, and Margaret is moved by 'conceptual enchantment'. I'd like to meet her some day!

(Back to looking for the quote that got me started...)

Improving Education (twitter-style)

tweet

(An experiment in Twitter-like communication in a more permanent medium, which Maria Droujkova pointed to at her blog, Natural Math.)

Learning Log #1 / Great Circles Conference at MSRI

With my bad memory, how am I ever going to remember all the cool new stuff I'm learning about lately?! I think I'll keep a log of what I've learned from the internet each week. Just this morning I learned:

• The nautilus shell is not a real-world example of the golden ratio, even though we've been told countless times that it is. ('We' would be anyone who reads much popular math exposition.) An email correspondent (here's his blog) pointed me to gowers, whose post on a book he compiled pointed me to the fascinating God Plays Dice blog, which pointed me to the Shallow Thoughts blog, in which Akanna Peck documents the mismatch of shells and golden ratio spiral. (I'd say the thoughts are anything but shallow. Her reference is to shallow sky, which is the solar system.)

• How to model a hand! Maria Droujkova's natural math google group included a discussion titled Changing Shapes with Matrices that pointed me to this youtube video on modeling a hand in 3D. My son and I also liked this one on modeling a head.

And then, of course, I learned more about tools out there on the Web. I signed up for twitter, even though I won't use it on a cell phone, so that I could read Maria D's notes on the Great Circles conference. And as much as I try to follow Maria's work, it took another site to point me to this slide show of hers about all the sorts of math-related social sites kids might want to participate in.

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Great Circles Conference at MSRI

With help from Maria's notes, here are some of the cool things those math circle folks are doing:

• Mary O'Keeffe, who founded the Albany Area Math Circle, spoke with passion about how she includes kids at different levels.

• Bob & Ellen Kaplan, from the Boston area, did math circles with younger kids. Their 'always collegial, never competitive' vision of math circles will be an important part of the anthology I'm putting together. (Their website is here.) I have already used Ellen's pancake problem (what is the greatest number of pieces we can get with n cuts through a circle?) with my Wildcat kids and some college students.

• Janet Metz's talk on Gender, Culture, and Math was riveting. She used the infamous remarks made by Larry Summers (formerly pres of Harvard, now economic advisor to Obama) that questioned women's ability to excel in math and the sciences, as a springboard for introducing some great statistical evidence to the contrary. Here's a paper she co-authored on this.

Summers imagined that men and women have the same average intelligence, but that men's varies more (more men are either kinda dumb or exceedingly smart), and imagined that this, along with women stopping to have babies, was why Harvard has no tenured women in math. (Here's a Slate article for more background.) I say imagined because the evidence is clear that the 3rd reason he gave, and then basically dismissed, namely discrimination, is clearly the dominant force. If you look at international data, for instance this page from World Economic Forum, you see that performance on math tests is closely linked to the Gender Gap Index. Of note is girls' performance in Iceland, where they do better in math than the boys do.

• Fred Smyth, of the Full Potential Initiative, gave another riveting talk on the psychological dimensions of women's under-representation in mathematics, titled Implicit Attitudes and Stereotypes matter in Math and Science.

• Tom Davis gave a great talk about using math to think about geography. His website is here.

• Zvezdelina Stankova talked about some of the stories in her new book, A Decade of the Berkeley Math Circle.

I'll edit this post as soon as I can find out who said this, but I can't leave it out. One of the programs described got kids analyzing art mathematically, and a bunch of them were standing in a museum discussing a piece. An adult walked up and asked what they were talking about, and... "Suddenly this kid, who was a failure in math, was talking to adults about topology in statues."

And 'Rithmetic, by Daniel Greenberg

I love this article, and now I can't find it anywhere on the web. So I've put it on scribd*. (There are permissions at the end for copying it.)

I was inspired to put it online by a discussion on the change.org education blog, about Sudbury schools. Someone asked about math, here's the answer.

Comments, anyone?








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*There are permissions at the end for copying it, so I know it's legal. But the folks at Sudbury school aren't comfortable with the fame this piece has gotten. So many people focus on this: "I found a book in our library, perfectly suited to the job at hand. It was a math primer written in 1898." And they want to know what book it was. But the book wasn't the important part of this story. The kids' choice to delve in was what made it work.

I asked for permission to include this piece in my book, and they wouldn't give it. So in the book I explained the important parts of the incident, and included a short quote.

Benezet - An Amazing Bit of History

L.P. Benezet was the superintendent of the Manchester, NH schools in the 30's. He decided that there was a big problem with math instruction and decided to delay formal instruction in math until 6th or 7th grade. In the earlier grades the children still learned about days of the week and month, time, money, estimating, measuring, and so on. They also focused more on reasoning and communication skills.

The complete articles, which were originally published in the Journal of the NEA, can be found here. I read the Math Forum, and saw Benezet mentioned. I posted a topic there to gather information, and another to discuss pros and cons. But the folks there insist on flaming each other, so it's a bit tedious.

Anyone here know anything more about his experiment?