Saturday, October 31, 2009
Also, can I moderate or prevent comments, or delete them? My students were brave enough to get videod while explaining math problems, and now someone has made a nasty comment.
Thursday, October 29, 2009
The Odds were at odds with the Evens. It never seemed fair to them that two Odds made an Even but two Evens didn't make an Odd. Fifteen fired the first salvo by stepping into the order twice. Sixteen managed to jump in, but then Eighteen disappeared. Seventeen and Nineteen were prime suspects. The Numerologist stepped in and told everyone to get back in the right order or ELSE. Unfortunately Eighteen was still missing. The authorities launched an investigation but there were so many factors involved that they never could get to the root of the problem.
Riddle: What do 10011, 23, and 13 have in common?
Gorgeous! The photo above, of an anglerfish ovary magnified 4 times, was taken by James E. Hayden, and won 4th place in the Nikon Small World photomicrography competition. You can see lots more winning photos here. More about Hayden at the end.
Getting down to business, allow me to welcome you to a math buffet. I think there will be something here to tickle everyone's palate. (Photo by skenmy.)
- Maria Andersen is at it again! She's redesigned her math for elementary teachers course, and shows us some of the results in Transforming Math for Elementary Ed. This post is full of links to work her students did, some of it exciting stuff.
- John Golden offers us a game called Area Block modeled on Blokus (one of my current favorites to play with kids). I haven't had a chance to play it yet, but I am definitely looking forward to it. (Here's a bit of math teachers at play trivia. John and Maria work within 30 miles of each other in my home state. What state is it?)
- Denise walks us through an algebra word problem, translating from English to "mathish". Nice! And Jason wants our help with teaching students how to read a few different types of problems, in "When vocabulary isn't the issue" and "A reading experiment". The puzzle given in that second post looks fun!
- Pat Ballew gets us thinking about geometry with his "Notes on Cyclic Quadrilaterals".
- Do we discover math or invent it? How we answer that question affects how we talk about the math of the ancients. Dan MacKinnon reviews a book and discusses this intriguing issue.
- A short review of The Little Book of Mathematical Principles is at We Overstep.
- Pumpkin Patch offers us another game, Sum Math.
- The descriptions don't always look accurate to me, but you may find some goodies among the "100 Incredible Open Lectures for Math Geeks". Some are audio only, some are videos.
If you haven't stuffed yourself yet, here are a few more tidbits I noticed over the past two weeks.
- Two mysteries solved, one a murder, and one the opening chord of Hard Day's Night.
- Sometimes there's a hard way to solve a problem, and an easy way...
- Does a bullet that's fired from a gun land at the same time it would if it were dropped? Built on Facts gives us some equations and Myths Busters weighs in on YouTube.
- On thinking before you compute.
- Like comics? Indexed on confusion as a function of information, and a graphic novel about Bertrand Russell.
Here's the interview Nikon did with James E. Hayden. I wrote and asked him how he uses math in his work. He said:
As for the math - it is, of course a large part of doing any kind of research. In a lot of my regular work, we work with analysis programs that quantify different aspects of the images we capture. Everything from simple counts (how many cells in the field of view?) to time lapse analysis - how fast are the cells moving? In what direction? At what angle to the original movement? Does the rate of movement change over time? Do we need to quantify the interaction of the cells in some way? Is there a change in the fluorescence intensity values? And other interesting things like that. My assistant, Fred Keeney (who won an Image of Distinction in this year's competition) has been taking computer programing classes to help us automate these kind of analyses.Submit your blog article to the next edition of math teachers at play using our carnival submission form. Past posts and future hosts can be found on our blog carnival index page. (Our schedule is changing to once a month. Denise at Let's Play Math! will host the next carnival on November 20.)
Saturday, October 24, 2009
I am personally sponsoring a contest for the best little (ie, very short) story written about how the numbers got mixed up this way. The winner gets their story included in the next MTaP, which comes out here on Friday, and gets a $10 gift certificate at Better World Books. I get to be the judge. :^) It could be funny, mysterious, intriguing, whatever will be memorable.
Deadline: Midnight (PST) on Wednesday
Why? Partly to have fun with our glitch, partly to 'make math our own' as Maria Droujkova likes to say. And partly because I came to math on the internet through 'living math' - Julie Brennan's (trademarked) term for math through stories, and for the list she runs for mostly homeschoolers and the site she maintains with gazobs of ideas for how to engage kids in mathematical thinking through math stories and finding math in any story.
Sunday, October 18, 2009
Last Thursday I posted a bunch of links, and included:
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?On Saturday night Joshua Zucker replied:
why 1782^12+1841^12? I don't know why to factor it, but of course it equals 1922^12 (just try it on your calculator, not at Walpha of course!)Well, I wasn't thinking of the consequences, and I believed him. I was impressed that he could use his calculator to factor the huge number you'd get from 178212+184112. I didn't reach for my calculator, being comfortably ensconced in my recliner. No, I reached for Google, and I googled Josh, because I was curious about what math he might lead me to.
I found a comment he made on Cut the Knot many years ago (in 2000). So I started exploring Cut the Knot, and thoroughly enjoyed some discussions about how math uses words differently from their common usage.
Eventually I remembered my original quest and googled "1782^12+1841^12". The very first thing I got was:
Fermat's last theorem. Statement that there are no natural numbers x, y, and z such that x^n + y^n = z^n, in which n is a natural number greater than 2. ...[Fermat's last theorem was proved in 1995 by Andrew Wiles. There are lots of whole number solutions to x2 + y2 = z2, like 32 + 42 = 52. But the theorem says there are no solutions to x3 + y3 = z3, nor to equations like that with higher powers.]
Huh? But Josh said 178212+184112=192212?? Next entry I clicked on was about a Simpson's episode:
In the 1995 Halloween episode of the award-winning animated sitcom The Simpsons, two-dimensional Homer Simpson accidentally jumps into the third dimension. During his journey in this strange world, geometric solids and mathematical formulas float through the air, including an innocent-looking equation: 178212 + 184112 = 192212. Most viewers surely ignored this bit of mathematical gobbledygook.Ahh, now I get it! And I finally had a vague memory of reading a post somewhere about how 178212 + 184112 and 192212 look exactly the same if you evaluate them on a calculator. (Try it!) That must be why I had originally gone to Wolfram Alpha with this.
On the fan discussion site alt.tv.simpsons, however, the equation caused a bit of a stir. “What’s going on, he seems to have disproved Fermat’s last theorem!” one fan marveled, referring to the famous claim by Pierre de Fermat—proved just months earlier—that for any exponent n bigger than 2, there are no nonzero whole numbers a, b, and c for which a^n + b^n = c^n. The Simpsons equation, if correct, would be a counterexample to the theorem, meaning that the proof had been wrong.
Meanwhile, here's the other treasures I found:
Friday, October 16, 2009
Educational research that doesn't acknowledge this messiness, that tries to buy 'scientific' cachet with control and treatment groups but frames the questions too narrowly, is more likely to reinforce the values of one group than to deepen our understanding of the learning and teaching enterprise. (Of course, we're each more likely to see flaws like this in research that doesn't resonate with our own values.) In this post, I want to dissect a flawed study, together with my friend Ben, and then give links to some studies I've enjoyed reading. I would actually appreciate it if any of you would like to point out flaws in some of those. (I'll start a new post for each one, so we don't get all tangled up.)
“The Advantage of Abstract Examples in Learning Math”, a study done by Jennifer A. Kaminski, Vladimir M. Sloutsky, and Andrew F. Heckler, was published in Science magazine in April 2008 (you need a paid subscription to see the article) and highlighted in the New York Times soon after. I had responded to this study before I started blogging, in email to colleagues. It was brought to my attention yesterday by Ben Blum-Smith's new blog, Research in Practice, where he writes a great critique of it. I agree with all he says (here and here) and want to add a bit more.
The people who did the research, along with the unquestioning NYT author, say that the research shows that students learn math better with abstract examples than with concrete examples. Ben and I are saying that their research design is severely flawed, and that they've shown nothing useful.
Ben gives a great description of what the research folks were supposedly trying to teach, which was the properties of a "commutative mathematical group of order three". That's a fancy name for something not much more complicated than clock arithmetic. Imagine a clock that only has three hours on it, and instead of a 3 at the top, it has a 0. So 1+1 is still 2, but 1+2=0 and 2+2=1. This is the most common example used when people are first learning about these groups. The examples used in the research seem very contrived in comparison.
Both the research folks and Ben described a tennis ball factory, where you're keeping track of how many balls you have in hand, after you've put as many as possible into those 3-ball cans, but Ben's description makes a lot more sense than the one used in the research. (When I originally read this study over a year ago, I never saw the tennis ball example. In the one 'concrete' example I was able to find details for, they used a full cup for the 0, or identity element, which I would find confusing.)
People who actually study groups like this sometimes do it without numbers. The 'elements' of the group might be labeled a, b, c instead of 0, 1, 2. There are properties that can be studied, like identity elements and inverses. (0 is the identity because adding it to other elements doesn't change them. 1 and 2 are inverses because 1+2=0, the identity. These properties can make sense even when the elements aren't numbers.) So the researchers 'taught' this using 'abstract' examples for some subjects and 'concrete' examples for others. They quizzed all of the subjects using a group consisting of a vase, a ladybug, and a ring. Although concrete, these strange elements fit much better with the 'abstract' example than with the 'concrete' example. It's not surprising that the subjects whose example was more similar did better when quizzed.
There is lots of narrowly focused research like this out there. It may be useful in physics to narrow a question down to one detail, when the interactions between the small parts is clear. But in social arenas, all the parts interact, in very complex ways. Research like this cannot tell us much of value, even when its design is less flawed.
Kaminski et al want to say that their research tells us children learn math better without concrete examples. Their claim is very political. To promote it, they have done a number of studies with minor variations. (Googling their names, I see work done in 2003, 2006, and 2008.) I'd rather see education research that addresses the big, messy picture. Here's the MAA president-elect's take on this, and here is an article interviewing one of the 3 authors of the study.
Research that I've found more interesting is much broader. It doesn't attempt a double-blind statistical power, which can only come through narrowing the questions until they become too artificial to be of use.
I've been reading Jo Boaler's work on the benefits to students at all skill levels of working together with one another. On first glance, it may seem that tracking would allow the best students to go farther, and allow slower students to get a more solid grasp on what they're studying, but tracking actually harmful to students at both ends and those in between. Boaler shows why, and shows how to make heterogeneous grouping work. One article is here, or you can read her book, What's Math Got to Do with It?
Alan Schoenfeld wrote the book Mathematical Problem Solving, in which he describes his very detailed research into the process followed by math students versus mathematicians while attempting solution of a hard problem. He taught a course in problem-solving strategies, using Polya's framework, and gave a pre-test and post-test to those students. Compared to students taking a more typical math course, these students' problem-solving skills improved significantly more. Here is an article of his on a different topic, how mathematical conversation in the classroom promotes learning.
Both Boaler and Schoenfeld compare groups with and without the 'treatment' they believe is effective, and show evidence for their belief, as Kaminski et al did. There are other sorts of research, which attempt to understand children's learning processes, but which don't have 'control groups'. One such project I'm interested in following is Measure Up, which introduces math through measurement and algebraic reasoning. Here's an article from that project.
Our biggest problem may not be understanding better how children learn, but implementing the good ideas that come from this research. When most elementary teachers are uncomfortable with mathematics, what we need to focus on is how to help them. Liping Ma's book, Knowing and Teaching Elementary Mathematics, compares elementary teachers in the U.S. and China. The Chinese teachers understand the math much more deeply. A good summary of the book is here. And an article by Ma is here. And here's a piece by another researcher, Hung Hsi Wu, on the depth of understanding needed by elementary teachers.
What math education research have you found valuable?
I think I had an interview at that time at Bechtel. (It was someplace with more security than I'm accustomed to.) I know I thought a lot about how most of the employers who might want my talents were doing things I didn't approve of. Code-breaking sounded fascinating, but back then code-breaking meant working for the government (or so I thought), and I was no happier then than I am now about my government's warring tendencies.
I ended up doing some computer programming for a very small company. It was business reports - not very exciting, but nothing terrible, either. After a bit more than a year, the company folded, and I headed toward teaching.
I haven't had to think about this issue much in the 30 years since then. I've seen much broader uses of math, and coding-breaking has come to be identified more with information security than with espionage. But of course the war-makers are still big employers of mathematicians, and that was made clear to me this morning by a post that looked really fun at first.
Liz, at STEM-ology, posted It's a Math World, After All, about a cool new 'ride' at Disney World called Sum of All Thrills, that lets kids (and adults?) design their own ride, and then "experience it on a giant robotic arm simulator." (It reminded me of the turtle geometry I was just reading about in Mindstorms.) I loved it!
But then I followed her link to the original Yahoo article, and saw this:
"Sum of All Thrills" sponsor Raytheon has nothing to offer the average consumer. But the high-tech defense and homeland security contractor does have jobs for those passionate about engineering...
I wasn't planning on going to Disney World any time soon, but Disneyland was the high point of a big trip my family took when I was 12, and I might be willing to take my son some day. This is a reminder to me of how much corporate propaganda is built into places like that. My comment on Liz's blog ended with "That's a show-stopper for me. War-mongers get way too much access to our children, and I have a problem with that..."
Have any of you struggled with math's less wholesome uses?
Thursday, October 15, 2009
Two of the posts I especially enjoyed:
John Cook has posted on the math behind musical scales in his posts Circle of fifths and number theory and Circle of fifths and roots of two at his blog The Endeavor.
Alison Blank has put together an inspired and inspiring Prezi presentation, Math is Not Linear, and posted about it on her blog Axioms to Teach By.
I added quite a few blogs from this issue to my already long list at Google Reader. I think I'm really going to like the quirkiness of this blog:
Vlad Alexeev shows us an impossibly small book of impossible figures in the post Mini Books of Anatoly Konenko at his blog Mathematical Paintings and Sculptures.Next issue is in two weeks, right here. After that, it looks like we're switching to a once a month schedule, on the 3rd Friday of each month (with Carnival of Mathematics taking 1st Fridays).
Wednesday, October 14, 2009
What's that got to do with math? Well, we won't have time for the cerebral pleasures of math if we're dealing with floods and droughts, famine and war.
Here's a site that give some numbers to help you think about it. I try to live simply, but if everyone lived like I do, we'd need over 2 planet Earths.
I wanted to write more, but this is already coming out late in the day...
Thursday, October 8, 2009
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.
Sunday, October 4, 2009
He starts the book with a story from his childhood, about how he was in love with cars, and at two knew about "the parts of the transmission system, the gearbox, and ... the differential" (more than I know even now). He adds:
I became adept at turning wheels in my head and at making chains of cause and effect: "This one turns this way so that must turn that way so..."But of course not all children will fall in love with gears the way he did, hence his "attempts ... to turn computers into instruments flexible enough so that many children can create for themselves something like what the gears were" for him. He points out many ways in which the gears encouraged his understanding of mathematics, including affect (he loved them), body knowledge (he could turn his hand or body the way the gear turned while he was thinking about it), and flexibility as a model for mathematical structures. His creation on the computer, the LOGO language, included a turtle on the screen (or a robotic turtle) that could be moved around.
Gears, serving as models, carried many otherwise abstract ideas into my head... I saw multiplication tables as gears, and my first brush with equations in two variables (e.g., 3x+4y = 10) immediately evoked the differential. By the time I had made a mental gear model of the relation between x and y, figuring how many teeth each gear needed, the equation had become a comfortable friend. (page vi)
He worked with Piaget for years, and has a similar clarity about the deep learning that must happen for children to understand things that seem very basic to us adults. He has differences with Piaget, though, and the most salient here is his conviction that the cultural environment makes a difference in when kids will learn things. To learn formal systems like mathematics, it helps for kids to have a fun "world" to play in that uses formal systems, like LOGO. So Piaget saw the 'formal reasoning' stage of development happening around 12, and Papert thinks much younger children can do formal reasoning if given the right environment.
He has a lot to say about the damage wrought by the culture associated with schooling:
Our children grow up in a culture permeated with the idea that there are "smart people" and "dumb people". The social construction of the individual is as a bundle of aptitudes. There are people who are "good at math" and people who "can't do math". Everything is set up for children to attribute their first unsuccessful or unpleasant learning experiences to their own disabilities. ... Within this framework children will define themselves in terms of their limitations, and this definition will be consolidated throughout their lives. Only rarely does some exceptional event lead people to reorganize their intellectual self-image in such a way as to open up new perspectives on what is learnable. (page 43)Of course kids in school hate making mistakes, and want to throw the mistakes away, or run away themselves. But if they're doing programming on a project they care about, the mistakes become bugs that need fixing, not testaments to their inadequacy, and they become willing to debug. The more they get into that habit, the more willing they'll be to deal with future 'mistakes' that way.
Most of us learned Euclidean geometry in high school, with its axioms, straightedge and compass, and our first taste of proofs. (There are alternates to this, non-Euclidean geometry and Origami geometry, that still use a system of axioms and step-by-step deductive proofs.) Analytic geometry uses the x and y coordinate system to connect algebra and geometry. Papert mentions those two and then talks about how turtle geometry is both easier for kids to connect with (tell the turtle how to move in a circle, by figuring out how you'd do it) and more sophisticated (it has a deep connection with calculus). Once a child has really played with turtle geometry, they're likely to feel more at home as they learn about other geometries. Papert goes into how using turtles to think about physics is likely to lead into some deep science learning, too.
Reading Mindstorms motivated me to find and download Scratch, a modern descendant of LOGO, and start learning it. Scratch has 'sprites' instead of the turtle. You can create as many sprites as you want, and give each one a script. This week I've brought my computer in to Wildcat, where I teach kids in a very free-form environment, so they can play with Scratch. They are loving it. I'll probably post soon about that.
While I was online, searching for more information about Papert's recent work, I discovered that he'd been in a tragic accident. While in Hanoi in December 2006 for a conference, he was hit by a motorbike and suffered a severe brain injury. There is hope he will eventually recover, but he hadn't yet as of July of 2008. Here's the news article from then. I've searched and haven't found anything more recent. I'm wishing him well.
I want to include so many quotes, but I think I'll just write more posts on this later. If you want to think deeply about how children (and adults) learn, read this book. If you want a fresh perspective on how computers might be used with children, read this book. If you want more reasons to shake your head over the current testing craze in the public schools, read this book.