Sunday, May 30, 2010

Big Money, Big Oil, and the Mess Corporations Make

Doug Noon at Borderlands has written another post that strikes just the right note for me.   This one is partly about a math skill we all have trouble with, comprehending big numbers. Do you really understand the difference between a million, a billion, and a trillion dollars?

What's a billion? Bill Gates' wealth increased by $50 billion dollars one year. If he wanted to share that income with every one of the 800,000 people in San Francisco, they'd each get $62,500. This video, from David Chandler, online at the L Curve, gives the best visualization I've seen yet of how skewed income distribution is in the U.S.

That kind of money buys whatever it wants. There are two legs or spikes on this graph, a horizontal one representing most of the population, and a vertical one representing most of the income. This graph represents income; wealth is even more skewed*.

David Chandler:
The horizontal spike has the votes. The vertical spike has the money. Who wins, when it comes to electoral politics? Who has influence? Whose interests are being represented in Washington? Can democracy meaningfully exist where the distribution of wealth, and thus the distribution of power, is this concentrated?
The Supreme Court recently affirmed the personhood of corporations (where much of this wealth sits). If BP were really a person, what would that mean? Pretty hard to put the corporation itself in prison for its criminal actions. And pretty hard to control a 'person' who may be immortal and can buy legislators, along with avoiding prison. Kinda like godzilla, maybe?

*Wealth is what you have; income is what you get, dw/dt (change in wealth with respect to time). The richest one percent are now estimated to own between forty and fifty percent of the nation's wealth, more than the combined wealth of the bottom 95%.

Wednesday, May 26, 2010

What must be memorized?

I say I have a terrible memory, and that's why I'm in math instead of science (with its myriad names for bones, muscles, organs, chemicals, etc.). I say there's very little to memorize.

But other people disagree. I used to think it was only my students who did badly at math who had the 'wrong' idea that math was about memorizing. Math people will tell you it's not about memorizing, but many of them do think memory has a bigger place than I give it.

What do you think?

Here's what needs to be in your memory, eventually:
  • addition 'facts' (if add 1, double, and add 2 to an even are easy, then there are 25 harder facts)
  • multiplication 'facts' (2 is doubling, 4 is doubling twice, 5 is easy, 9 can be easy; only 9 facts left)
  • circumference and area of a circle (I know which is which by using dimension - circumference is 1D and has r, area is 2D and has r2. If I'm unsure of the formula, I can also check reasonableness of my answer by estimation.)
  • Pythagorean theorem (a2+b2=c2 for right triangles)
  • quadratic formula 
  • [edited to add ...] sine and cosine, tangent and reciprocal identities (the rest of trig pretty much follows, 6 facts)
That's a pretty short list. What would you add to it?

When I started teaching, I had a BA in math and still didn't have 7x8 memorized. I also didn't have the quadratic formula memorized. Whenever my topic of the day would include quadratic formula, I'd put it at the top of my notes. 7x8 I could figure out in 2 or 3 seconds.

To me, math is all about connections. When I think of slope, I think of a rise and a run (and a little right triangle under the slanty line, showing them). I know steeper lines want a bigger slope, so the rise has to be on top in the ratio that compares rise and run. To me, that's not memorizing.

So tell me what I'm forgetting.  ;^) What more do we have to memorize in math?

    Tuesday, May 25, 2010

    Logarithms and Ropes (as found in Mathematician's Delight)

    I recently got a copy of Mathematician's Delight, by W. W. Sawyer. I had loved his book Vision in Elementary Mathematics, so I knew I'd like this one. I found out about it through a blog I stumbled upon in my wanderings, where the blogger included this:*
    Nearly every subject has a shadow, or imitation. It would, I suppose, be quite possible to teach a deaf ... child to play the piano. ... [The child] would have learnt an imitation of music, and would fear the piano exactly as most students fear what is supposed to be mathematics.

    What is true of music is also true of other subjects. One can learn imitation history - kings and dates, but not the slightest idea of the motives behind it all; imitation literature - stacks of notes on Shakespeare's phrases, and a complete destruction of the power to enjoy Shakespeare.
    I think that idea, of a shadow subject, will stick with me, and become more powerful for me over time.

    I've told my students logarithms were invented in a time when calculators didn't exist, and scientists were looking at lots of data about the planets, trying to discover patterns. Napier invented a way to do multiplication by adding and division by subtracting, a second application of which allows powers and roots to also become questions of addition and subtraction. I don't think this is enough of an introduction to this strange concept.

    How did Napier dream this up? Sawyer gives us a glimmering of the sort of inspiration Napier might have had, with this marvelously concrete model for logarithms:
    We are all familiar with machines which [we] use to multiply [our] own strength - pulleys, levers, gears, etc. Suppose you are fire-watching on the roof of a house, and have to lower an injured comrade by means of a rope. It would be natural to pass the rope round some object, such as a post, so that the friction of the rope on the post would assist you in checking the speed of your friend's descent. In breaking-in horses the same idea is used: a rope passes round a post, one end being held by a person, the other fastened to the horse. To get away, the horse would have to pull many times harder than the person.

    The effect of such an arrangement depends on the roughness of the rope. Let us suppose that we have a rope and a post which multiply one's strength by ten, when the rope makes one complete turn. What will be the effect if we have a series of such posts? A pull of 1 pound at A is sufficient to hold 10 pounds at B, and this will hold 100 pounds at C, or 1000 pounds at D.

    Thus, 108 will represent the effect of 8 posts. ... The number of turns required to get any number is called the logarithm of that number. ... So far we have spoken of whole turns. But the same idea would apply to incomplete turns. ... Accordingly, 101/2 will mean the magnifying effect of half a turn. ... The logarithm of 2 will be that fraction of a turn which is necessary to magnify your pull 2 times. (page 70)

    I had to put the book down here, to ask myself why half a turn wouldn't magnify the pull 5 times - half of ten. As I thought about that, I wanted to know if there would be an easy way, either a thought experiment or a very simple physical experiment (i.e., no special equipment), to prove that this relationship must be multiplicative. That is, how do we know the friction of the rope doesn't just add to our pulling force, so that a certain amount is added at each turn? (Can anyone help me with this?)

    If we've decided that the relationship must be multiplicative, then we know that two half turns must multiply to have the effect of one whole turn, and that would mean we need the number that multiplied by itself gives ten. To get to this thought, I had to imagine two posts near one another, with the rope halfway around one, and then halfway around the next.

    Why haven't I seen this before?!

    I haven't read any more of the book yet, because I keep needing to think more about this cool idea. I look forward to more pedagogical delights as I keep reading this book, and maybe others he wrote. (One list is at the bottom of this page.)

    *W. W. Sawyer wrote this book in 1943, long before feminists began to analyze the effect of using the male for the generic. Although Sawyer uses 'man' and 'he' in a generic sense in other sections (which I've taken the liberty of changing in the second quote I've used), perhaps he was trying to avoid that in this story by calling the deaf child of his music example 'it'. I had real trouble with that, and didn't know how to fix it without messing with his meaning, so I left the meat of the example out. You can go here to see it.

    Sunday, May 23, 2010

    3rd Annual Math Circle Teacher Training Institute

    This will be my third year attending this fabulous program, hosted by Bob & Ellen Kaplan (founders of the Boston area math circles, and authors of Out of the Labyrinth: Setting Mathematics Free and many other intriguing math books),  Amanda Serenevy (founder of Riverbend Community Math Center), and Leo Goldmakher (joining them for the first time this year).

    I love it. The first year I was amazed that I could do math all day long and never get tired of it. There's lots of freedom to explore in whatever way works best for you. Notre Dame provides dorm housing, dorm food that's better than you'd ever imagine dorm food could be, and a beautiful setting (including gym facilities, I swam just about every day).

    Arrive: Sunday, July 4 (or as early as you can on Monday, July 5). Notre Dame is in South Bend, Indiana, 2 hours from Chicago.

    Five days of math circle mania:
    • we participate in math circles in the mornings,
    • we run one math circle for kids late one afternoon, and watch our colleagues try it out on the other afternoons,
    • we discuss the ins and outs of math circles and other fun ways of doing math in the early afternoons,
    • we play with Amanda's collection of math toys and browse her collection of math books, whenever we want,
    • we plot, plan, and socialize in between all that.
    Depart: Saturday, July 10

    Total cost: $800

    Email kaplan at if you're interested in joining us.

    Friday, May 21, 2010

    Math Teachers at Play #26

    Math Teachers at Play #26 is out at Math Hombre. John has more interesting little facts about 26 than you could have imagined. And so many of the posts are by bloggers I hadn't known about yet - I'll be exploring for days.

    John included a great puzzle:
    Hmmmm.  This will be the only Math Teachers at Play ever directly after a square and before a cube.  (Proof.)  Of course, in a 3x3x3 cube, only 26 cubes are visible, so you're really taking the 27th on faith.  I guess that makes 26 the third Rubik's Cube number?   If 8 is the previous and 56 is the next, what is the fifth Rubik's Cube number?  What is the closed form for the nth Rubik's Cube Number? 
    I liked Princess Kitten's post on playing an online game called Cops and Robbers, and from a comment there, I found this cool site - ThinkQuest: "Over 7,000 websites created by students around the world who have participated in a ThinkQuest Competition". I really like seeing math content from kids.

    I loved finding out in this post that someone has used a bit of math (or engineering) know-how to help with the difficult situation the people of Haiti are facing.

    Lots more goodies over there - Check it out!

    Wednesday, May 19, 2010

    Reading a Math Book

    Students really stumble here. They prefer to just look at examples, to help them with the problems. Actually reading a whole section is daunting for most of them. On my syllabus, in a section on preparing for class, I've included this for years:
    Read the next section. (Reading a math book is different from other reading. You may have to read a section 3 or 4 times in order to fully understand it.) Take notes, using your own words; I’ll check them a few times during the term. (This gets easier over time, and you’ll become a more independent learner.)
    But that's not enough. Last year, I read How to Read Mathematics, and loved it. The level was way too high for most of my students, though, so I made my own handout. I'm still working on it.  Maybe you have suggestions?

    Here it is:

    How to Read A Math Book

    Have you noticed that you read differently when you’re reading different sorts of books? Good fiction takes you away, you can’t put it down. But sometimes just the opposite happens with a good non-fiction book – you have to put it down, so you can think more about the ideas. Math books are extreme that way.

    Don’t forget poetry and plays, where reading aloud can be what makes it work, or cookbooks, where every word makes a difference (that’s like math!), or reference manuals, where you wouldn’t want to read straight through, but really need to skip around, and try out the suggestions.

    So how do you read a math book? Here are some ideas. I’ve mixed my own ideas together with ideas from an article by Shai Simonson and Fernando Gouvea ( They wrote for higher-level math students, so I’ve just kept the nuggets that seemed approachable for this course. Their words (sometimes modified) are indented.

    When we read a novel we become absorbed in the plot and characters. We try to follow the various plot lines and how each affects the development of the characters. We make sure that the characters become real people to us, both those we admire and those we despise. We do not stop at every word, but imagine the words as brushstrokes in a painting. Even if we are not familiar with a particular word, we can still see the whole picture. We rarely stop to think about individual phrases and sentences. Instead, we let the novel sweep us along with its flow and carry us swiftly to the end. The experience is rewarding, relaxing and thought provoking.

    Novelists frequently describe characters by involving them in well-chosen anecdotes, rather than by describing them by well-chosen adjectives. They portray one aspect, then another, then the first again in a new light and so on, as the whole picture grows and comes more and more into focus. This is the way to communicate complex thoughts that defy precise definition.

    Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. A mathematical piece and a novel are both telling a story and developing complex ideas, but a mathematical piece does the job with a tiny fraction of the words and symbols of those used in a novel.

    1. Look for the Big Picture

    “Reading mathematics is not at all a linear experience ...Understanding the text requires cross references, scanning, pausing and revisiting.” (from Emblems of Mind, by Edward Rothstein)

    Don’t assume that understanding each phrase will enable you to understand the whole idea. This is like trying to see a portrait painting by staring at each square inch of it from the distance of your nose. You will see the detail, texture and color but miss the portrait completely. A math article tells a story. Try to see what the story is before you delve into the details. You can go in for a closer look once you have built a framework of understanding. Do this just as you might reread a novel.

    2. Be an Active Reader (and Expect to Read Slowly)

    A math article usually tells only a small piece of a much larger and longer story. The way to really understand the idea is to re-create what the author left out. Read between the lines.

    Mathematics says a lot with a little. The reader must participate. At every stage, s/he must decide whether or not the idea being presented is clear. Ask yourself these questions:
    • Why is this idea true?
    • Do I really believe it?
    • Could I convince someone else that it is true?
    • Why didn't the author use a different argument?
    • Do I have a better argument or method of explaining the idea?
    • Why didn't the author explain it the way that I understand it?
    • Is my way wrong?
    • Do I really get the idea?
    • Am I missing something?
    • Did this author miss something?
    • If I can't understand this can I understand a similar but simpler idea?
    • Which simpler idea?
    • Is it really necessary to understand this idea?
    • Can I accept this point without understanding the details of why it is true?
    • Will my understanding of the whole story suffer from not understanding why the point is true?

    Putting too little effort into this participation is like reading a novel without concentrating. After half an hour, you wake up to realize the pages have turned, but you have been daydreaming and don’t remember a thing you read.

    Reading mathematics too quickly results in frustration. A half hour of concentration in a novel might net the average reader 20-60 pages with full comprehension, depending on the novel and the experience of the reader. The same half hour in a math piece buys you a page or two, depending on the topic and how experienced you are at reading mathematics. There is no substitute for work and time.

    3. Make the Idea your Own

    The best way to understand what you are reading is to make the idea your own. This means following the idea back to its origin, and rediscovering it for yourself. Mathematicians often say that to understand something you must first read it, then write it down in your own words, then teach it to someone else. Everyone has a different set of tools and a different level of “chunking up” complicated ideas. Make the idea fit in with your own perspective and experience.

    Monday, May 10, 2010

    Day of the Teacher: Who Helped You Grow?

    The way NCLB is set up, underachieving schools are blamed - instead of help, they get dismantled. Instead of recognition that the kids whose families don't have enough money will move more, be hungry sometimes, and have lots of other life events interfering with their learning, we get blame for the teachers who didn't bring their test scores up, and mass firings in Rhode Island, approved of by a Democratic president who won through slogans like Hope, and Change Is Possible. President Obama, that kind of blame is not helpful, and I want more change than I've seen so far! Your Race To the Top is in the same vein as NCLB, and the competition it fosters could be worse.

    Meanwhile, May 12 is Day of the Teacher (according to Kelly Kovacic). I'd like to spread the message of respect and appreciation. I'm going to add my thanks to a few of my favorite teachers from long ago, and maybe you can too.

    Thank you, Mr. West, for the first science class that really challenged me to think. I loved dissecting worms, starfish, frogs, fish, and all that. I learned from the world, instead of a book. (Now, 40 years later, I'm wondering if animal rights readers will be upset with me for unquestioningly loving that.) I was eager to take physiology and anatomy from you, where we got to dissect a cat, and our tests were lab practicals. I don't remember much detail, but I have images in my mind, even now, from your two college-level course I got to take in high school.

    Thank you, Mr. Scalabrino, for your poetry lessons. You took us to the mall, a cemetery, and a bunch of other unusual writing locations.  You made us sit in the front hall to watch other students pass by. You taught me to observe details. I wrote my adolescent poems about tasting the morning, and all the senses anger reaches. When I started writing more mature poetry twenty years later, I took myself seriously in part because you had taken us seriously.

    Thank you, Miss Purvins, for taking us to Western Michigan University, to see a Shakespeare play. Thank you for facilitating discussion in your class that challenged me to think. I still remember being amazed at that the level of discussion there, my first experience of real literary engagement.

    Thank you to a philosophy T.A. in college whose name I don't remember. (Contemporary moral problems, around 1978, University of Michigan) You sat with me at Dominick's, and we recorded our conversation about euthanasia. That helped me write a much better paper. You also helped me work on editing those run-on sentences. (I haven't been completely cured, I know...) Now I'm working on a book. I wish I remembered your name!

    Thank you, George Piranian. I wasn't quite ready for the rigor of your honors calculus class, and had to call you out on "every man should climb a mountain, and write a poem". (You responded gracefully, I believe.)  I worked 4 hours a night on those problems you gave us, and this time the challenge was sometimes too much. (I might just pull that notebook out, and show some of those problems to the boy I'm tutoring now.) I wasn't used to low grades, and came to you for help. You couldn't make that mountain easier to climb, but you let me know you were willing to walk alongside me.

    Thank you, Robin Jacoby. Throughout high school, I thought history classes were about wars and presidents, and keeping track of dates and titles. In your Theories of Feminism course, I learned the history of feminism through source documents, and we discussed why things happened when they did, what it meant for women, and so much more. I learned that history was fascinating.

    Thank you, Gisela Ahlbrandt and Kim Rescorla, for letting me know (once again) that I really do love math. U of M had convinced me that I didn't like it, and you two, my first teachers at Eastern Michigan University, got me over that delusion.  Gisela, your classes were the hardest classes I ever loved. Dr. Rescorla, when I took your course in Linear Algebra, I was so proud to be able to understand everything and ask questions that went a little beyond the text. My thanks to all the other great teachers I had at EMU, too. I wish I remembered everyone's names.

    When I began teaching, I made a list of all my favorite teachers, to try to help me figure out what great teachers do. I noticed an interesting divide. The great male teachers I'd had were great performers, and the great female teachers I'd had were great facilitators. I wanted to be both. I'm still learning.

    Dear Readers, would you all write about your favorite teachers sometime soon, and link to it here? Let's toot each other's horns.

    Thursday, May 6, 2010

    Link Love: Aunty Math

    I was exploring Tiff's old Math Monday posts, and came across her post on using Delicious. That led me to Aunty Math (aka Aunt Mathilda Mathews), who shares a bunch of cute little stories with a math challenge in each one.

    James Tanton would like this quilt story, though for him the puzzles came packaged in a pressed tin ceiling*. And this camping trip story will be perfect to take to my son's school, since they just got back from a camping trip. The problem solving strategies are also done nicely.

    If I'm Math Mama, does that make Aunty Math my sister?  ;^)

    Let me know what you think of it.

    * James tells this story in his introduction to Thinking Mathematics, Volume One:
    My career as a mathematician began at age ten. I didn't realize this at the time, of course, but in retrospect it is clear to me that my journey into the rich world of mathematical play - and I use the word play with serious intent - was opened to me thanks to a pressed-tin ceiling in an old Victorian-style house.

    I grew up in Adelaide, Australia, in a house built in the early 1900s. The ceiling of each room had its own geometric design and each night in my bedroom I fell asleep staring at a 5x5 grid of squares above me, lined with vines and flowers.

    I counted squares and rectangles in the design. I traced paths through its cells and along its edges. I tried to fit non-square shapes onto the vertices of the design. In short, I played a myriad of self-invented games and puzzles on that grid of squares as I fell asleep.

    Monday, May 3, 2010

    Backing Up (We interrupt this blog for a Public Service Announcement)

    I have a whole book sitting on my computer, and nowhere else. What precious data do you have on yours?

    I have two external hard drives, and was trying to keep one at my office, the other at home. I was also trying to backup weekly. But I probably backed up about 5 times total. When Sam mentioned on his blog, I decided to give it a try. (It is free, after all.)

    I'm at 88% 79%* of the free storage limit, without having uploaded my photos, so the free option may be a bit small. (It's 2G.) But it's working great for me so far. My laptop and desktop now stay in sync with each other on all the stuff I included. (I left out my personal business folder, and my photos - didn't want that stuff floating around online. I also didn't try to include application files.) There's a third copy of all my information online, too. It's all automatic now, and I'll be able to grab stuff from my computer when I'm on vacation.

    I think I like it enough to be willing to pay for more storage when I hit my free limit.

    If you're interested in (I'm sure there are other good options out there), you can give me some more free storage by using this link to join. I think you get an extra 250Meg for using a link like this, too.

    I once had a hard drive crash and burn, on an almost new computer. The warranty got me a new hard drive, but the papers I had written for some lit courses I was taking - gone forever. However you like to do your backups, if you haven't done it recently, go do it!

    *I just now set up a file sharing folder, put my manuscript in it, and invited someone to look at it. That was the 5th thing in a list of steps they'd recommended. It got me another 250Meg free.

    Drip Logic

    Me: Clean up your drips!
    R: There aren't any.
    Me: I see at least one.
    R: You see at most one.

    (Lovely! And now he's cleaned up that one drip, too.)
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