Math Without Words, by James Tanton

Way back in the spring James Tanton kindly sent me seven of his books to review.  Ever since then I've been trying to look them over thoroughly enough to do a good job reviewing them. I think I'd better do the reviews one at a time, or I'll never get to it. The books are so full of goodies it might take me years to feel like I've looked them over properly.

You may have noticed a few interesting sheets of puzzles in the video of my math salon. Here's one:

These came from Math Without Words, a compendium of 75 delightful puzzles. Each puzzle has one or a few sections that are filled in to show show the puzzle works. Then there are more sections ready to be thought about and puzzled over. The puzzles get you thinking mathematically in new ways, and range from easy enough to engage five year olds, to hard enough to stump the grownups.

Folks at the math salon really loved these puzzles, and I think you will too. You can buy a print copy of the book for $27.50 or download an electronic version for $19.50. Both are available at Lulu. I just now bought the electronic version, so I can more easily make copies for the groups I work with.

That first puzzle may look too easy. But for young kids it really is a great puzzle. Here's another puzzle we enjoyed at the math salon:

And I'll leave you with a more challenging puzzle that we haven't tried at the salon yet:

If you'd rather have a dozen of the puzzles in wall calendar form, that's available too. Next up for review is Thinking Mathematics, Volume 1: Arithmetic = Gateway to All.

Fun Math Books: I Love Math! Series

Although these books are long out of print, there are inexpensive copies available online of most of them. (They were published in 1992 and 1993.) This list gave me the twelve titles below. (If anyone here knows of more, please let us all know.) I recently picked up a bunch of them, because my son really liked the two we already had.

• Alice in Numberland: Fantasy Math
• From Head to Toe: Body Math
• How Do Octopi Eat Pizza Pie? Pizza Math
• Look Both Ways: City Math
• Play Ball: Sports Math
• Pterodactyl Tunnel: Amusement Park Math
• Right in Your Own Backyard: Nature Math
• See You Later, Escalator!: Mall Math
• The Case of the Missing Zebra Stripes: Zoo Math
• The House that Math Built: House Math
• The Mystery of the Sunken Treasure: Sea Math
• The Search for the Mystery Planet: Space Math

My son and I especially enjoy the stories (in every volume) about Professor Guesser, a cat detective who solves mysteries using mathematical reasoning. She's featured in the title of The Case of the Missing Zebra Stripes: Zoo Math. Some of the zebras are missing their stripes, and Professor Guesser figures out what's really going on.

The books are full of stories, games, mazes, riddles, and lots of math.

Changing the Way We Teach Math - 3 Good Books

I've been reading math teacher books for years. Most of them refer to elementary education, and I've struggled to see how I could use their insights in my college classrooms. Reading teacher blogs over the past year or so has given me lots to think about, much closer to my situation.

Of course there weren't any blogs to be found 15 years ago, when I started teaching full-time. There were two very special books though, which addressed both elementary and secondary math.  I read both volumes of What's Happening In Math Class, edited by Deborah Schifter, about 12 years ago, and have been re-reading them this week. Both books are full of wonderful teacher stories, and totally remind me of my blog reading, except that there is more of a mix of elementary and high school level teachers writing these. (I've searched for elementary teachers blogging about math, and haven't found much.)

It was wonderful to re-read these and rediscover the treasures that have been living in the depths of my memory all these years. This is where I first heard of the Green Globs program (are any of my readers using that still?) and XMania (an extended exercise in creating and using a base 5 number system, used in a number of teacher ed programs).

As I reread ‘Third Graders Explore Multiplication’, by Virginia Brown (volume 1, page 18), I suddenly remembered reading this story so long ago. Remembered the group of kids who figure out the commutative property of multiplication, that 3x9 will be the same as 9x3, and in fact this re-ordering will work for any numbers. Before reading this the first time, I had always thought that was obvious. But it’s not obvious to most kids, and it can be an exciting discovery.

These books also introduced me to good questioning as a method of teaching mathematics. Almost every one of the teacher chapters in these books includes lots of dialogue, both between students and between teacher and students.

Between the two volumes, 22 teachers tell their stories, of working to change their classrooms, and of lessons that engaged the kids and taught them more about how their students learn math. Reading these books, like reading teacher blogs, reminds me that the teacher is always learning in a good classroom, right alongside the students.

Volume 1 focuses on what a good mathematics classroom might look like, and volume 2 focuses more on the struggle the teachers went through as they attempted to change the way they taught. Ruth Heaton wrote (volume 2, page 74), "In the culture of teaching, it is unusual to find practitioners willing to discuss how confused and frustrated they feel about their work." Ahh, but that has changed dramatically with the current crop of teacher blogs! Plenty of teachers are baring their souls, discussing their bad days, their confusions, and their ongoing struggles. So many of us have been hungry for this, and technology has given us our wish. Now we can read more teacher stories than we have time for, and can then narrow it down to the ones that really speak to our personal struggles.

I sometimes find that I think better when I'm reading a book than when I'm reading online. So these two books have been a great resource. I found one more gem, written by Deborah Schifter and Catherine Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform. Like volume 2, this book shows teachers struggling to change the way they teach, although this time through the eyes of Schifter and Fosnot. All three books came out of the SummerMath for Teachers Institute, held at Mount Holyoke College. This institute is described (page 106 of Reconstructing...) as an entirely new experience for the elementary teachers attending:

At SummerMath Institute … many [teachers] experience mathematics for the first time as an activity of construction, evaluation, and exploration, rather than as a finished body of results to be stored away. And for the first time they sense that mathematics instruction can be an invitation to the exploration of ideas, rather than a laying on of facts, rules, and procedures.

Let's all explore ideas in our math classes this fall, and if you're struggling, these books are good company to have (along with all your blogger friends, of course).

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.


Logarithms
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, 10^1/2 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.

The Cat In Numberland, by Ivar Ekeland

I love this book! I wrote about it before, in my post on A Dozen Delectable Math Books, but it was out of print then. It has come back out, and I've just gotten two copies, one for my son's school, and one for a friend. I think I'll need to buy about 5 more copies for friends with kids, and one for my niece, who's 15 and enjoys math.

Here's what I wrote before:

The Cat in Numberland, by Ivar Ekeland (ages 5 to adult)
The cat who lives in the Hotel Infinity gets confused when the hotel is full, and the numbers are all able to move up one room to make room for zero. This story is charming enough to entertain young children, and deep enough to intrigue anyone.

David Hilbert, a mathematician interested in thinking carefully about how infinity works, and different sizes of infinities, first made up the basic story. Many others have embellished on it. In this version, Mr. and Mrs. Hilbert run the hotel, and with a little help, find rooms for all the guests who come to visit, even when the hotel is already full. Mr. Hilbert gets worried when the fractions show up (doesn't everyone?), and Zero helps him see a solution.

John O'Brien's illustrations are delightful.

You Can Count on Monsters, by Richard Evan Schwartz

Schwartz is a mathematician and an artist, and his monsters are the numbers from 1 to 100. Number one is sad because it can't play the factoring game with the other numbers. He shows why, with factors trees that have 1's included, and can just go on forever. (His factor trees have the branches going up; mine have always gone down.)

The prime numbers are the basic monsters, and the other monsters are made from strange conglomorations of their prime factors. Searching for the factors in the composite monsters is sometimes easy (that's the 10 monster to the right) ... 






Sometimes hard (that's 99 to the left)...

And always interesting.






My 7-year-old son was confused by 8. He thinks of 8 as four 2's, but there are only three 2's in the picture. I think it's a good confusion. My guess is that he'll eventually get how the composite monsters are made from other numbers, and then the 8 monster will suddenly feel right.

At the end, Schwartz gives lovely explanations of how to find the primes, and why they go on forever.



Available for $24.95 at AK Peters (the publisher); cheaper at Amazon (but check that shipping charge to make sure).

_____
[Transparency: I got my copy of this book for free, as a review copy. But I think I might just buy 2 or 3 copies to give as presents. Yum!]

Hannah, Divided, by Adele Griffin

Over the past year I've learned a new term: 2e, or twice exceptional, is a term used by advocates of kids who are exceptionally smart, along with having exceptional learning differences. (I'm paraphrasing Tiffani, who blogs beautifully at Child's Play about 2e issues. Another blog I've enjoyed on the subject is Life Among the Gifted.)

Hannah, Divided is the sweet story of a girl who would be designated 2e nowadays. Growing up during the depression on a farm, she's not too worried about her struggles with reading, and the comfort she takes in numbers is very personal. She doesn't much expect either her learning trouble or her gift to take her away from milking cows and sharing the chores with her family. But they do.

Her teacher, Miss Cascade, has prepared the one-room schoolhouse and all its students for a visit from a possible benefactor. On the day Mrs. Sweet arrives, she takes an interest in Hannah's math abilities and quizzes her after school. On the way home, Hannah is so fired up, she just has to run, and count.

Finally, she took this year, 1934, and divided it by two. Over and over, skipping the decimal point like a checkers piece until it stood at the front of the line.

Granddad McNaughton encouraged her mathematics. Sunday afternoons, they passed gleeful hours inventing games with figures and sums, making up riddles and puzzles to solve.

She's given a chance to go to Philadelphia to study math, and takes it. Leaving home is difficult for her, and she leans on her need to pace her room 32 times, get each item in exactly the right place, and tap her paper 32 times. As hard as it is, the math she's able to learn from her tutor at Ottley Friends' School makes all the hardship worthwhile.

As I read this simple story, I kept thinking of how moving it might be for my young friend Artemis, and other kids like Hannah. I hope some of you have a chance to enjoy it soon.

Gifts for math lovers

I posted back in June about my favorite math books. Any of those would make a great gift. But I'm excited about a few books I've read recently, and wanted to share them here for those of you who like to give books as gifts. Both are biographies of mathematicians.

Carry On, Mr. Bowditch, by Jean Lee Latham (1955) is written for younger readers, but will charm many adults too. It's a fictionalized account of the life of Nathaniel Bowditch, who loved math, but had to leave school when his family needed his help. He was indentured to a ship chandlery for 9 years, which dashed his hopes of someday going to Harvard to study math. But he spent his spare time learning everything he could on his own - he learned Latin so he could read Newton's Principia Mathematica, and then learned French so he could read another book recommended to him.

After his indenture ended, he sailed with a merchant ship, and became interested in the mathematics of navigation. He was incensed at the errors in the book of tables used for navigation, and began the laborious work of correcting them.

"You don't 'cast your eye' over navigation tables!" Nat barked. "When I checked that one table of Maskelyne's, I worked every figure three times, just to be sure I was right!"
"Three times? Every figure? But why in ..."
"Why not? Nat roared. "Mathematics is nothing if it isn't correct! Men's lives depend on those figures!" (page 161)

He also taught the crews how to "do a lunar", a startlingly egalitarian action back then. In this passage he's talking to a young woman who later becomes his wife, but this is much like the lessons he gave the crews he sailed with:

Nat said, "That's the North Star. If you think of the North Star as the middle of your clock face, and the line from it through those other stars as the hour hand, you can tell time."
"It says about one o'clock. Is that right?"
"No, this clock runs backwards."
"Is it eleven o'clock?"
"No, there's one other difference. It takes twenty-four hours for the Big Dipper to swing around the North Star. So every hour space on the clock face stands for two hours." (page 85)

Bowditch eventually decided to write his own book, which he hoped would be error-free, and would also include navigation lessons and general information needed by sailors. His book, the American Practical Navigator, published in 1902, is still carried on every U.S. naval vessel (according to Wikipedia).

Bowditch was born in 1773 in Salem, Massachusetts. I enjoyed reading about the early days of the U.S. as an independent nation from his perspective. My only concern with a book like this is that I'll mix up what's fiction and what's true. The astronomy lesson I've quoted is a bit oversimplified, but apparently close.


The Man Who Knew Infinity: A Life of the Genius Ramanujan, by Robert Kanigel, would be unbelievable if it were fiction or even slightly fictionalized. A friend of mine, who works with gifted kids, describes their learning style as sitting in the eye of a hurricane and grabbing at the ideas whirling by. As I read about Ramanujan's mathematical discoveries, I keep coming back to that image. Other mathematicians who worked with him were astounded by his process. Bruce Berndt noted that, although Ramanujan's proofs were often full of holes, his results were almost always correct, and suggested, "We might allow our thoughts to occasionally escape from the chains of rigor, and, in their freedom, to discover new pathways through the forest." (page 183)

Ramanujan grew up in South India and attended school sporadically. (In his younger years, he preferred learning on his own. Later, he couldn't deal with exams in subjects outside mathematics, and was kicked out of university.) It took him years of working as a clerk to support himself before he managed to catch the attention of a famous mathematician in England, G.H. Hardy, whose interest in him suddenly changed his life. He went from just scraping by with a job he had little interest in, to a paid position as a research student in mathematics at Presidency College in Madras, India. A year later he would sail to England to begin with Hardy the work of making his mathematical results comprehensible to others.

I'm only halfway through, but have learned much already about India, mathematics at Cambridge in the late 1800's, and the history of mathematics. Kanigel does a good job of giving us enough background so that we have some chance of understanding different cultures and different times. (At times, his own bias shows, but subtly.) His description of Hardy's writing (readable, clear, cogent, almost suspenseful) makes me want to go to my local math library and borrow his 1908 Course in Pure Mathematics.

One of the delights for me is the sweet trivia I'm picking up. Here are two bits I enjoyed.

• When young, Ramanujan played Goats and Tigers, a traditional game in India in which:

  Three "tigers" sought to kill fifteen "goats" by jumping them, as in checkers, while the goats tried to encircle the tigers, immobilizing them. (page 18)

  
  
• It is so hot traveling through the Red Sea that cabins on the east side of the boat, which can cool down away from the heat of the afternoon sun, are quite a bit nicer than those on the west side. Hence the acronym POSH for Port Outward, Starboard Homeward (page 199).
  

I've bought both these books to give to my young friend Artemis. (If you prefer giving games as gifts, I highly recommend Blink (under $10), Set (under $15), and Blokus.) I'm always searching for ways to enjoy the holiday spirit and still consume less. Used books are part of my solution to that conundrum. And I like getting them from Better World Books because of their interest in global literacy and taking care that old books don't end up in landfills.

May your holidays be peaceful.

Math Education Research

Education is a complex and messy endeavor. We all have our own ideas about how children should be raised, about how learning happens, and about what is important for children to learn. The schools have to deal with parents on all sides of every political spectrum demanding what they think is needed for their children.

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.)

Thumbs Down
“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.


Thumbs Up
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?

Mindstorms: Children, Computers, and Powerful Ideas

I have a new hero. Seymour Papert writes so brilliantly about math, learning, and how it all fits together, I think I'll have to read his book a few times to absorb it all. He wrote Mindstorms: Children, Computers, and Powerful Ideas back in 1980. (Why I never read it until now is a mystery to me. I've taught programming and math since the early 80's and could have used these ideas.) I expected a book about computers from the 1980's to be pretty severely dated, but the ways in which it's dated are surprisingly trivial. Papert's notions about why programming a turtle is valuable are still true, powerful, and not widely applied. But the book goes way beyond programming turtles.

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..."
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)

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.

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.

A Dozen Delectable Math Books

Reading about math was not like this when I was young.

Here are my most favorite books - all yummy!

... from board books to adult stories, from number pairs to infinity and surreal numbers. (I've just guessed at the ages.)


(Photo by Foxtongue)


Quack and Count, by Keith Baker (ages 2 to 7)
This is a board book, so it's good for the youngest child who will sit and listen to a story. But it stays good because it's so luscious. Great illustrations, fun rhythm and rhyme, cute story, and good mathematics. 7 ducklings are enjoying themselves in every combination. “Slipping, sliding, having fun, 7 ducklings, 6 plus 1.” (And then 5 plus 2, etc.) It would be great to have a book like this for all the number pairs that make 8, and one for 9, etc. If I ever get to teach math for elementary teachers again, I'd love to get my students to make books like this one.

Anno's Counting House, by Mitsumasa Anno (ages 2 to 7)
Everything I've seen by Mitsumasa Anno is delightful. There is so much to see in his books, many of which have no words. In this book, ten people are moving from one house to another. In each two-page spread you can see one more person who's moved from the left house to the right, and lots of furniture and other small items. In Anno's Mysterious Multiplying Jar, there is one island with two counties, with three mountains each, ..., until we get to ten jars within each box - a lovely, very visual representation of factorials. Anno's Magic Seeds does have words, and tells a fascinating story, of a plant whose seed, when baked, will keep you from being hungry for a full year. The plant grows two seeds in a year, and one needs to be used to grow a new plant... He's written over 40 books, most available in English.

How Hungry Are You?, by Donna Jo Napoli and Richard Tchen (ages 3 to 12)
There are lots of great of great books on sharing equally. Until recent, my favorite was The Doorbell Rang, by Pat Hutchins, but this one is even more delightful. The picnic starts with just two friends, rabbit is bringing 12 sandwiches and frog is bringing the bug juice. Monkey wants to come, "My mom just made cookies. I could take a dozen." They figure out how much of each goody each friend will get. In the end, there are 13 of them, and the sharing becomes more complicated.

The Cat in Numberland, by Ivar Ekeland (ages 5 to adult)
The cat who lives in the Hotel Infinity gets confused when the hotel is full, and the numbers are all able to move up one room to make room for zero. This story is charming enough to entertain young children, and deep enough to intrigue anyone.

The Number Devil, by Hans Magnus Enzensberger (ages 7 to adult)
The Number Devil visits Robert in his dreams, and gets him thinking about the strangest things! Rutabaga numbers and prima donnas (roots and primes) are just the beginning. Anyone who'd like a gentle introduction to lots of interesting math topics will enjoy this one.
Powers of Ten, by Philip and Phylis Morrison (ages 6 to adult)
The first photo shows a couple having a picnic. It's shot from one meter above them. The next is from 10 meters, then 100. After we've traveled to the edge of the universe, we come back to the couple, and zoom in. Each page has one large photo, and explanatory text about what can be seen at that level.

The Man Who Counted, by Malba Tahan (ages 6 to adult)
Written in Brazil, set in the Middle East, these stories follow the adventures of Beremiz, an accomplished mathematical problem-solver. He uses math to settle disputes, solve riddles and mysteries, and entertain his hosts.

Mathematics: A Human Endeavor, by Harold Jacobs (ages 12 to adult)
This one is a textbook, and it's delightful. The first chapter, on inductive and deductive reasoning, uses pool tables to get the student thinking about patterns. Chapters on sequences, graphing, large numbers, symmetry, mathematical curves, counting (permutations and combinations), probability, statistics, and topology round out an introduction to a wide variety of math topics, accessible to beginners.

Uncle Petros and Goldbach’s Conjecture, by Apostolos Doxiadis (adult)
Uncle Petros is a recluse. Our hero, his nephew, is trying to discover his secrets. It seems he was close to solving Goldbach's conjecture, that every even number greater than 2 is the sum of two prime numbers. There is just a tiny bit of math in this, but lots of (slightly twisted) history of math.

Euclid in the Rainforest, by Joseph Mazur (adult)
Logic, infinity and probability are the topics. Adventures in Venezuela, Greece, and New York furnish the background. Mazur has wide-ranging interests, and skillfully brings the math to life.

Chances Are: Adventures in Probability, by Michael and Ellen Kaplan (adult)
History, philosophy, science, and statistics all come together in this delightful exploration of probability.

Surreal Numbers, by Donald Knuth (adult, with well-developed math skills)
This book requires lots of work, doing the math, and what fun work it can be. Alice and Bill are enjoying their extended vacation on an isolated tropical beach , but are getting a bit bored, when they discover a rock with two 'rules' on it. Conway has invented number through these two rules, and Alice and Bill (and the reader) are sucked in, trying to figure out how it all works. This is higher math.

(An overlapping list is at Nerdy Book Club. A more complete list is on my Math Books page.)

Math is Like Mountain Climbing

It must be true; I've read it in 3 different places! :^)

I like the analogy: they're both very hard work, and both give their enthusiasts lots of pleasure in what they achieve, along with a view of the world that most people don't get.

~~~

In Out of the Labyrinth: Setting Mathematics Free, a book full of wisdom gleaned from their experiences conducting math circles, Robert and Ellen Kaplan write:

Those who love to climb mountains have a very different view of them, and it may be no accident that so many mathematicians are also mountain walkers and climbers. It isn't just the exhilaration of solving the rock face, but the fresher air along the way and the long views from the top that draw them on. ...

We aim to take acrophobia away by having our students do the climbing however they will, with us as their Sherpas. We bring up the supplies and peg down the base camp; we point out an attractive col or a dangerous crevasse; but they do the exploring on a terrain we've brought them to. (page 11)

In an earlier description about why math might be particularly difficult to teach well, they write:

Fall from a ledge and the odds are slim that you'll climb back up to and past it. (page 8)

The handholds seem to grow fewer the higher you climb. Mathematics is all ledges. You no sooner acclimate yourself to breathing the thin air at this new height than the way opens up to one still higher... (page 10)

~~~

In The Art and Craft of Problem-Solving, one of the first treatments of problem-solving I've found that doesn't just regurgitate Polya's lovely four step process*, Paul Zeitz writes:

You are standing at the base of a mountain, hoping to climb to the summit. You first strategy may be to take several small trips to various easier peaks nearby, so as to observe the target mountain from different angles After this, you may consider a more focused strategy, perhaps to try climbing the mountain via a particular ridge. Now the tactical considerations begin: how to actually achieve the chosen strategy. For example, suppose that strategy suggests climbing the south ridge of the peak, but there are snowfields and rivers in our path. Different tactics are needed to negotiate each of these obstacles. For the snowfield, our tactic may be to travel early in the morning, while the snow is hard. For the river, our tactic may be scouting the banks for the safest crossing. Finally, we move onto the most tightly focused level, that of tools: specific techniques to accomplish specialized tasks. For example, to cross the snowfield we may set up a particular system of ropes for safety and walk with ice axes. The river crossing may require the party to strip from the waist down and hold hands for balance. There are all tools. They are very specific. ... (page 3)

As we climb a mountain, we may encounter obstacles. Some of these obstacles are easy to negotiate, for they are mere exercises (of course this depends on the climber's ability and experience). But one obstacle may present a difficult miniature problem, whose solution clears the way for the entire climb.For example, the path to the summit may be easy walking, except for one 10-foot section of steep ice. Climbers call negotiating the key obstacle the crux move. We shall use this term for mathematical problems as well. A crux move may take place at the strategic, tactical, or tool level; some problems have several crux moves; many have none. (page 4)

After so richly developing his metaphor, Zeitz uses it to explain mathematical problem-solving. For example:

Let us look back and analyze this problem in terms of the three levels. Our first strategy was orientation, reading the problem carefully and classifying it in a preliminary way.Then we decided on a strategy to look at the penultimate step that did not work at first, but the strategy of numerical experimentation led to a conjecture. Successfully proving this involved the tactic of factoring, coupled with a use of symmetry and the tool of recognizing a common factorization. (page 6)

I haven't finished this book. It's full of hard mathematical problems that I can come back to over and over - a whole mountain range I can carry with me!

~~~

Each of these 3 authors has used the metaphor to a slightly different purpose. Mike South's piece (here) reminds me of Lockhart's Lament. They're both about how destructive 'teaching' can be in math. As a teacher, I continue to struggle with this conundrum.

I'll attempt to tantalize you with the beginning of his essay:

On the distant planet of Lanogy, all of the population centers are in sight of ... mountains. It's not just because there are lots of mountains on the planet, although that is also true. There are certain resources which are only available in the mountains. You need them to build cities, hence the proximity. But not only that, the mountains are the source of resources needed to facilitate trade in Lanogian economies, so all trade that takes place is near mountainous areas out of convenience. In addition to that, many other technologies turn out (some times unexpectedly) to benefit dramatically from the resources the mountains have to offer.
Now, interestingly enough, despite how useful the mountains are, almost no one on Lanogy likes them. Spending time in the mountains voluntarily is, to almost anyone you talk to, such a laughably improbable concept that it would only occur to them in jest. Everyone knows that builders and commerce agents have to do their share of mountaineering as part of their jobs (in fact, that very fact encourages a lot of people to eschew those professions), but the only people that would ever spend most of their time there would be the mountaineers. These very rare and very peculiar people (those whose only job is to climb mountains) might, possibly, do it voluntarily. But what they would do, why they would do it, and, indeed, what they do professionally is a complete mystery to the rest of the Lanogians.
Now, the reason for this general dislike of climbing in and retrieving resources from the mountains could be due to the simple fact that most people are really not very good at it. Now why, in a society that can obviously see the value of the resources obtained from the mountains, people still aren't good at climbing them, is widely disputed.

I am not into mountain climbing myself. Too scary. But I can wish I were braver, and I can understand better so many people's fears of math by reading these pieces. I can also get better at math myself by using Zeitz's strategies, tactics, and tools.

---
Polya's work, written back in 1944, was so helpful most of us mere mortals haven't discovered anything much more to say. I wrote a problem-solving handout for my classes, in which I updated Polya's language, and added a few ideas too basic for him to have included, like: Write 'Let x =' the quantity you're trying to find. Maybe I can write a separate post on Polya, and include it there.

Book Review: The Teaching Gap, by James Stigler

The U.S. has not done well in international comparisons in math. Our students score well below students in a number of other countries on TIMSS*, which tests students in 4th and 8th grade in dozens of countries. Stigler was part of the group of researchers who conducted an in-depth analysis of classroom videos associated with the 1995 TIMSS. They were looking for differences in classroom practice that would help to explain differences in scores. This book, published in 1999, is a fascinating description of their research results.

Classrooms in the U.S., Germany, and Japan were compared, and the main insight that came out of the study was that the culture of the classroom was very different in these 3 countries. Although it's an oversimplification, what they saw was something like this: In Germany, the teacher directs the students in developing advanced procedures, in Japan, the class works individually and in groups on structured problem solving, and in the U.S., the teacher leads the class in learning terms and practicing procedures (pages 25-46). One researcher said he had trouble "finding the mathematics" (page 26) in the videos of U.S. classrooms. (Yikes!)

Classroom culture is hard to change, according to Stigler, because much of it is deeply imprinted in us, as what school is. "The scripts for teaching in each country appear to rest on a relatively small and tacit set of core beliefs about the nature of the subject, about how students learn, and about the role that a teacher should play in the classroom." (page 87) Many teachers who've tried teaching with more of a problem-solving focus (including yours truly) can attest to how much resistance students put up: "That's not how math class is supposed to work! Just tell us how to do it!"

After viewing the Japanese lessons, a fourth-grade teacher decided to shift from his traditional approach to a more problem-solving approach such as we had seen on the videotapes. Instead of asking short-answer questions as he regularly did, he began his next lesson by presenting a problem and asking the students to spend ten minutes working on a solution. Although the teacher changed his behavior ... the students, not having seen the video or reflected upon their own participation, failed to respond as the students on the tape did. They played their traditional roles. They waited to be shown how to solve the problem. The lesson did not succeed. The students are part of the system. (page 99)

Which makes it clear that, however cool we think those Japanese classrooms are, we can't just bring their style over here as is. What we might be able to use here, however, is their lesson study process, modified to suit us. Teachers plan one lesson together in great depth, over a long period of time.

During lesson study, the teachers discussed what problem to start with, what materials to give students, what solutions and thoughts the students might come up with, what questions to ask, "how to use space on the chalkboard (Japanese teachers believe that organizing the chalkboard is a key ingredient to organizing students' thinking and understanding)", timing, working with different levels, and how to end the lesson. (from page 117, paraphrased)

Then they all watch in the classroom while one teacher plays out their plan with the kids. Afterward they all discuss some more, modify, and try it again in another teacher's class.

"Virtually every elementary and middle school in Japan is engaged in kounaikenshuu [lesson study]." (page 110) What Dan, Kate, and others are doing online (here and here, for example) might come close. Wouldn't it be great if we could start our own kounaikenshuu movement here?!


Here are some quotes I liked:

page 49:
He [Japanese math teacher] concludes by posting the goal for mathematics: "To learn to think logically while searching for new properties and relationships." He asks students to repeat this goal several times and memorize it.

page 75: (paraphrased)
The chalkboard as used as a visual aid that helps focus students' attention in the U.S. versus as a cumulative record of the day's lesson in Japan.
page 93: (regarding chalkboard use)
[Japan] Apparently, it is not as important for students to attend at each moment of the lesson as it is for them to be able to go back and think again about earlier events, and to see connections between the different parts of the lesson.

page 89:
Teachers were asked what was the "main thing" they wanted students to learn from the lesson. 61% of U.S. teachers described skills they wanted their students to learn. ... 73% of Japanese teachers wanted their students to think about things in a new way... to see new relationships between mathematical ideas.

page 90:
[In the U.S. view,] practice should be relatively error-free, with high levels of success at each point. Confusion and frustration ... should be minimized; they are signs that earlier material was not mastered.

page 91:
[In Japan] frustration and confusion are taken to be a natural part of the process, because each person must struggle with a situation or problem first in order to make sense of the information he or she hears later. Constructing connections between methods and problems is thought to require time to explore and invent, to make mistakes, to reflect, and to receive the needed information at an appropriate time.
...
Students will learn to understand the process [of adding unlike fractions] more fully, says the [Japanese teachers'] manual, if they are allowed to make this mistake [of adding denominators] and then examine the consequences.

page 94:
Japanese teachers view individual differences as a natural characteristic of a group. They view differences in the mathematics class as a resource for both students and teachers. Individual differences are beneficial for the class because they produce a range of ideas and solution methods that provide the material for students' discussion and reflection. The variety of alternative methods allows students to compare them and construct connections among them. It is believed that all students benefit from the variety of ideas generated by their peers. In addition, tailoring instruction to specific students is seen as unfairly limiting and as prejudging what students are capable of learning...

In Japan, classroom lessons hold a privileged place in the activities of the school. It would be exaggerating only a little to say they are sacred. They are treated much as we treat lectures in university courses or religious services in church. A great deal of attention is given to their development. They are planned as complete experiences - as stories with a beginning, a middle, and an end. Their meaning is found in the connections between the parts. If you stay for only the beginning, or leave before the end, you miss the point.

page 119: [Japanese teacher speaking]
Conceptually it's easy to break 6 down into 5 and 1, and it's easy to break 7 into 2 and 5, but it's really hard for first-grade students to break 7 down into 3 and 4. [!]

[During a lesson study meeeting] the teachers consulted some of the teachers' manuals and found 5 common ways of solving simple subtraction problems with borrowing.

page 127:
[Japanese] culture genuinely values what teachers know, learn, and invent, and has developed a system to take advantage of teachers' ideas: evaluating them, adapting them, accumulating them into a professional knowledge base, and sharing them.

-----
*The letters originally stood for Third International Mathematics and Science Study, which was conducted in 1995. At the National Center for Education Statistics website, the letters now stand for Trends in International Mathematics and Science Study, which is conducted every 4 years.