Friday, June 26, 2009

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 Teachers at Play #10

Math Teachers at Play #10 is now up at Homeschool Math Blog.

I think my favorite part is the paper folding thought experiment, followed closely by How I Taught My Mother Binary Numbers. Come to think of it, the paper folding is at heart an example of the power of binary numbers.

P.S. I'll be hosting the next Math Teachers at Play, #11, on Friday, July 10.

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.

Saturday, June 20, 2009

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.

Monday, June 15, 2009

5 Finds

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

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

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

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

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

Saturday, June 6, 2009

Richmond (CA) Math Salon - Saturday, June 13, 2 to 5pm

Anyone local reading this, and want to join us?

Richmond Math Salon
Saturday, June 13
2 to 5pm

• All ages welcome. (Fun for kids 5 to 90.)

• Explore math in a fun, safe environment,
where no one will judge or grade you.

• A family math event:
You and your kids can explore math
in a way that works for each of you.

This is the 8th meeting of the Richmond Math Salon. We’ve circled around one full year, so our topic this month will be circles. As always, we’ll start with math-oriented puzzles and games, and then do a variety of activities, accessible to all ages, related to circles.

This monthly event is currently held in a small home, so please RSVP if you plan to come. The Richmond Math Salon is hosted by Sue VanHattum, a math professor at Contra Costa College.

Interested? Email me at, or call me at 236-8044 (before 8pm).
Math Blog Directory