Last week I posted Myths About Math. This is the second installment in my series of posts busting those myths. Mind Over Math's 7th myth was "It's bad to count on your fingers."
When I'm teaching I often think out loud. Talking about powers, I often count on my fingers: 2 (first finger) times 2 (second finger) is 4, times 2 (third finger) is 8, (fourth finger) 16, (fifth finger) 32. I know my students haven't memorized powers of 2, so the quickest way to figure them is to multiply repeatedly by 2 and keep track on our fingers.
Of course, students who haven't memorized their times tables, but who add well, can do the same. Fives are the easiest: 7 times 5 is... 5, 10, 15, 20, 25, 30, 35 (and we stop when our 7th finger pops up).
So counting on our fingers is useful any time we're trying to figure something that needs a repeated step. Perhaps the thing I want to figure can be memorized. But if I haven't memorized it yet myself, the most efficient way to figure it will likely involve fingers.
Learning math is a process. Young children can add 2 blocks and 3 blocks by touching each one and finding the total. At some point, they learn that they can imagine the blocks and count them by touching a different finger for each one. And those fingers are one of the first steps towards the power math has to generalize.
What happens when a person becomes embarrassed about counting on their fingers? If they still want to think, they'll hide it. That's the better option. A worse option happens to way too many students: This may be the point where discomfort with not living up to someone else's expectations makes them give up on math, and then they just guess.
What do you think?
Thursday, July 30, 2009
Saturday, July 25, 2009
Math Teachers at Play #12 ...
... is up here, at The Number Warrior. Jason has started out with a riddle about the number 12. Last issue, I included some online information about the number 11, but I like the riddle idea better.
I think my favorite bit is Mark Dominus' piece on error-correcting codes at the doctor's office. It's in the advanced math section, but it's mostly straightforward.
I think my favorite bit is Mark Dominus' piece on error-correcting codes at the doctor's office. It's in the advanced math section, but it's mostly straightforward.
Friday, July 24, 2009
Myth-Busting: What's Gender Got to Do with It?
Mind Over Math's Myth #1 was: Men are better at math than women. Sigh... I hope my grandchildren will think of that as quaint nonsense.
Let the myth-busting begin! I've gathered together 4 different perspectives on the issue. Mostly we're looking at nature versus nurture here, but there are lots of angles to this.
We all know what drives this myth. There are lots more male mathematicians (PhD level, anyway), engineers, and statisticians. And at least in the U.S., there have been big differences in math comfort level and skill level between women and men. But there is some great research on how these differences are connected to the broader culture (thereby having nothing to do with innate ability).
International comparisons show the cultural variability
The World Economic Forum has created a measure of the gender gap in each country. It turns out that, in countries with a smaller gender gap overall, there is usually also a smaller gender gap in math performance, or even a reversal - in Iceland, girls do significantly better than boys in math. Unfortunately for those of us who live here, the U.S. is not even near the top of the list for gender equality.
Of course, there are naysayers who doubt that this is the whole story. Larry Summers, then president of Harvard, made the claim that perhaps the average ability wasn't much different between men and women, but men, so he claimed, have more variability, and so there are both more dunces and more geniuses among men. (Slate wrote a good article on the Summers fiasco.) Interesting hypothesis, but wrong. Janet Mertz gave a talk at a conference I attended in April (see my original post here), on a paper she co-authored, which looks at that very top level of performance. In countries with greater gender equality, there are more women in these top ranks, and the link to culture is clear.
Stereotype Threat
So, in general, the more equality overall, the better women do in math. But how does this operate on an individual level? Here's some psychology research on that. The term 'stereotype threat' refers to the fear that people have that their behavior will confirm a stereotype. That fear interferes measurably with performance on tests, when gender or race is brought to attention before the test. Actually, the reverse can happen too: Men, reminded that men generally do better in math than women, will do better than they otherwise would on a math test.
I learned about this at that April conference, where Fred Smyth, of the Full Potential Initiative, gave a great talk about it. Here's another good summary of the research, from NYU.
Not different abilities, but perhaps different learning styles
Of course, that's just one piece of the puzzle. Here's another. There are differences overall in learning styles among girls and boys, according to Jo Boaler. In her book What's Math Got to Do with It? she discusses girls' stronger need to understand why. In classrooms where that need isn't honored, girls are turned off to math, more than the boys are. Boaler also describes how classes that focus on the 'why' are better for both boys and girls. She quotes a great book, In a Different Voice, by Carol Gilligan. Gilligan claims that women are more likely to be 'connected' thinkers and men are more likely to be 'separate' thinkers. (But we may want to ask whether that difference comes from nature or nurture.)
More Myths to Bust
About a decade ago, when I read Women in Mathematics: The Addition of Difference, by Claudia Henrion, I was so relieved. It was what I'd been looking for for years. All the other books I'd read on women in math were either collections of biographies or cheerleading (we-can-do-it, rah rah). Finally, in 1997, Henrion brought us some explanations. She described the myths about what it means to be a mathematician, and then gave examples of women mathematicians whose lives disproved those myths. I found a great review of this book at Thus Spake Zuska. She listed the myths this way:
Plain old sexism
"Maybe you're just not cut out for this." I heard that from a math prof when I was an undergrad at the University of Michigan. I never knew whether it was sexism or not. Maybe he said it to struggling male students too. But plenty of women have shared stories with me of blatant sexism directed at them in math class or from a math teacher. Yes, it's still happening. And it's still having a destructive effect. But perhaps this is an example of exponential decay - we'll always see a bit of it, but maybe it's heading closer and closer to zero.
Myth #1 has had more effect on my life than any of the others, but it's not the biggest problem in most people's math lives. More myth-busting coming soon!
Let the myth-busting begin! I've gathered together 4 different perspectives on the issue. Mostly we're looking at nature versus nurture here, but there are lots of angles to this.
We all know what drives this myth. There are lots more male mathematicians (PhD level, anyway), engineers, and statisticians. And at least in the U.S., there have been big differences in math comfort level and skill level between women and men. But there is some great research on how these differences are connected to the broader culture (thereby having nothing to do with innate ability).
International comparisons show the cultural variability
The World Economic Forum has created a measure of the gender gap in each country. It turns out that, in countries with a smaller gender gap overall, there is usually also a smaller gender gap in math performance, or even a reversal - in Iceland, girls do significantly better than boys in math. Unfortunately for those of us who live here, the U.S. is not even near the top of the list for gender equality.
Of course, there are naysayers who doubt that this is the whole story. Larry Summers, then president of Harvard, made the claim that perhaps the average ability wasn't much different between men and women, but men, so he claimed, have more variability, and so there are both more dunces and more geniuses among men. (Slate wrote a good article on the Summers fiasco.) Interesting hypothesis, but wrong. Janet Mertz gave a talk at a conference I attended in April (see my original post here), on a paper she co-authored, which looks at that very top level of performance. In countries with greater gender equality, there are more women in these top ranks, and the link to culture is clear.
Stereotype Threat
So, in general, the more equality overall, the better women do in math. But how does this operate on an individual level? Here's some psychology research on that. The term 'stereotype threat' refers to the fear that people have that their behavior will confirm a stereotype. That fear interferes measurably with performance on tests, when gender or race is brought to attention before the test. Actually, the reverse can happen too: Men, reminded that men generally do better in math than women, will do better than they otherwise would on a math test.
I learned about this at that April conference, where Fred Smyth, of the Full Potential Initiative, gave a great talk about it. Here's another good summary of the research, from NYU.
Not different abilities, but perhaps different learning styles
Of course, that's just one piece of the puzzle. Here's another. There are differences overall in learning styles among girls and boys, according to Jo Boaler. In her book What's Math Got to Do with It? she discusses girls' stronger need to understand why. In classrooms where that need isn't honored, girls are turned off to math, more than the boys are. Boaler also describes how classes that focus on the 'why' are better for both boys and girls. She quotes a great book, In a Different Voice, by Carol Gilligan. Gilligan claims that women are more likely to be 'connected' thinkers and men are more likely to be 'separate' thinkers. (But we may want to ask whether that difference comes from nature or nurture.)
More Myths to Bust
About a decade ago, when I read Women in Mathematics: The Addition of Difference, by Claudia Henrion, I was so relieved. It was what I'd been looking for for years. All the other books I'd read on women in math were either collections of biographies or cheerleading (we-can-do-it, rah rah). Finally, in 1997, Henrion brought us some explanations. She described the myths about what it means to be a mathematician, and then gave examples of women mathematicians whose lives disproved those myths. I found a great review of this book at Thus Spake Zuska. She listed the myths this way:
- Mathematicians work in complete isolation
- Women and mathematics don't mix
- Mathematicians do their best work in their youth
- Mathematics and politics don't mix
- Only white males do mathematics
- Mathematics is a realm of complete objectivity
Plain old sexism
"Maybe you're just not cut out for this." I heard that from a math prof when I was an undergrad at the University of Michigan. I never knew whether it was sexism or not. Maybe he said it to struggling male students too. But plenty of women have shared stories with me of blatant sexism directed at them in math class or from a math teacher. Yes, it's still happening. And it's still having a destructive effect. But perhaps this is an example of exponential decay - we'll always see a bit of it, but maybe it's heading closer and closer to zero.
Myth #1 has had more effect on my life than any of the others, but it's not the biggest problem in most people's math lives. More myth-busting coming soon!
Thursday, July 23, 2009
Myths About Math
I've been thinking lately about how different people have very different answers to the question "What is math?" Recently, the number one math myth I've been bumping up against is:
1. It's all about arithmetic. (Elementary school math, at least.)
and its corollary,
2. Gotta memorize those time tables.
Why do I call these myths?
1. Math is so much more than arithmetic. It's shapes, logic, problem-solving, and lots more. Arithmetic is one piece of a huge puzzle that can keep us engaged all our lives if we don't get discouraged or bored by narrowing it down.
2. Sure, they'll need to know their times tables, for all sorts of reasons. But if someone doesn't memorize easily, give them something more intriguing to think about, where they get slowed down, but not stopped, by not knowing their times tables. The skill will develop in this need-to-know context.
So I've been curious what all the math myths are that are floating around out there. Here's a list I found online, that originally comes from Mind Over Math, by Kogelman and Warren, a great book for overcoming math anxiety:
Twelve Math Myths
1. Men are better at math than women.
2. Math requires logic, not intuition.
3. Math is not creative.
4. You must always know how you got the answer.
5. There is a best way to do math problems.
6. It's always important to get the answer exactly right.
7. It's bad to count on your fingers.
8. Mathematicians do problems quickly, in their heads.
9. Math requires a good memory.
10. Math is done by working intensely until the problem is solved.
11. Some people have a 'math mind' and some don't.
12. There is a magic key to doing math.
The word myth doesn't have to mean untrue, but in this context I think it does mean that - inaccurate things lots of folks believe. And yet, there is often some way in which the belief is true, which is why it gets its power.
I am planning to do a series of posts on these myths (and any others my readers bring up), in which I explore them one by one. What's true? What's not? Why do people believe these things?
What math myths have you run into that aren't on this list?
1. It's all about arithmetic. (Elementary school math, at least.)
and its corollary,
2. Gotta memorize those time tables.
Why do I call these myths?
1. Math is so much more than arithmetic. It's shapes, logic, problem-solving, and lots more. Arithmetic is one piece of a huge puzzle that can keep us engaged all our lives if we don't get discouraged or bored by narrowing it down.
2. Sure, they'll need to know their times tables, for all sorts of reasons. But if someone doesn't memorize easily, give them something more intriguing to think about, where they get slowed down, but not stopped, by not knowing their times tables. The skill will develop in this need-to-know context.
So I've been curious what all the math myths are that are floating around out there. Here's a list I found online, that originally comes from Mind Over Math, by Kogelman and Warren, a great book for overcoming math anxiety:
Twelve Math Myths
1. Men are better at math than women.
2. Math requires logic, not intuition.
3. Math is not creative.
4. You must always know how you got the answer.
5. There is a best way to do math problems.
6. It's always important to get the answer exactly right.
7. It's bad to count on your fingers.
8. Mathematicians do problems quickly, in their heads.
9. Math requires a good memory.
10. Math is done by working intensely until the problem is solved.
11. Some people have a 'math mind' and some don't.
12. There is a magic key to doing math.
The word myth doesn't have to mean untrue, but in this context I think it does mean that - inaccurate things lots of folks believe. And yet, there is often some way in which the belief is true, which is why it gets its power.
I am planning to do a series of posts on these myths (and any others my readers bring up), in which I explore them one by one. What's true? What's not? Why do people believe these things?
What math myths have you run into that aren't on this list?
Monday, July 20, 2009
Monday Math Madness #37: Spiral Numbers, Part 2
I really like this contest, held at Wild About Math! every other Monday. Maybe because I won a prize the first time I entered. Or maybe because the first person I encouraged to try one of these got it, after just enough effort to make his victory sweet.
Background from #36 (part 1):
Background from #36 (part 1):
Imagine arranging the positive integers in a spiral pattern.
The numbers from 1 to 16 look like this in the spiral pattern.
10 9 8 7
11 2 1 6
12 3 4 5
13 14 15 16
The location of each number corresponds to an X,Y Cartesian coordinate where the number 1 is at the origin: (0,0). 2 is at (-1,0). 3 is at (-1,-1). 4 is at (0,-1). 5 is at (1,-1). 6 is at (1,0). 7 is at (1,1) and so on.
Here's this week's contest problem:- Come up with an algorithm that tells what number is at an arbitrary X, Y coordinate.
- Come up with an algorithm that tells the X, Y coordinates for an arbitrary positive integer.
Thursday, July 9, 2009
Math Teachers at Play #11
(by L. Marie)There's a whole bunch here. Get comfy, settle in, and enjoy!
What's Special About This Number? says the Maoris used base 11, 112 = 30 + 31 + 32 + 33 + 34, 113 = 32 + 192 + 312, and 11 x 11 = 65 + 56 (palindromic equality). And of course Wolfram Alpha (WA) has something to say about 11, too.
For this issue of Math Teachers at Play, we have games, geometry, arithmetic, logic, and some test prep.
Games
Dana, at School for Us, brings us Score 21.
GeometryJohn Golden, at Math Hombre, brings us his game, Michigan Smith (cousin to Indiana Jones, doncha know), which involves measurement, dice, and a bit of strategy.
Clemencia Rosado, at Storytime and more, brings us M for Math Games.
Mike, from Walking Randomly, asks "Can anyone think of interesting extensions or variations to this system of wheels?" His Wheels on Wheels on Wheels is a spirograph extravaganza. (And here's another at Learning in Mathland.)
I've seen a lot of What Can You Do With This? posts all over the mathblogosphere. (Does that make it a meme yet?) Ryan, over the water at Maths at SBHS, brings us another What can you do with this? It's a visual illusion done with perspective and enlargement. Very cool.
And here's one from Watch Math on Slopes of Perpendicular Lines.
John Cook, from The Endeavor, brings you Three rules of thumb. I've converted feet to miles, but never seconds to nanocenturies!
Tony, at This Young Economist, brings us 30-20-10 Pricing. How much do you really get off when you get a 30% discount, and another 20% discount, and another 10% discount? Not as much as you might expect...
Kendra, at Pumpkin Patch, also joins us from far away to give us a living math lesson using dominoes.
Denise, at Let's play math!, brings us Solving Complex Story Problems II. She says, "Diagrams make it easy to model a multi-step word problem. When I was in school, we wouldn't have seen this sort of problem until algebra, but with these models, a 5th-grader can solve it." I've been hearing a lot about Singapore's bar diagrams. I'm looking forward to trying them out with some kids.
Denise, at Let's play math!, brings us Solving Complex Story Problems II. She says, "Diagrams make it easy to model a multi-step word problem. When I was in school, we wouldn't have seen this sort of problem until algebra, but with these models, a 5th-grader can solve it." I've been hearing a lot about Singapore's bar diagrams. I'm looking forward to trying them out with some kids.
Logic
Misty, at Homeschool Bytes, says, "Make math interesting by mixing regular household items with math concepts, like a handful of candy, two circles, and the concept of venn diagrams." See her post on Venn Diagrams.
jd, over at JD2718, presents 5 logic puzzles. I think logic puzzles are my favorite mathy pastime. Thanks for some doozies!
Test Prep
Mr. D, at I Want to Teach Forever, has a question for us at the end of his post on 3 Ideas to Prepare Students for College Placement Exams. Can anyone help him find an answer?
And I thought I'd throw in one last bit. A few weeks ago I wrote a post about math and mountain climbing, and just now found out about this lovely post about math being like trees.
[Next time I host, I'll try to add a bit more spice. I've put this together in the middle of a marvelous week-long institute on running math circles, put on at Notre Dame by the folks from the Boston area Math Circle. That concludes this edition. Submit your blog article to the next edition of math teachers at play using the carnival submission form. Past posts and future hosts can be found on our blog carnival index page. Technorati tags: math teachers at play, blog carnival.]
And I thought I'd throw in one last bit. A few weeks ago I wrote a post about math and mountain climbing, and just now found out about this lovely post about math being like trees.
[Next time I host, I'll try to add a bit more spice. I've put this together in the middle of a marvelous week-long institute on running math circles, put on at Notre Dame by the folks from the Boston area Math Circle. That concludes this edition. Submit your blog article to the next edition of math teachers at play using the carnival submission form. Past posts and future hosts can be found on our blog carnival index page. Technorati tags: math teachers at play, blog carnival.]
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.
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 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:
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:
---
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.
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. ...In an earlier description about why math might be particularly difficult to teach well, they write:
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)
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)After so richly developing his metaphor, Zeitz uses it to explain mathematical problem-solving. For example:
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)
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.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.
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.
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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!"
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.
"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:
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*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.
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.
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*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.
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