Monday, April 26, 2010

Blog vs Book

I've been neglecting this blog because I've been working hard on my real job for this year - a book, Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet. We don't have a publisher yet, but I'm still hopeful. I'm pulling together material from about 20 different people who help students learn math in innovative ways. I think it's pretty exciting.

I thought I'd mention the book just so you all know I haven't gone away. (I'm also working on a book review to post here. I hope to get that out soon.)

Here's a current draft of the table of contents. If I've asked you if I can use your blog post in the book, and you don't see it, don't worry, I still have to get that part organized. After the table of contents are some questions I have for my readers.

[draft] Contents

Preface 4

Introduction 6

Section 1. Math Circles, Clubs, Centers, Salons, and Festivals
Section Introduction 13
Sue VanHattum, Richmond Math Salon
Julia Brodsky, The Art of Inquiry: A Math Circle for Young Children Jamylle Carter, The Oakland Math Circle: A First Iteration
Maria Droujkova, One Day At the Math Club
Amanda Serenevy, Riverbend Community Math Center
Mary O’Keeffe, Creation of the Albany Area Math Circle: Great Circles Conference 2009
Colleen King, The Game of Math
Nancy Blachman, Inspiring Mathematical Interest: The Julia Robinson Mathematics Festival
Bob and Ellen Kaplan, Letters to a Young Student
A Young Voice:

Section 2. Homeschoolers
Section Introduction 35
Julie Brennan, Learning As We Go 36
Holly Graff, One and a Quarter Pizzas: An Unschooling Adventure 46
Pam Sorooshian, Radically Sensible Ideas 50
Sue VanHattum, Recommendations (needs a title)
A Young Voice: Lavinia Karl, An Unschooler Goes to College

Section 3. The World Online
Section Introduction 58
Maria Droujkova, Making a Math-friendly Internet [not yet written]
Denise Gaskins, Let’s Play Math [hoping to interview]
Colleen King, The Game of Math Goes Online 59
Rebecca Zook, Zook Tutoring blog [post yet to be selected]
Kate Nowak, Blogging towards better teaching 63
A young voice:

Section 4. Classrooms
Section Introduction 67
Alison Blank, Math Is Not Linear
Chris Shore, Textbook Free: Kicking the Habit 68
Dan Meyer, Be Less Helpful
Ed Cruz [not yet written]A young voice? [Algebra Project students?]

Section 5. Diversity and other Public Policy Issues
Section Introduction 77
Patricia Kenschaft, Racial Equity Requires Teaching Elementary School Teachers More Mathematics
Danny Martin, Students’ Mathematical Identities [not completed] 84
Sue VanHattum, Girls and Women, Doing Math [not completed]
Melanie Hayes, Learning From My Kids: Letting Gifted Children Bloom 87
Recommendations: Bringing Passion into the Classroom [not completed]


A Collection of Puzzles & Problems 97
Jonathan, jd2718 blog, A Little Math Magic 98

Bibliography 99
Part 1. Fun Math Books For All Ages
Part 2. Recommended Books About the Teaching and Learning of Math
Part 3. References

Finding or creating local math alternatives
A note on online sites
Author Biographies 103

  • How's it looking?
  • What's missing?
  • Do you know a young person who might be interested in writing a short piece for the end of the classrooms section? (I may need someone for the Internet and Math Circle sections too.)
  • Anything else you want to tell me?

Wednesday, April 21, 2010

Testing, Accountability, and Teachers

It seems so obvious to me -  high-stakes testing and 'accountability' are not the way to improve education. (High-stakes testing currently takes up over a quarter of the time in some schools.) Neither is firing all the teachers. But President Obama apparently approves of all that. His Race to the Top requires merit pay, even though we all know that teachers' primary motivation is not money. (In Florida, lawmakers passed a bill requiring that teacher pay be based on test scores, the people stood up and said no, and the governor vetoed it.) I want to believe that President Obama cares about the children in this country, but he is continuing and expanding Bush's destructive policies. Why?

Lois Weiner gave a speech recently that helped me understand, and chilled me. Lois' response to Diane Ravitch at a panel at NYU spells out details Diane has left out of her popular new book, The Death and Life of the Great American School System. Doug Noon, at his Borderlands blog, transcribed her talk. Here are the parts that struck me most:
... if we are going to defend public education, we need to have a very different analysis. And so the analysis that I’m going to offer tonight, I think, takes two sets of blinders off – that we have to take off. The first set of blinders separates educational reform from what’s going on in the economy. The other set of blinders says that we can look at education in this country separately from what goes on in the rest of the world. Because what I’m going to lay out tonight for you is a perspective that says NCLB, all these policies that Diane just described, are neoliberalism coming home.
I want to unpack for you this neoliberal ideology. And if you really want to understand it, you ... have to go to the way that the World Bank talks about it. Because in the World Bank documents, they present it in it’s unvarnished form. So I’m gonna quote for you from something called... The World Development Report 2002. And, of course they don’t use this exact language, but this is the analysis: The market is the best regulator of all services, and the state, the welfare state causes problems by intruding on free choice. Next, the global economy requires that workers from every country compete with others for jobs. And since most people will be competing with workers in other countries for jobs requiring little formal education, money spent on a highly educated workforce is wasted.  In other words, most jobs are in Walmarts. ...we’re all going to be competing for these jobs that require a seventh or eighth grade education.
And think about this, because we don’t need a highly educated workforce, we don’t need highly educated teachers. Therefore, we can have a teaching force that’s a revolving door. Teachers will use standardized scripts.

You can read the rest at Borderlands, or you can read what she says at her blog, where I was struck by this:
Education is the last service that is still mostly public – and unionized. Teacher unions are the most stable, potentially powerful foe of the neoliberal project and are therefore frequently and viciously attacked as impeding school improvement.
She's also helped edit a book on these issues. I'd rather be thinking about how to teach math, but if we're working in the schools, we should know who's pulling the strings.

Tuesday, April 13, 2010

Maria Droujkova is visiting the Bay Area

I'm very excited that Maria Droujkova is here in Richmond and Berkeley for a few days. She'd like to meet with anyone who's interested, to chat about math and anything related. We'll be at Genki, in Berkeley, on Wednesday evening, from 5 to 7:30pm. I'm looking forward to it.

Genki is a reasonably priced sushi restaurant, on San Pablo at Cedar (1610 San Pablo Avenue, Berkeley, CA 94702, 510-524-1465). If you mention the math group, the person who greets you will point you to our table. If you can rsvp to me (suevanhattum at hotmail, or 510-236-8044) by 4 on Wednesday, I can give Genki a sense of how big a table we'll need.

It would be great to meet folks who are reading my blog.

From Maria:
I will be at a sushi place talking math with whoever is interested. I hope to brainstorm about the "Family Math Methods 101" I plan to run online - an open college-level elementary teaching methods experience for parents and community educators. If you would like to join the conversation, please come.
Maria Droujkova

Make math your own, to make your own math.

Thursday, April 8, 2010

Eric Mazur on Physics, Teaching, Learning, and all that

"The lecture method is a process by which the lecture notes of the professor get to the notebooks of the students without passing through the brains of either."

Part I. Teaching Physics
The video (Confessions of a Converted Lecturer) is over an hour; it had crappy sound quality on my laptop (not so bad on my desktop); it's mostly just one person talking and showing a few slides of his data analysis; and it's not even about math. But it was totally worth it.

Eric Mazur is a physics professor at Harvard. He teaches physics to pre-med students. They don't have an interest in physics, but they are very good students. Eric was, by conventional standards, a very good teacher. But then he got his hands on the Force Concept Inventory (FCI), which has conceptually-based physics questions with no calculations, and it changed him. His students were doing well when measured by conventional physics questions, but could not answer questions that physicists see as much easier. His students could recognize which formulas to use for which problems, and could then work their way through long calculations, but they still didn't understand some of the most basic principles of physics.

As he described this process students use, of learning just the procedures, he said:
"Some students told me that physics is boring, and I could never imagine it. But, imagine that it's been reduced to following a recipe, that you don't even understand. Yes, then it would be boring."
He said that most students come in, even after getting a 5 in an AP physics course, as Aristotelian thinkers. (Aristotle codified some of the ideas the ancients had about physics. Those ideas were often wrong.) After a conventional Harvard physics course, and after scoring well on a conventional exam, most of the students are still 'Aristotelian thinkers' - "they haven't understood the material in week two, on which everything else depends."

I remember one thing like this from my college physics course. I remember a question about something rolling off a table, and what its path would be. It's hard for me to believe I didn't always imagine a parabolic path. But I do remember being surprised. I  think my previous conception was that the object would go straight down. It sounds so wrong to me now... Apparently, there are lots of basic concepts in physics like that, where people have firmly entrenched wrong ideas, until they've really understood the principles involved. Then they have trouble believing they ever had the wrong picture in their heads before. We don't remember not knowing...

Is math like this too? Can you imagine not knowing that 3+11 is the same as 11+3? Kids don't know that at first, and when they get it, it's a powerful tool, since counting up 3 from 11 is way easier than counting up 11 from 3. Are there misconceptions that we all had? I can't think of any right now. (Anyone who works with little kids, are there wrong ideas that come to us naturally?)

When Mazur gave a pre-test to his Harvard students, using the Force Concept Inventory, and told them how low their initial scores were, they wanted to spend time going over each question to understand it. There wasn't time during class, so he arranged to meet with interested students in the evenings. In this voluntary, ungraded environment, perhaps they felt more comfortable expressing their confusion. One evening, he gave what he thought was a good explanation of why the forces a light car and a heavy truck exert on one another in a crash are equal (as they must be, if one understands Newtonian mechanics), but what he got was a roomful of confused looks. He tried again, adding acceleration into the picture, and saw he'd only increased the confusion. He knew that about 40% of them already understood this idea, and, giving up, asked them to explain it to each other. Out of the chaos was born Mazur's version of "peer instruction". 

Nowadays, when he teaches, he explains for just a few minutes, and then asks a concept question. In a lecture hall with hundreds of students, they all begin to talk. Here's the process:
  1. He shows the question and explains it,
  2. Students silently think for 1 to 1 1/2 minutes,
  3. Individuals answer (using clickers, which allow immediate tallying of anonymously given answers),
  4. Discussion ensues among small groups of students (he doesn't try to control the groupings),
  5. Students answer again, after their group has come to some consensus,
  6. He explains the answer (and goes on to repeat the process for the next small bit of lecture).
I was impressed with the ways Mazur dreamed up to convey his data about this. His graph is unusual, and he takes a few moments to explain it (32 minutes into the video). He shows both the pre-test and post-test scores on the FCI from classes at very different sorts of schools. The Harvard students started with higher scores, so had less room to improve, and improvement was what he was measuring. He showed that across all the schools, the total gain (in FCI score, after one physics course) was only a quarter of what it could have been. When he started using peer instruction, that gain went up to half. Over time, he increased it more by learning to pose better conceptual questions.

There's more to savor in the video. If you can find the time to watch it, I think you'll be glad you did.

Part II. Teaching Math
So how do we use this for math? There is some work being done along similar lines. Jerome Epstein has developed a Calculus Concept Inventory, and found similar results - that conventional courses don't help the students to truly understand the concepts.* And Cornell has the Good Questions Project (which I mentioned before, here).

Good questions hook in to a student's personal experiences. This one may work at Cornell (the author's students really liked it), but would definitely not work at my college:
An article in the Wall Street Journal's Heard on the Street column Money and Investment (August 1, 2001) reported that investors often look at the change in the rate of change to help them get into the market before any big rallies. Your stock broker alerts you that the rate of change in a stock's price is increasing. As a result:
a. you can conclude the stock's price is increasing cannot determine whether the stock's price is increasing or decreasing
c. you can conclude the stock's price is decreasing
My students definitely do not have stock brokers, nor do most of their teachers. As I glance through the Good Questions file I found here, I don't see others like this one. Perhaps someone has thought about the issues of privilege already. Here are a few questions from the 53-page list:
True or False. At some time since you were born your weight in pounds equaled your height in inches.

True or False. As x increases to 100, f (x) = 1/x gets closer and closer to 0, so the limit as x goes to 100 of f (x) is 0. 
Your mother says “If you eat your dinner, you can have dessert.” You know this means, “If you don’t eat your dinner, you cannot have dessert.” Your calculus teacher says, “If f is differentiable at x, f is continuous at x.” You know this means
(a) if f is not continuous at x, f is not differentiable at x.
(b) if f is not differentiable at x, f is not continuous at x.
(c) knowing f is not continuous at x does not give us enough information to deduce anything about whether the derivative of f exists at x.
I'm hoping all these good questions will help me to teach a much better section of calc II this fall. (It sure would help if I could get a 'smart classroom' to teach in for the fall, so I could have students use those clickers...)

I'd like to think about how to use these same ideas at the algebra level, too. How do we ask the questions that will get students thinking about concepts rather than just focusing on the procedures? There's lots of good work being done by the bloggers I read. Jason Dyer, at Number Warrior, has a Fractions Concepts test that I'll use when I review fractions at the start of the term with my beginning algebra students. Hmm, maybe we could all work together to create an Algebra Concept Inventory...

[Hat tip once again to Dan, whose blog, Math for Love, has been very inspiring.]

*This is not available online, since they'd like to keep the content of the questions private, in order to retain their validity as a test instrument. I have emailed the author, asking for a copy to use in my calculus course in the fall.

Wednesday, April 7, 2010

Ron Eglash on African Fractals

For some reason fractals have never excited me as much as they do most people. Yeah, there's lots of pretty pictures, but I don't see the math very clearly. (I've had a similar reaction to the notion of 'ethnomathematics' - great idea, but the material is often flimsy.) Ron Eglash's TED talk satisfies me on both accounts - I can see the math!

[Hat tip to another Dan, who blogs at Math for Love, for his pointer to this video and much more!]

Monday, April 5, 2010

Guest Post: Maria D on Love and Memorization

Online, Maria Droujkova has founded the Natural Math site, the Natural Math Google group, the Math Future wiki site, and a series of talks on Math 2.0. In person, she leads unclasses in Cary, NC, playfully introducing children of all ages to math mixed with art, drama, and love. I think of her as a math fairy.  Thanks, Maria, for offering us this gem!

My daughter K has been memorizing more complex poetry lately, which is a lot of work. She commented that sometimes memorization work makes a poem "fall apart" and lose meaning or beauty.

If you love the poem a whole lot, spending more time and attention on it through memorization just creates additional lovable meanings, connotations, and associations. The poem becomes deeper and richer instead of falling apart.

Thus we must select poems we love for memorization. But to be coherent and stable, every community needs shared social objects that absolutely everybody in the community knows well: anthems, oaths, prayers, songs, movies, books, formulas, jokes...

In K's educational utopia, small groups of people work on separate content of their choice. Still, what if you want to belong to a group through friend and family connections? You will be forced to memorize their content, whether you love it or not.

I have a solution to this problem that works reasonably well. I believe love is something we build. When I help others learn, I start by helping them build much strong love for the object through appreciating its beauty, connecting it to their meaningful contexts, finding it in interesting media and so on. Only when love is strong, memory work can follow.

Yesterday, we worked with mirror books. L took notes for the story. Meanwhile, I just want to mention, as an example, that the incredible coolness and beauty of mirror books generate much strong love for multiplication.

When should kids memorize times tables? When their love for multiplication is strong enough.

Saturday, April 3, 2010

Murder Mystery: A Project for Logarithms

[Note added on 2-7-14: This is one of my most-read posts. If it helps you design a good project for your class, that's great. Please do not use it as is. Make adjustments for the age of your students (mine are in college), and fill in the details where I've put notes in brackets. Make sure you try each part out yourself before using this with students! To do otherwise is to ask for trouble.]

There've been some great discussions on teaching logarithms recently at JD2718 and f(t). At JD's blog, I mentioned a project I do in my classes, and nyates asked for the details. Here it is - just enough hook to get students working pretty well in groups. (One class decided we were CSI at CCC.)  :^)   But in my opinion, they're still mostly following the formula. If you have any ideas for improving this, I'd love to hear them.

Years ago, I shared my office with a chemistry teacher. When she complained that we math teachers often did a bad job in Intermediate Algebra courses with the (vital to chemistry) topic of logarithms, I decided to try to do it better. It was right at the end of the course, which always means neglect, so I moved it up a bit somehow. I also wanted to pull the students in more, so I made this project up.

This project uses Newton's Law of Cooling, though I don't mention that. (I often get students coming in the second day with that information. They've searched online, and found it. Good and bad - I like the research skill and initiative, but they want to use the formula they've found, instead of reasoning it out.) I work with them to figure out:  body.temp  = air.temp + * b ^ t, where body.temp is a function of time, air.temp is constant, is how much hotter the body was than the air when first measured, b < 1 (exponential decay), and t is measured in hours or minutes. The only log work necessary is: , so this doesn't get them practicing the other log properties.

To help them remember when to use logs, I tell them that historically logs helped people multiply, divide, and find roots, but that now calculators do all that, so "the purpose of logarithms is to get the variable out of the exponent." (What do you think? Is that problematic?)

I have them get in groups, and give them maybe half an hour each day to work on the assignments. They've worked with exponential functions, and I start this project as we begin working on logarithms. 

I play the theme music from Gilligan's Island to start class on the day we begin, and I tell them...
"Our whole class has gone on a cruise together and been shipwrecked. There's plenty to eat and drink, so no one's too stressed. But then, our classmate John Doe is murdered! (He was so quiet, you may not remember him.)"

Then I give them this:

Shipwreck and Murder

You’ve all been shipwrecked on a tropical island - a wonderful place, with bananas, coconuts, fish, and a pretty constant temperature of 80°. Your classmate, John Doe, has been murdered. You know there’s no one else on the island - it was one of your classmates that did it! None of you will sleep peacefully in your flimsy grass huts until the murderer is discovered.

You have watches, thermometers, and other simple tools, but no experts on murder investigations. The day John was murdered, everyone was walking around the island by two’s, noting its features, in hopes that would help you all figure out where you are. It turns out that everyone walked by the spot where John's body was found, and recorded the time when they were at a nearby spot (where they could see a volcano on the next island). Figuring out the time of death would likely narrow down the suspects to four or less.

Angel had a hunch that knowing the body temperature would help determine the time of death. So, at 1 pm, she checked the temperature of John’s corpse. It was 96.1°. Then at 2 pm, it was 91.7°.

Finding the murderer will be our goal. It might take us a few days. (I hope you don’t mind some sleepless nights...)

One more clue… this comes from Daniel, “I read a lot of murder mysteries. In one of them, this detective says, ‘A dead body cools off just like a hot cup of joe.’ I don’t know if that helps or not…”

~  ~  ~  ~  ~  ~  ~  ~

[Change Angel and Daniel to the names of students in your class.] After they've read this, and asked me questions about fingerprints, how he was killed ("looks like a coconut to the head"), and a few other distractions, I ask them to do the following assignment.

Assignment 1

[Do each of these assignments in groups of 3 or 4. If you want, you can turn in one copy per group. It will be important later to be able to describe how your understanding of the problem changed over time, so each person should keep neat notes on the case.]

We want to think about how a hot cup of coffee cools off.

1. What would be a reasonable starting temperature?

2. After about how long would it be cold?

3. About what temperature is it when it’s cold? (Why?)

4. Now let time be the x-axis (t-axis) and temperature (T) be the y-axis, and (on graph paper) graph temperature versus time for a cop of coffee, using what you know from common sense. Does a straight line graph make sense for this?

~  ~  ~  ~  ~  ~  ~  ~

I get a few volunteers to promise to actually measure the temperature of a cooling cup of hot water, and point out that old fashioned mercury thermometers will break and other body temp thermometers might break - they'll need a lab thermometer or a cooking thermometer. No matter how many people promise to do this, I know I might have no data by the next day, so I've saved data from an old class.

Assignment 2

Sarah says “We need some numbers here.” And she boils some coffee up over a campfire and measures its temperature with a thermometer Jessica provides.

Here’s what she gets:

The coffee starts out at 176 degrees, and cools off like this…

Min Degrees
1       169
2       162
3       156
5       146
10     125
15     111
20     101
30      90
60      81

1. If he measured the coffee at 2 hours and 3 hours, what temperature would it be?

2. Graph this data, and connect the points with a smooth curve.

3. So we can conclude that this graph has what line as an asymptote?

4. Give an example of a function with this asymptote.

~  ~  ~  ~  ~  ~  ~  ~

Assignment 3

John Doe’s dead body was lying near the viewing spot for the volcano, and it turns out that there was only one path going by that spot. So, after checking with each other, and remembering who passed whom, you all agree that the murderer was most likely one of the people at that spot right before or after the time of death.

Below are the times that each pair walked by the volcano viewing spot:

Colleen & Mouang 11:15am
Tiana & Armoriana 11:38

etc... (all class members listed)

Paolo & Vithaya 1:24pm

When you figure out the time of death, you’ll know the 4 most likely suspects.

Time of Death:

[Note: Before typing up this list in assignment 3, I've figured out the time of death, checked with the class to find out whether anyone objects to playing the killer, and made sure no one who'd be uncomfortable with it will be a suspect. The idea is that the time of death will be between two of the times listed, and the 4 people listed for those 2 times are the suspects. To find the time of death, they'll solve the equation body.temp  = air.temp + * b ^ t for the time that the body was 98.6 degrees, which will be the time of death. If we've let the first temperature measured be at time 0, then the time we get from this equation will be negative. That's a nice switch, since some of them think story problems can never have negative answers.]

[Note added on 2-7-14: If you are a teacher planning to use this, the idea is to use your own students' names here. Make sure none is walking by at the exact time of death, so that the pair just before and the pair just after are both suspect, and have pairs walk by every 5 to 15 minutes.]

~  ~  ~  ~  ~  ~  ~  ~

Assignment 4: The Next Day

Of course, all 4 suspects swear they’re innocent. The next day, Rasha finds me dead, left lying right in the clearing.

My body temperature is 93.1°, and it’s 9:46am. You check at 10:16am, and it’s 87.2°.

Here’s everyone’s alibis:
• Colleen, Tiana, Robyn, Pardeep, Maureen, Jianfei, and Tayyaba were all swimming together from 8am to 9:30am.
• Brandon, Adeyinka, Cristina, Ash, Cookie, and Paolo were all looking for clams together from 8:30am to 9:30am.
• Mouang, Armoriana, JoAnn, Adam, Denisha, Edith, and Angel were all gathering coconuts together from 9am until they heard Kevin’s screams.
• Daniel, Josue, Natalie, Danielle, Dwight, and Vithaya were hiking from 9:30 until they heard the screams.

Please find the killer before someone else is murdered!

[Note: Students often like to accuse me of being the killer, so I get killed next. I tell them I think it's because I knew too much. The list of names puts each of the 4 suspects in a different group. The one without an alibi is the serial killer.]

I do this project in Intermediate Algebra (the community college equivalent of a high school Algebra II course, done in one semester) and in Pre-Calc. In Pre-Calc, I often end with an assignment to each write up their closing arguments as prosecuting attorney, explaining to the jury how we know the time of death, and why that means the prime suspect is the killer. A lot of them have fun with that assignment.

I give them a problem like this on their next test, and it's not great. I wish I had collected data on success rate on that question. It's probably less than half of the students. Worse than other questions...

[Notes added on 2-7-14:
  1. Well, I'm not sure when it shifted, but they do better on this test problem these days.
  2. If you plan to use this, you will need time to fix up the two lists of names. Work the problems out ahead of time, so you know how it will play out in class.]

Please comment, critique, and suggest improvements.
Math Blog Directory