And 'Rithmetic, by Daniel Greenberg

I love this article, and now I can't find it anywhere on the web. So I've put it on scribd*. (There are permissions at the end for copying it.)

I was inspired to put it online by a discussion on the change.org education blog, about Sudbury schools. Someone asked about math, here's the answer.

Comments, anyone?








______
*There are permissions at the end for copying it, so I know it's legal. But the folks at Sudbury school aren't comfortable with the fame this piece has gotten. So many people focus on this: "I found a book in our library, perfectly suited to the job at hand. It was a math primer written in 1898." And they want to know what book it was. But the book wasn't the important part of this story. The kids' choice to delve in was what made it work.

I asked for permission to include this piece in my book, and they wouldn't give it. So in the book I explained the important parts of the incident, and included a short quote.

Math Teachers at Play #5, and ...

Yesterday and today I attended the Great Circles workshop being hosted by the Mathematical Sciences Research Institute in Berkeley. It was marvelous! I'll write up my thoughts about it as soon as I can. (I hope to include links to lots of exciting sites, mention some research on gender issues in math, and mention some books I discovered while there.)

Today marks the 5th Math Teachers at Play blog carnival, posted at Let's Play Math! I didn't submit anything this time, but Denise included my Students Learning Through Teaching post. I guess she must have liked it. Personally, it feels too experimental for me to want to promote it yet. Will it help my students? I'm not sure. What do you think?

Students Learning Through Teaching

I teach beginning algebra at Contra Costa College. Most students in a course like that do not like math. I work hard to create an atmosphere of working at it and having fun with it. However, I still feel like I'm too tied to the textbook. Reading blogs like f(t) and dy/dan really inspires me. (I must admit, I also feel like such a slow learner when I see all the amazing things they're doing.) But there's more... The burst of energy I've gotten this semester from all the possibilities available on the web is great. A student in another class recommended khanacademy.org and then we heard about mathtv.com. After I watched the videos on that site, I figured, "We could do that!"

On Thursday two students presented one problem each. And then on Friday 3 students presented a problem each. All of them were recorded using the video capacity on a digital camera. The student whose camera it is has put them all on Youtube. (I haven't learned how yet.) And I've linked to them here.

We had a great time with these first few presentations. Students are asking more questions, and listening better. When I showed how a step on one problem could be done in a more elegant way, the student who had presented it said "That's beautiful, Sue!" I was delighted. "Bet you never thought you'd say that about a math problem!" (And we're talking rational expressions here, one of the ugliest topics in the beginning algebra curriculum.) ;>

They're all up now: Here's Hira, Stanley, Tulawna, Brandon, and Nailah. You guys are great! When we get back from Spring break, I'll have to be as brave as you all, and get on YouTube too. In search of a tripod...

Math Teachers at Play #4

My favorite way to explore the nets.

Blog carnivals are a way for people who follow one blog to find out about lots of other blogs on the same subject. Denise's Let's Play Math! was the first blog I followed regularly, and Math Teachers at Play is the blog carnival she got started. Looks like it's scheduled to appear every two weeks. And my brand new blog is in today's carnival, Math Teachers at Play #4!

I'm off to a weekend contra dance camp, so I'll have to wait until Sunday to explore all the great blogs gathered there. I hope you all enjoy mine.

Benezet - An Amazing Bit of History

L.P. Benezet was the superintendent of the Manchester, NH schools in the 30's. He decided that there was a big problem with math instruction and decided to delay formal instruction in math until 6th or 7th grade. In the earlier grades the children still learned about days of the week and month, time, money, estimating, measuring, and so on. They also focused more on reasoning and communication skills.

The complete articles, which were originally published in the Journal of the NEA, can be found here. I read the Math Forum, and saw Benezet mentioned. I posted a topic there to gather information, and another to discuss pros and cons. But the folks there insist on flaming each other, so it's a bit tedious.

Anyone here know anything more about his experiment?

My Ideal School

My ideal school
Is part of an ideal community
People aren’t separated by wealth and poverty
The richest have a little extra, the poorest still have what they need
People aren’t pulled apart by race
And they’ve learned to respect the glory of differences
If gender differences still exist, the ones who don’t fit are celebrated
They all mix together in a public school that they,
Neighborhood by neighborhood, control

My ideal school
Is close to home
The kids can visit whenever they want
Perhaps a dedicated mentor lives there
(Can’t say teacher, it makes the wrong image)

My ideal school
Has a garden and a kitchen and hot yummy food
And a beautiful place to sit and eat together
(Is it calm? Or is the excitement of the children too much for ‘calm’?)

My ideal school
Is full of resources that draw the kids’ interest
Is staffed with adults who know
That children have their own ways of thinking
That each child moves through learning in their own way
That there must be safety, both physical and emotional
That there must be affection and loving and hugs

In my ideal school
The children see adults learning
They see adults getting stuck, and then getting it, frustration and joy
Here is a woman learning cello
Here is a man learning to knit
Here are 3 grown-ups talking about a book

My ideal school has traditions
They go camping in September
They make Stone Soup together in January
Each day begins with music, someone is playing guitar and many are singing
Most everyone gathers together at lunch time and shares their food
The day and the year both have a rhythm

At my ideal school
When two kids fight
Bigger kids come help them to use words to solve their problems
The big kids help to build a deck or a chicken house, or a new classroom,
trek through the mountains and fix bikes,
take responsibility for the gardens, chickens, and maybe a sheep or a goat

My ideal school
Is part of a network of schools
That crisscross the community like a spiderweb
And each is different
So each family can find
A haven for their children
That resonates with their values.

My ideal school might not be called a school
We need a break from the past, we need a new word for a new place
Maybe it’s the Children’s Center
Except there are lots of grownups there, too,
Learning as much as the kids

Joyful Learning

I wrote the thoughts below last September in response to E, who wrote this on Living Math forum:
>I'll have to admit, making learning fun seems like such a foreign concept,
>because school was never fun for me. Learning and fun never went hand in hand.

That shook me. I love learning new things. Sometimes it makes me bite my nails, sometimes I think I'm never gonna get it, but there's almost always a moment where my angst becomes so worth it, where I'm just glowing inside from the joy of a new skill or a new, broader understanding.

When I was in school, I was often bored, but I knew it was school that was boring, not learning. Luckily, the academic subjects came easily for me. (Except penmanship. I got a D in that in 4th grade, and I still struggle with it sometimes.) But school made me think I couldn't sing. I love to sing, so I work at it. Sometimes I do great, and sometimes not so great. I'm a slow learner with music, and many physical things. So I like lots of practice. When I was in a Unitarian church choir, I wanted to practice each song about twice as many times as what seemed reasonable to the others there. Being a slow learner shouldn't be a problem... But in school it is.

I teach math at a community college, and I give my students a "Math students bill of rights." One right is the right to learn at your own pace. I tell them that schools make that impossible, because we have to work as a group. (I do answer all homework questions, and modify each class for the group I'm working with. But if you're the one person who's slower than the group, it isn't your pace...) I've always been torn between my love of teaching, and my understanding that the best learning comes from within, and may not need a teacher.

Waldorf teachers are supposed to be learning new things all the time themselves; I assume that's so that they are in touch with both the struggles and the joys that come with learning. Maybe picking something you want to learn yourself, E, and working on it, would be a good way to find some joy in learning, so you can pass that along to your son.

One response to your post had said 'math is a different story'. I think I disagree. Math is different for homeschooling parents who are comfortable winging it with every other subject. But for the child math is just like all the rest, they learn the most if they approach it at just the right moment for them. My close friend who homeschooled did everything except math in unschooler style, trusting her daughter. But she was scared of math, so she followed a school curriculum for that, and it was the one thing her daughter didn't like. The daughter did go on to learn calculus at the community college, because she planned to be a veterinarian, but she never found as much joy in it as she could have if she hadn’t been pushed...

OK bloggers, I want to start a Joyful Math Movement, with Lockheart's Lament (www.maa.org/devlin/LockhartsLament.pdf) as one of our founding documents. Who wants to join?

Call for Submissions: Anthology on Learning Math Outside the Classroom

Do you…
… run a math center, math club, math circle, or other math program which people attend by choice, and at which something other than tutoring and/or homework help is provided?
… homeschool, and have exciting ideas about how kids learn math?
… do new sorts of work helping people learn math through the web?
… have ideas about how to integrate these experiences into public school classrooms, or public policy perspectives?

If you do, perhaps you’d like to add a chapter to Joyful Math: Learning Outside the Classroom, and In.

Why this anthology is needed:
Math is seldom taught well in American schools, and most people end up quite uncomfortable with it. Throughout the history of public education in this country math has troubled most students. Is it different in other countries? Can it be different here?

A number of ground-breaking projects have sprung up over the last decade, with this common thread – most are happening outside the traditional classroom setting. This anthology will include exciting reports from the frontlines of what its editor thinks of as the “joyful math movement”, and will help parents, teachers, and math enthusiasts everywhere think more clearly about how to help students learn math.

We already have an exciting core group of authors, including Julie Brennan (host of Living Math Forum and livingmath.net), Maria Droujkova (naturalmath.com), Amanda Serenevy (riverbendmath.org), and the published authors Robert and Ellen Kaplan (Out of the Labyrinth: Setting Mathematics Free). Now we need to find more voices. Please send this call for submissions along to all lists, groups, blogs, wikis, sites, etc, where people doing projects like these might see it.

How to join in the fun:
If you have something to add to this anthology, we’d like to hear from you. Please give enough detail to make your unique perspective clear, and write your proposal in the same style you’d use in a chapter you’d submit later. Or simply send a first draft of your proposed chapter. Send proposals, questions, and requests for more information to Sue VanHattum at mathanthologyeditor at gmail.com. Deadline for proposals: May 30, 2009. Tentatively accepted contributors will be notified by June 30; chapters will be due by September 30, 2009.

Some questions for contributors:
• Describe your project. Tell stories from daily experiences. Also tell what you’ve learned from this experience. Have you had ‘aha!’ moments?
• What hurdles have you overcome in this project? How has your vision for what you’re doing changed over time?
• In an ideal world, how could public schools use your ideas/practices?

Math Salons and Base Eight

I've been doing monthly math salons at my home since the fall, where parents, kids and other adults play with math. At the last session I had decided to play with base eight, and had told participants we'd be doing 'alien math'. I waited until the last minute to look for good activities, and couldn't find anything I really liked. I did find two good activities related to binary, though, and prepared those.

The night before the salon, I laid in bed thinking up a children's story about my eight-fingered aliens. I got up early and wrote it, and then printed it in booklet form. On the first page I asked the readers to illustrate it for me. The kids really liked doing that.

In the story I claimed because their system is based on 2x2x2, these aliens are really into doubling. I also claimed that the kids figured out a code for counting up to eight using just the 3 fingers from one hand. I figured binary would come pretty easily to the aliens, and mentioning this would be the tie-in with the binary activities we'd do at the salon.

Here's the story:

Eight Fingers, by Sue VanHattum & Friends

[Dear friends, I wrote this story, but I wasn’t sure how the people looked, except for their fingers. Can you help me illustrate it with pictures? Thanks! Yours, Sue]

Once upon a time, on the planet __________, there lived people who looked very much like we do, except for one thing. On each of their two hands, they had a thumb and only three fingers.

Longer ago than anyone there remembers, they used to use stones to keep track of things. But then, just like us, they began to use their fingers to count.

Eventually, though, they starting writing down the numbers they were counting on their fingers. Just like us, they had a symbol for each number from zero up to seven. Their 0 and 1 looked like ours, an empty circle of nothing, and a tally mark for one. But the others were a little different…

But then they wrote this 1 0 to mean all eight fingers. Or to mean eight horses, or eight flowers, or eight yummy strawberries.

Do you know how they wrote nine? _____

Now, eight is a very special number, even here on Earth, and they discovered long, long ago how special it was. They liked doubling even more than we do.

If you started with one finger, and doubled once, you could hold up your two fingers in a circle. Then if you doubled again, you had all the fingers of one hand up. The next time you doubled you had all your fingers up. If you did it again, all your fingers and toes were up. Again, and you had to have a friend put all their fingers and toes up with yours. Most kids started giggling so much when they did that, it made them fall on the ground laughing.

Now all that doubling helped them discover a very neat pattern. Kids liked to tell each other the secret code that used just three fingers, and showed all the numbers from zero up to seven. Their fingers were a little more flexible than ours, and they had great fun flashing the codes for each number to each other. I wonder if you can make a code like theirs …

There’s much more to their story, of course, but I just haven’t written it yet…

There is no end

Math Poems

I was in a mood one day, and this was my response to a math question...
(Do you know any math poems?)



Imaginary Numbers Do the Trick
by Sue VanHattum
(written on December 15, 2008)


In an email group of 4,000 homeschoolers,
a member wrote:
   My son asks, “The square root of 1 is 1,
   So what's the square root of -1 ?”
This was my reply to her…

What we call the real numbers
is everything on a number line,
positive, negative, zero.

If you're thinking about those real numbers,
on that number line,
none of the negative numbers can have a square root,
because anything times itself will come up positive
(or zero).

But, once upon a time (for real),
mathematicians dueled
by giving each other lists of thirty hard problems.
The winner got recognition
and perhaps a job.

All this dueling led to
a solution for cubic equations:
these mathematicians
created a formula
that would find the numbers
that would solve a thing
like 2x3-3x2+4x-5 = 0.

But that formula was a problem!
It came up with square roots of negative numbers,
which drove the mathematicians wild.
No, no, no. There is no such thing!
Well, maybe there could be…
and if there is,
what would it look like?

With a wave of the magic wand of imagination,
These mathematicians
made up a new number,
which later got the name i.
(Imagine it written in fancy script.)
i is the square root of -1,
so i squared must equal -1.
i is the first step in creating …
the imaginary numbers.

Picture, if you will, a new number line
of imaginaries
crossing the line of real numbers at 0,
with the real number line horizontal,
and this new one vertical.
(It looks just like x and y axes,
but now it's all one number system,
a bit more complex.)

i sits one step above zero.
Another step up this imaginary number line,
we see 2i,
2i is the square root of -4.
(It is?!
Why yes, 2i times 2i equals 4 times i squared,
and i squared equals -1,
so we get -4.
Cool, huh?)
And on it goes.

Now all of this wouldn't really solve much
if there were no square root of i,
and that seems too weird to think about.
But, once you study trigonometry
(how'd that get in here?!),
the solution to that little problem
is actually quite elegant.