Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Sunday, January 25, 2009

Hooray for Administrators Who Ask and Listen!

The second semester duty schedule came out today, and instead of telling my students that I can't be available Mondays after school because I am supervising detention, I can tell them I'm available in the library every day during sixth period. (And Mondays after school.) Hooray!

Thursday, January 22, 2009

Leave Those Kids Alone

My friends and mentors Bob and Ellen Kaplan founded a very collaborative, fun version of a Math Circle in Cambridge. It gives young children the opportunity to work as real mathematicians do - on broad, unfamiliar problems in a playful atmosphere. In my work, I have dealt with hundreds of parents, only a few like this. I wish I had responded so eloquently.

Dear all,

Some parents in Chicago, having heard of us, wrote to ask how they could best get their 8-year-old onto a yet faster track and drive him over higher and higher hurdles. We thought you might be interested in our answer (with their names disguised). Alex has written a great e-mail to them, offering him a place in her Math Circle.

Dear F and S,

Your son sounds wonderful, and it is very important to do the right things for him now. Please read our e-mail carefully, and re-read it, since some of it may not be what you expected to hear.

You want J to grow up loving you, and math, and his friends; having a life full of joy and the pleasures of the mind and the world. You want him to appreciate all the beautiful things in the world the way an artist does, and be someone who will be a benefit to those around him, and who will in turn delight in the achievement of others and of our amazing species. The last thing you want is for him to be driven by doubt and rivalry, living the sad life that athletes often do, never satisfying their own expectations or those of their family.

Mathematics is an art, and like all art, it needs peace of mind, leisure, lack of pressure to let invention and imagination grow. Tests and contests are exactly the wrong setting for doing real math: they foster competition where cooperation is wanted; they take the deep pleasure of mathematics away and replace it by the shallow and short-lived pleasures of winning and then having to compete again: the surest way to neurosis, ill heath and dissatisfaction with oneself, one’s family and the world. This road to ruin you see being followed by so many in our high anxiety world.

Let J’s real delight in mathematics grow by having him enjoy, at leisure, with no time pressure or pressure to succeed, real mathematics: not this problem and that (short-winded exercises that reach down to no depth), but exploring geometry, or algebra, or calculus, slowly, with questions and mulling things over, trying different approaches (many of which won’t work - but that’s part of the fun of exploring!). This is why we wrote our book, The Art of the Infinite, which sets the reader out on paths that go deeper and deeper into the heart of things. This is why we began our Math Circles, where people talk together about serious problems in a non-competitive way, as philosophers and artists do, trying to understand, not triumph over, the world and the human mind.

All parents want the best for their children, but often rush them instead headlong into lives of endless dissatisfaction. Let J have the time to fool around, to try this and that, to get things wrong – because this is the way all great mathematicians have begun, focusing not on themselves or petty successes but on mathematics itself, which needs imagination to live with happily. And imagination grows in a life where the cares of the day are seen to be unimportant. Think of the Eight Immortals of the Wine Cup, laughing at the moon.

With our very best wishes,

Bob & Ellen Kaplan

Friday, January 16, 2009

How to Bounce a Ball Part 2 - Solution

Read Part 1 Here

I did my best during this experience to let the kids figure out as much for themselves as they could. It is a pleasure to watch them turn into little scientists when there was something about which they are really curious. Through the course of this investigation, they had to conjecture, test, and discover several things that those of you reading this probably take for granted:
  • horizontal motion can be considered independently from vertical motion
  • angle of incidence = angle of return if no spin on the ball
  • the distance from a person to a wall is the shortest distance
  • the distance from a person to a wall is perpendicular to the wall
  • a diagram helps make sense of relationships between distances
  • a neat and labeled diagram makes it easier to discuss
  • the easiest way to measure distances in a room with a square grid tile floor is by counting floor tiles
  • triangles that have the same angles are the same shape
  • triangles that have the same shape have proportional sides
Without further ado:

Once they had a good diagram, they had to find a way to partition the wall in the same proportion as d:d'. All of this discussion was in terms of number of floor tiles. They marked the nice-acting ratios first:

2:1 marked 2/3 of the way
1:2 marked 1/3 of the way
1:3 marked 1/4 of the way

I had them do a bunch of these just to give them practice with some tangible fractional amounts.

After we had marked several of these and proved they worked, the kids were honestly kind of over it. I wanted to talk about any distance, and we eventually got here: a = (d/ (d+d'))w. But that last part, I felt like I was just doing it for them. That was ok, though.

The Teacher Has Her Fun
I was also interested in modeling the path of the ball with an absolute value function. Here is my derivation.
So we have y = c*abs(x - a) + d, where a is given above in terms of d, d', and w, and since this thing has to go through the origin, c = -d/a. I don't know what I'd do with this, but I was trained as an engineer and I couldn't help myself. Thanks for reading! I hope you enjoyed my little adventure.

Thursday, January 15, 2009

How to Bounce a Ball Part 1 - The Problem

Rubber Band Ball of Death

This all started because I left my rubber band ball lying out. Inevitably, a boy picked it up and started bouncing it. First he was bouncing it against the floor. Then he was bouncing it against the wall, playing a little solitare game of catch. Then his friend arrived, and tried to catch it off the bounce. They starting counting how many times they could bounce it to each other without dropping it.

Then their math teacher noticed what they were doing. She pointed out that this game wasn't very hard, because they were standing right next to each other.

They moved away from each other laterally, maintaining their equal distance to the wall. "This is easy! See? We just have to throw it in the middle."

Then she asked one boy to take two steps forward. The game changed. They could still get it to each other, but it was trickier. They tried standing various distances from the wall. Classmates joined in. The teacher wondered out loud if someone could mark an X on the wall for any two people, such that throwing the ball to the X would guarantee a catch.

This is how my exponents unit was interrupted for a week, how 12 9th graders solved a problem and didn't care they were doing math, and why my north wall is sporting a galaxy of multicolored X's.

Give up? The solution is here.

Things It Took Me Way Too Long to Learn

When you start in this profession you get plenty of advice... Some of it doesn't make sense at the time, some of it is crap, and some of it sounds good but you don't really get it. These are things that even if I heard them early on, they didn't mean anything until I realized them for myself. The nutty thing is, I will probably have another list that will be obvious after another four years of this.

1. Keep questioning. In the course of one lesson, Year 1 Me would do ten different examples. Year 4 Me asks ten questions about the same example.

2. Sit down. In one scenario, I sit behind them while we all look at, observe, and discuss an image. In another, the kids work while I scoot around in a rolling chair. Most times, anything is preferable to standing in front of them.

3. The best way to have a conversation with a teenager is to stand next to him, as you both stare off into the middle distance.

4. Take the few minutes out of class a few times a week for something fun, interesting, or entertaining, whether or not it's related to your content. It's worth it.

5. It works better if they say it. My best teaching colleagues are 15 years old.

6. If you really want her to do something, give her a choice of 2 things that are acceptable to you. I hear this works with toddlers, too.

7. They care if I like them. I should care if they like me. Not because I want to make a bunch of friends, but because they will work harder and learn more, and my work day is nicer when I get to hang out with people I like.

8. Concrete first, abstract later.

9. Except for summative assessments, always publicize answers. I don't like to answer "Is this right?", ever, if possible.

10. Students practicing a new skill need time for: arguing, comparing conflicting methods, explaining why right methods are right, explaining why wrong methods are wrong, checking, troubleshooting and finding errors, and summarizing their learning. This is why they should have a way to check the answers on their own. This means they might only get through three problems on my sheet of twelve. That's ok.

Ten seems like a nice arbitrary place to stop. I'm sure I'm not telling any of you seasoned vets anything you don't know. This was just on my mind today and I wanted to get it all down.

Wednesday, January 14, 2009

The Class You Dread

One of my classes is normally the lowlight of my day. It's full of generally well-behaved and capable kids. But something about the chemistry is terrible. I hate it. They hate it. We all feel like we are grimly marching to the bell. Eyes are rolled. Sighs are sighed. Comments are muttered under the breath.

Something had to be done.

I talked over the situation with a school psychologist friend for a good hour. Many people only see school psychologists in their role working with learning disabled students, but helping teachers problem-solve in their classroom is under their umbrella as well. I didn't want to take stabs in the dark with this group. A failed attempt would have been worse than no attempt at all. I wanted a professional to endorse the plan.

There were a few little things first. I changed the seating chart, to jostle them out of their habits. I papered over the little window in the classroom door, which is a frequent distraction (there is some little punk who likes to stand out there that period and make faces or something). I gave them a little speech about our goals, their futures, and positive learning environments and the behaviors I saw that indicated their attitudes were all that positive.

The big "intervention" (the psychologist's word of choice these days, it seems) was: Each day before class, I write on the whiteboard the list of the problem numbers assigned for homework that night. While class is going on, I am looking for positive learning behaviors (taking out materials without being asked, attempting practice problems, volunteering to answer questions, asking questions, etc). If all goes well for 5-10 minutes, I quietly X out one of the homework problems. By the end of the period they may have 3 or 4 less problems to do that night.

If you are savvy to basic behavioral psychology, you'll recognize this as negative reinforcement. In a nutshell, removing an undesirable consequence in order to promote good behavior. Which, psychologist friend informs me, research has shown to be more effective than punishment or positive reinforcement.

It's also a random and variable reinforcement interval. I didn't tell the students I'd delete a homework problem exactly every 5 minutes, for example. The intervals will change and be unpredictable. Also promoted by research.

He used a bunch of other big words that I don't remember.

He warned me that I would have to stick with it for a while. The kids didn't trust it at first. They wanted to know what the catch was. They were sure they'd have to do the deleted problems later, or something. They wanted to be the ones to choose the problem to delete. They also tried to start arguing with me about when they deserved a deletion. All of this has taken some time.

But it's been pretty painless. And, the environment in the classroom has certainly improved. More kids are contributing more often, kids are more willing to engage, and I don't dread going to this class anymore.

Sunday, January 11, 2009

Cool Video!

I am only posting this because I haven't seen it on anyone's blog yet! A colleague shared the link with me on a listserv. It shows a man juggling inside a big equilateral triangle made out of wooden struts. It makes me think about how my 8th period freshmen (9/12 of whom are boys) like to spend the few minutes before the bell rings bouncing my rubber band ball against the classroom wall - I try to get them to stand at different distances away and talk about where they have to bounce it to get it to each other.

Anyway...this video could be the start of a beautiful geometry lesson.


If people watch this and get ideas about where to take a lesson from it, I'd love to discuss it in the comments!