Math Be Brave

Yantra in Math Class

2011-09-16T20:37:00.000-04:00

Today was our first pop quiz. If you've read anything here in the last year, you know that means meditation and cookies. I have been itching to meditate for the last 6 days with these kids, just waiting for an excuse, not feeling quite ready, finally caving and just resting in the structure from last year.

But this year I wanted to be a bit more thoughtful about it. Here was the lesson...

I. Writing prompt: What do you know about stress? They write for about a minute.

II. Class conversation: Raise your hand if you've ever felt stressed. Raise your hand if you've felt stressed today. Raise your hand if you get stressed out about friendships. About school. About math class. What happens when you get stressed, what does it feel like? What does it look like? What do people do about it?

III. Introduce Jesse as a stress-reduction (I didn't actually use the word meditate) instructor: In addition to teaching math, I also teach adults and my students to techniques to manage and reduce their stress so that they can feel better and focus on whatever it is they want to do. When people get stressed, as you guys described, at best it's uncomfortable and distracting. Sometimes stress about a test can cause a kind of panic that makes us forget everything we've learned. So in this class, we'll experiment with some different techniques to help us focus and relax at the same time.

IV. Pop quiz announcement: We're starting this today because we have a pop quiz, and I want you to experiment with not being stressed about it.

V. Introduce the yantra: There's a picture of a yantra up on the smartboard. I explain: in India, people have been making these kinds of geometric images for thousands of years, and using them to relax and concentrate. Today I'm going to tell you a kind of story about the yantra while you look at it, and you can just see what happens to your body and mind. There's no pressure, just see what happens. Keep your eyes open and look at the yantra. Listen without speaking as I tell you the story...

VI. The story: The dot in the very center, called the Bindu, represents unity, our connectedness to each other, our families, our neighbors, the whole of the universe. It's small and hard to see, but see if you can just focus your eyes on that one small point. (wait time...) Now let your vision expand so that you can see all the red triangles. They say that the triangles that point up represent the masculine or male energies in the universe, and the triangles that point down represent the feminine or female energies. (wait...) Now look at the circles around those triangles. These circles symbolize that which is constant, infinite, cyclic in our lives; time, which has no beginning and no end, our breath which pulses in and out, 60,000 times a day without our effort, gravity which pulls on us all the time, the universe which is infinite in it's vastness, always expanding, no starting point, no end point. (...) Now look at the petals of the lotus flower around the circles. The lotus flower is so important in India, and in this yantra it represents your understanding, your knowledge, which is opening, always expanding, growing. (...) Finally, let your eyes see the squares at the outside of the yantra. These squares are your identity, your separateness, your individuality, which enclose everything else. (...) Look at the Bindu again, that point in the middle. Relax your eyes, so that you can focus on the Bindu but see the whole yantra at the same time. (...) Close your eyes and see the after image on the lids of your eyes. (...) Open your eyes and see the yantra again. (...) Close your eyes again. Just see how much of the yantra is still visible to you.

VII. Debrief: Write for one minute about what you experienced. What did you feel, what did you notice?

VIII. Pop quiz: ...And if you feel stressed, experiment with seeing the yantra in your mind's eye and just see what happens. See if it helps you relax and focus. And if it's not quite enough, there will be cookies.

----

It was awesome to do a meditation with kids that doesn't require them to close their eyes. I hadn't planned this part, but as it turned out it was so much easier to get them to do this practice than any closed eyed meditations I've done in the past. It was awesome to introduce meditation to kids with such a high voltage geometric image. They had cool experiences with it. I want to get them to make their own and put them on their binders so they can remember and keep that intention of focused relaxation, relaxed attention, anytime they're in math class (and beyond!)

2) My incredibly brilliant co-teacher found himself somewhere between lazy and curious, and when it came time to teach his lesson (which we usually share) he began...

"So this isn't really my thing. This is really Jesse's thing, and she's really crazy. So it's ok if you think this is weird, because I do, but we'll try it and then we'll decide if we ever want to do it again. We'll wait to talk about how weird it is til after she's gone."

He did the whole thing. I happened to walk in when he was introducing the yantra, and he asked me to lead the storytelling part, but he was totally prepared to do it himself. Amazing. We are learning so much from each other. Which I will continue to tell you about...

first day of year six

2011-09-08T19:49:00.003-04:00

Today was the first day, and the two things I want to share are about my new coteacher/department cofacilitator.

1- He regularly counts how often I say awesome. Or maybe how often I say awesome and amazing. In meetings and I think maybe in class too. Then he gives me responsibilities because I'm awesome.
2- I'm so excited about our curricular collaboration, and want to say right now, whatever comes after, that what we're up to seems to be somewhere between strictly awesome and downright revolutionary. Always.

We make a strange and really good team, I think. I'm curious how it will develop and excited to share our work with you!

Jennifer Abrams on coaching...

2011-08-26T16:17:00.000-04:00

In Jennifer Abrams' most recent newsletter, she wrote about a bunch of cool stuff, and I encourage you to read the whole thing here. In particular, I keep wanting to go back and read this paragraph on dramaturgy and how it connects to coaching. As I contemplate my new coaching role at school, this gives me language for what I aspire to do, and will help keep my eyes on that vision.

from Jennifer:

Definition of a Dramaturg
This July I spent time at the Kennedy Center participating as a new board member of the National New Play Network as NNPN ran their summer MFA Playwrights’ Workshop. Theater professionals from all over the country came to support six new playwrights as they brought their plays more fully into being. The playwrights were connected with a director, a dramaturg, other support staff and a cast of actors. After one week of amazing collaboration, we experienced a reading of their work. Of all of the parts people played during the week, the role that intrigued me most was that of dramaturg. We don't have dramaturgs in the teaching profession or do we? In a nutshell, a new play dramaturg has three roles. One - to be a champion for the play itself. Two - to be a brainstorming partner for the playwright. Three - to be a liaison between the audience and the play. If done well, the dramaturg is a coach, a muse, a healer, a connector and much more. We who are leaders at any level in our organization or school might consider ourselves as resident dramaturgs - champions of the work we are trying to do, brainstorming partners for our colleagues, and liaisons between our schools and the ‘audiences’ we work with - students, families, communities.

Your appreciation for all things is off the charts...

2011-08-22T23:09:00.000-04:00

"Every person whom you come into contact with, every exploration into every place on your planet, every interaction with anyone who comes to you will be one that has the potential of uplifting them because you represent the best of all that exists in all of the universe. And you know it because you feel frisky. You're full of yourself. Your eyes are bright. Your body feels good. Your body is functioning the way you want it to. Your heart is lifted. Your mind is clear. Your wit is sharp. Your fun is full. Your eagerness is pronounced. Your love is apparent. Your appreciation for all things is off the charts." - Abraham Hicks

off the charts people...let's go...

Videos of Self Teaching

2011-08-04T17:06:00.001-04:00

I was just going through a bunch of unlabeled discs, most of them DVDs with films I made back when I was an experimental filmmaker making films. I was getting into the routine of recognizing the films, labeling them, throwing them away.

Then all of a sudden there's me, 4 years ago, in clothes and hair and glasses I remember but haven't seen since then, welcoming in a classroom of kids that I taught my first year. I actually experience not believing what I'm seeing...I have no memory of this being recorded, of this particular day, of those conversations, of that way I'm moving or speaking. I know those faces, relatively sweeter and younger than I would have thought, utterly familiar. And there I am, so serious, holding my clipboard, hardly looking up, but when I do serious in that too, connecting with such conviction and intensity.

I see simultaneously the significant and undeniable presence I am in that room, just because I am who I am, despite all my obvious mistakes and inattention to the important stuff, I am there and these kids that I remember being so difficult know me and are comfortable there. I imagine them seeing me being so serious, keeping track of their failures, so clinically observing their shortcomings. That's what I recognize there. It takes me back. I remember the feelings I had about those kids. I feel grateful I'm not still living that life, not because it was so hard, though I remember it was, but just because I'm grown and full in new ways now, and am glad for those transformations in me.

I wonder if I could have had any objectivity if I had seen this when it was shot. I would wish for myself to recognize the fundamental worth of that teacher in the room, and be able to also see how easy it could be to shift the dynamic in that room. Those kids deserved to be seen, watched, loved through simple attention to their thinking and presence in the room.

I'm humbled and honored to have seen this just before school begins again. May I learn how to attend to what's important, may I find the balance between humility and sincere confidence and rest there. I think next year I'll use video (sped up?) more in my coaching. Immediate insight!

A Teacher's Job

2011-07-19T20:45:00.000-04:00

Someone must have suggested this idea to me, but I don't know who, when, where it came from.

Let's imagine a new image of teaching:

There are bundles of packaged and organized mathematical concepts, ideas, or units.

It's the teacher’s job to get that bundle unpacked before the students come in, so that they can experience the natural questions and packing and organization that they do themselves.

Rather than simply describing the still packed bundles, or unpacking and repacking them in a powerpoint.

OC...moments from the year

2011-07-19T20:21:00.001-04:00

I am taking time today to go through any notes I wrote during the year and incorporate them into my planning for next year. It's nice to try to do something with my great ideas! I'm having a good time, enthusiastically diving into work during vacation. It takes discipline and presence to really do this with joy.

I found some quotes from a student, OC. I didn't share them before because they felt too sweet to tell anyone else, but now I think there's something to learn here.

"It's like God took everything amazing and put it together (hands gesturing, pulsing towards each other with enthusiasm) when he made you."

"Jesse, when you get married can I come to your wedding?"

"Jesse is way too kind and adorable for anything bad to happen to her."

"Oh, I know that I can't make you unhappy. Nothing can bring you down, Jesse. If you'd been around in the 1930s the Great Depression would never have happened."

I really felt connected to OC, like he got it, he got me, he got the math, not just the content but the beauty, the incredible awe of it all. He is an amazing kid, sophisticated, with remarkable resonant maturity in a 14 year old body. I was totally professional with him, but there was such tenderness and sweetness in our interactions, it felt like it was too much. And indeed, by the end of the year he had really pushed me away, doing his adolescent thing beautifully, if painfully for me. He laughed and scoffed at me, cut my class, apologized but avoided conversation, actual sincere interaction with me at the end.

I took this kid's crush on me personally, and I think that made school harder for him. In order to find balance, he had to push extra hard against me towards the end of the year, so that it felt like there was nothing between us at all.

At the end of the year this year, I felt really sad, more than ever before. My aversion to saying goodbye, to stop seeing their faces everyday, was so strong that I didn't want to come to school in the morning. It was funny too, because they were never more annoying to me than at the end of the year and so being with them was joy and sorrow, annoyance and delight all at the same time. We were terminating, as they call it in the psychoanalytic approach to group dynamics, and it was painful for all of us, probably more so for them than for me. I taught 9th and 10th graders last year, so I will see them again in the halls, but it will not be the same. This is the thing that I know now that I didn't before: they will see me and we will know each other, some will hug me and say hi, others will not, in any case it will not be the same. I will become obsolete in their experience of high school, just by virtue of being in their past.

That's the way it should be. It's normal, there's nothing at all wrong with it. But I felt it this year for the first time, as we were ending. I felt the finality of the goodbye, the end of seeing them every day, for better and for worse, when we get along and do amazing work and when we are frustrated constantly and accomplish nothing. I am living a new life now, without them, without their mysterious adolescent energies in my life. Next year when I return that energy will return to my life but they will not.

This stuff is interesting, right?! As I experience more of these intense emotions in my relationships with kids, I get better at holding space for them to be where they're at, and to support them throughout their pushing and pulling. That's what I'm there for. I think this experience with OC will allow me to better see what's going on with a kid without taking their affection or mistrust personally, and will give them space to grow and learn in the safety of my conscious presence. I think my awareness of how painful it is to say goodbye will allow me to better appreciate them while we are together, and I trust that my experience of enjoyment and enthusiasm will serve them well.

Thank you OC, and all of you, my previous students...through you I have learned so much.

Reading and Writing in Mathematics

2011-05-22T16:41:00.002-04:00

Also, we've been studying up on Literacy in our school this year, and our librarian just shared this resource with us. Haven't tried these yet, but they're thought provoking for me as a relative novice in teaching writing and literacy.

CUNY requirements, for all you NY State folks...

2011-05-22T16:40:00.000-04:00

Just discovered that these have changed, and thought I'd pass on the heads up.

The Rapture

2011-05-21T10:01:00.000-04:00

The last week has been an intense one: rain every day, anticipating the end of the year and on top of that the world was supposed to end today.

My 10th graders didn't mention it once, but my 9th graders were obsessed. Everyday discussing with humor and dramatized panic how their lives would end, loudly derailing all attempts at mathematical instruction. From my point of view it's a reasonable question: I'm not sure I would choose to be in school if in fact we really knew we were living our last few days.

Yesterday was the pinnacle. RJ came into school and wrote "THE END IS NEAR" on the whiteboard. I inserted "of school" between "end" and "is," but no one noticed. Every student was asking me what I believed, but they didn't care about the answer. Whether they agreed with me, were reassured or skeptical, it didn't change their desire to talk about this and nothing else.

By math class two periods later, they were saying goodbye to each other. ST asked, "Jesse, will we still have math class in heaven?" DM shouted out, "Jesse I love you, I'm going to miss you," and then asked the class if they would miss him. "Clap if you're going to miss me, class." Everyone clapped. "Clap if you're going to miss ST." Everyone clapped. "Clap if you're going to miss Jesse." Everyone clapped. Very sweet.

Then out of the blue, OF loudly shares, smiling, "I'm going to die a virgin." The class notices but the chatter continues. I feel like I'm the only one who has really appreciated the vulnerability and generosity of this comment. He is so sincere, expressing in his innocence the loss of all the unknown ahead of him. He doesn't sound like he is eager to remedy the gap in the next 24 hours, just that he's sad, or even nostalgic, for a future he won't live to see. Amazing.

The funny thing is that all week I've been trying to figure out how to terminate with these students. From all that I have learned about in studying the theory of group dynamics, termination, or the ending of a group, is a hugely important opportunity. We all re-live our experiences of loss and abandonment when things end. If the facilitator of a group makes this ending transparent and gives the group time to process their feelings, to experience an end without surprise, the group can both experience less trauma in the ending itself, but also heal their past injuries. What this looks like in my classroom is simple. I tell them every day how many days we have left. I keep bringing the end to mind and then when they have things to say about it, I listen.

This week I learned that with the kids who know already that they have to repeat the year and are disinclined to come to school for academic reasons, the group can still be a reason to come to school. They are a part of something, whether they pass or fail, whether they behave well or badly and get kicked out of class. It matters that they are here at the end; it matters that they are a part of this group that is ending. We all want them there, and it's not the same without them.

So when this major endings conversation arose, I didn't stop it. This rapture thing is getting my kids to do all this processing around endings and it's awesome. So "No, we're probably not going to die," I say. "But this group is going to end in 3.5 weeks."

It helps to have time to say goodbye. To tell each other that we're going to miss each other, to acknowledge that even when we see each other again next year it won't be the same. Some people will stay and others will go, and our group will not exist anymore.

Powerhouse.

Resilience, process, curiosity and relationships

2011-05-09T23:05:00.000-04:00

One backdrop, two stories, one conclusion.

Backdrop: These four words describe our most important values across our school. They are the threads that connect the islands of our classrooms, the values that we aspire to teach. I like them. I think they are basically the most essential ingredients for a happy life as well, and the fact that we came up with them to align our curricula is compelling.

Story 1: I saw one of my favorite seniors, VC, this morning at the deli. She looked exhausted and discouraged. I carelessly told her she was doing great, that it would all be over soon.
And it's true. She is, and it will.

But as I walked out of the deli, I remembered that it won't stop. High school will end, sure. She'll do fine, graduate and all that. I'm not worried about her. But the stress and exhaustion and dissatisfaction that she is experiencing, those aren't over when high school ends - at least they weren't for me. College will begin for her, with all it's responsibility and the challenge of that independence. Fun, liberating, exciting, stressful, exhausting, the whole gamut. Graduation, job hunting, career finding and having, family...there's a progression to move through if she wants to, that will just keep going and providing her with opportunities to feel stressed and exhausted or not.

I want her to enjoy this part too. To enjoy the intensity and excitement and grief and fear and know that she's exhausted because life is so grand, so big and full. Or if it's too painful to enjoy, at least to feel it fully and know that she is really living. That her life is already what she makes it, and that she shouldn't wait 7 weeks until graduation to live it fully, or even 5 days.

Story 2: This morning I forgot to check our student teacher GL's worksheets for mathematical hotspots. We talked a ton about instructions and formatting but I failed to see that her quadratics had integer vertices, and they didn't, and so the kids didn't know how to do it. It was fine, I told them it was my fault and we'll fix it for tomorrow. Then I taught GL how to come up with nice systems of quadratics and I'm sure we'll do fine tomorrow.

But GL didn't look so pleased. She's really hard on herself and felt unsuccessful because of that mistake. No big deal, I know that feeling. But watching her do that, watching her align her success as a teacher with each individual success in the classroom...it was so obviously not true, beside the point, and most importantly, a total distraction from the actual work of improving our teaching. There is a reflectiveness that's necessary to grow and learn, but the self-judgment that comes afterwards...that's unnecessary.

Conclusion: Those four words from the top come in: resilience, process, curiosity, relationships. I want to speak about these words with both of these young women. To plunge into their deep resilience, not just surviving while they anticipate some future that will be easier but deeply resting in knowledge of their own self-worth, connected to this day, this difficulty as a part of their process in growing and learning, curiosity about how that will unfold and fierce optimism about how awesome it will be. Because they deserve it, and it will be...may my relationships with both of them support their ever more and more loving relationships to themselves. May they find ease and enthusiasm in their work and lives now and always, pulsing in and out of their seeming successes and failures with curiosity and acceptance.

A few things...

2011-05-02T21:07:00.000-04:00

By the chance that you haven't already read this, it made my whole day...
http://researchinpractice.wordpress.com/2011/05/02/why-we-need-a-better-word-than-accountability/#comment-549

And also I'm super interested and excited by open space technology, which I just learned about this evening. Check it out!
http://www.openspaceworld.org/cgi/wiki.cgi?AboutOpenSpace

animated phi = awesomeness

2011-04-26T20:49:00.002-04:00

Nature by Numbers from Cristóbal Vila on Vimeo.

Vi Hart

2011-03-31T09:05:00.003-04:00

Just a cursory glance at this, but it looks awesome! Go take a peek.

Jesse, can we meditate?

2011-03-29T21:31:00.000-04:00

It's afternoon, last block. We have an hour left. The kids are rowdy, rambunctious. The weather is SO NICE and we all would rather be outside. My co-teacher is away at a conference in California and everyone is jealous, even though the weather is nice here too. We imagine him beaching and sunning and relaxing and doing nothing, even though of course he is working hard and northern California is probably colder than New York.

And I'm thinking to myself, this is the kind of class that could feel bad. But I don't want to feel bad, and I've been practicing letting other people off the hook for making me happy. I've been practicing letting my students off the hook for making me happy. That's a revelation. I'm thinking, whatever they do today, I am going to be happy.

So it's from that place that I ask them, "What can I do? Here we are in math class, with all these understandable but non-mathematical circumstances distracting us. What can we do today?"
I'm sincere, not frustrated, and that feels good.

So TD says, can we meditate? He sounds so eager, frustrated almost, like "alright already, can we just do something useful?" We've been meditating and eating cookies every test day all year. But this is just regular math class. WHAT?! This is the best day yet. Yes, we can meditate. I turn out the lights. The kids put their heads down or close their eyes. I tell them it's just fine if they fall asleep when we're meditating. We'll wake them up gently later. They don't have to do anything right now, they don't even have to listen to me. They just have to close their eyes and relax.

I lead them in, slowly, lots of space between sentences. Like I'm just telling them a bedtime story...

"Relax your bodies, just feel what it's like to be in your own skin right now, antsy or hungry or tired. However you are, just notice that, really feel it.
How happy are you?"
(someone calls out something that sounds like 65%)
"OK, so if you're 65% happy, let's say. Is it possible for you to increase your happiness to 70%? Just by focusing on it, just by choosing to be a little bit happier? Just see. Is happiness something that you can decide to change?
Use your imagination to envision something that you want, something that would make you happy. It could be small or big, it doesn't matter. See yourself having this thing, let the image be really vivid, and feel what it's like to have this thing, feel the joy of having it, feel it in your fingers and feet.
What does it do to your happiness to just imagine having this thing that you want?
If you can, I'm here to tell you that this is the single most important thing: to be as happy as you can be. You all deserve to be so happy.
Now feel how even if the room were totally silent, and I stopped speaking, that even with your eyes closed you would know that you aren't alone, you can feel that there are people around you. That's a mystery, that connection. Feel it, even imagine that there are lines connecting you to everyone else in the room.
Send happiness down those lines. Make a wish that the people around you be happy. Make a wish that the people around you receive whatever would make them happy.
Then feel the people around you wishing you happiness.
What does that feel like? Let yourself just take that all in. You deserve all those good wishes and more. Just soak it up.
And when you're ready you can pick your heads up."

Then I rang the singing bowl and just let it sing. Usually I gong it at the end. They were all waiting for that. I didn't. When it stopped resonating I told them that the sound of that simple resonance is the reason for the bowl: it's used to begin and end meditations, and I use it to get their attention because I wish that joy for them always.

Then, folks, seriously...then they all did the most beautiful work I've ever seen. Maybe it was just chance, but I walked around the room and everywhere all I saw was perfect, diligent work. The most beautiful graphs I've ever seen these kids do.

Nice day.
May you all be as happy as possible all the time.

June 28: Tau Day

2011-03-18T00:24:00.000-04:00

Watch this and enjoy.

NYTimes: Let Kids Rule the School

2011-03-16T19:33:00.000-04:00

Read this and enjoy.

Two Student Stories

2011-02-22T13:28:00.000-05:00

I remembered these today and laughed out loud.

-----
"Jesse, do you know have one of those pornographic cameras?"
"Do you mean panoramic?"
"No."
"Photographic?"
"No." But he looks a little confused.
"You do know that you're asking me if I have a camera that videotapes people having sex."
"Yeah, I mean it's so awesome that it's pornographic!"

I think it's worth noting that he was describing one of the first 24p cameras, awesome about 8 years ago but truly old school now. Huge and not even HD.

-----
"Jesse, can I go outside to fart?"
"No." I continue with the lesson. In front of the class.
"OK, well, I'm gonna fart."
I ignore him and move on.
"I just farted."
I crack up. I can't help it. Everyone thinks that I'm laughing because it smells but really I just can't keep up my adult front anymore. I am a teenager at heart.

----

Letter of recommendation

2011-02-18T18:30:00.000-05:00

I wrote this yesterday afternoon for a student that received a safety transfer from our school today. The last time I interacted with this kid, he looked me in the face and said he would rather go home than stay in school, even though he knew that I was right to encourage him to stay. It was an incredibly disappointing interaction. I felt the loss of all our good work together, like I was watching him go to the dark side and everything I was doing was just doggy paddling in zero gravity: futile exhaustion.

I thought I would have some negative things to incorporate somehow in there, but when I started writing, all that came was this, and it couldn't be truer.

I wish he could read it. In my desire to open a window on my classroom to the world, I wanted to share this here.

-----

To whom it may concern,

This is a general letter of recommendation on behalf of one of my all time favorite students, C.A. I hope this letter will loudly advocate for him in whatever his future calls for and whatever he pursues.

I met C.A. during the middle of 9^th grade last year. Our 9^th graders travel as cohorts throughout the year, and C.A. was one of a few students moved mid-year to help the cohorts thrive. I don’t know why he was moved into my class, but it was one of the most positive changes I’ve ever experienced in a classroom. Always, changes in classroom groups are difficult transitions, leaving the group open to redefine themselves, either for the better or the worse. Often the change is difficult enough that it’s for the worse.

This was not the case for C.A. He brought enthusiasm, intellectual vigor and a candid honesty that inspired me and our class to be more courageous in our class conversations, more demanding in our expectations of each other and more diligent in our work. C.A. was a constant contributor to our conversations and also reminded people what they were capable of with humor and generosity. He was resilient, had a positive attitude (as frequently as one can expect from any adolescent) and worked hard.

This year C.A. has had another teacher for math, and so I have only seen him in the hallways in passing. We always speak to one another with great affection and enthusiasm. We continue to have an open and honest relationship and he has told me both about his frustrations with his teachers and peers as well as his own his failings.

Though I will miss C.A. a great deal, I hope he will find in his new school a place that challenges him and recognizes his great value. He is ready for leadership and intellectual companionship that exceeds what most inner city high school classrooms have to offer. C.A. is a powerful and incredible young man and deserves the best of chances in whatever school he is in.

------

May it be so. May the light call him forth even in the darkness. May he know his own worth and advocate for himself even when others do not. I am so grateful that he was in my life, in that classroom.

Doodling in Math Class

2010-12-16T20:34:00.000-05:00

Enjoy this. I have so much to tell and so little time, but this is great and makes me want to doodle and study math all the time.

Bieber, yeah

2010-11-18T00:32:00.000-05:00

Lately I've been oh so bummed out about teaching. Why is it still so hard after all these years? I'm so ready to enjoy it so much more, to have it be so much easier, whatever.

My sister is in town, and this afternoon she stopped by office hours and met three of my students, all studying for a quiz tomorrow, even though it's the first week of 2nd quarter and they've never come to office hours before, and it's after 4pm, etc. etc.

I have been forgetting that 75% of the job is awesome. Exactly what I want, exactly what I am good at, exactly what will help me grow into being more of the person I want to be, exactly that thing (one thing, anyway) that will serve the world.

This morning in my most adorable class, one of my most adorable students asked me if I had dressed up as Justin Bieber for Halloween. I was like, "Yo, I'm dressed up like Justin Bieber every day" and started singing Baby, Baby, Baby. He asked me to do a head toss like Justin. I did it. It was hilarious and adorable. Have I mentioned how cute my students are?

Remember the good stuff, people. Especially now as it gets dark and Thanksgiving blues attack our sweet ones. Take time to connect with the kids you love, the kids who love you. You're doing great. You're doing great. You're doing great.

Voting Math

2010-11-01T22:11:00.000-04:00

If you're looking for an election day lesson, check this out. The brilliant Patrick Honner and the NY Times Learning Network pair up to bring you the best of mathematical curriculum writing.

Just a little patience

2010-10-06T07:38:00.001-04:00

Last night at meditation class I taught about patience. I had been thinking about patience as a virtue that had to with time, but when I started contemplating it last night, it took on a new meaning that has only to do with any given present moment. When I am impatient, I am resisting whatever it is that's happening right now. When I practice patience, I am looking directly at what's in front of me and making my decisions from there.

All the waiting, all the holding back, that is impatience too. Patience is receiving, compassion, acceptance, welcome. Patience is loving, paying attention.

As a meditation teacher, talking about patience is really important because people need to be patient with themselves in order to practice. If all they time they are aspiring to be in some kind of mental silence, and impatient with themselves when they're not, it's going to be a rough ride. In my classroom, it's the same way.

Yesterday I got so impatient with my students for not being as fully prepared as I thought they should be. Maybe they should have been, maybe not. My impatience just made me unhappy and ineffective. I just refused to accept that they hadn't done what I'd asked them to do. Amazing.

Today whatever happens, I'm hoping I can remember this universal patience that receives what there is in the present moment and responds accordingly. I hope to be patient with my students and myself, and to experience and extend a kind of grand support from that. We'll see how it goes.

The 5th year

2010-10-02T12:02:00.000-04:00

I did it: yesterday, I gave quizzes in all my classes, and we meditated beforehand and while they worked I walked around with cookies. They were polite and if it wasn't magic it was fun and it felt good to me. I felt like I was paying tribute to my dear old teacher VanA, and that I was honoring the youth and hearts of my students.

I've been wondering if our most difficult students are easier than in previous years. That's what it feels like to me and I'm curious if it's me, if it's them, if it's some combination.

Four years ago I had these two boys, GN and GM in one of my 9th grade classes. I tried everything I could think of, but if those two were there, we couldn't accomplish anything. That was my 2nd year teaching. These were the two that came to mind when I was remembering the terrors of past years. Behavioral nightmares, disrespectful and unwilling to do any work, distractions to the whole class, reminders of my despair*.

When I remembered them deliberately and imagined them coming into my room this year, I realized that I have just changed beyond recognition. This job has given me patience beyond what I ever thought necessary. It has clarified my vision, allowing me to see the tenderness and lovability of adolescents. I have learned to observe, I have practiced not taking things personally, investing myself in the lengthy creative process of a year or four rather than the daily proof of my failure or success.

My new 9th graders are testing this version of Jesse the teacher. That's hard, but I think it's a good sign. I can pass this test.

And people, I'm thinking all the time about the mathematics. I've got all sorts of questions about it and ideas and inspiration about the wonders of this textbook and the mystery of good pedagogy. But at the moment, if the relationships I am participating in and facilitating with and among my students feel safe and positive then I can work out the rest. The relief is palpable.

Have a great weekend. Enjoy fall.

*Just for the record, I ended up having great relationships with both these kids once I didn't teach them. They matured and GM in particular impressed me with his biting insights and sincere intelligence.

It's Wednesday

2010-09-29T16:55:00.000-04:00

The Introduction of a Textbook
Have I mentioned that I'm using a textbook for the first time ever? I work at a school that values teacher creativity and student personalization, and though the school is full of textbooks, I have never used one for planning, and when I've tried to adapt something from a textbook for teaching, it's been hit and miss. The math department bought it's last text specifically because it provided lots of practice problems, so we could focus on pedagogy and not coming up with equations that have nice integer solutions, for example. It's good at that, but that's not what I've been focused on and so I haven't used it. I was prepared to never use a textbook.

Until now.

I'm using the CME Project's Algebra 1 text, and I'm excited and inspired and full of faith and hope and also utterly perplexed by how to integrate it into my classroom, not because it's limited but because I just don't know where textbooks fit in. Up until now I've been creating my own worksheets and activities, specially designed most every day for the class I'm about to see and set up to make on task work visible.

I have so many questions it's crazy. The main thing I want to share today is that, despite the fact that I'm a little disorganized as I try to figure out how to integrate this book, and my students may be doing less work than they are capable of, I am so satisfied by what my kids are doing when they are working. This book is just all about conceptual understanding, every single question provides multiple access points and demands real thinking. There is something interesting to talk about every day. That blows my mind.
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Teaching Meditation
I teach meditation twice a week at yoga studios in Brooklyn. These classes are awesome and it's an amazing thing to be able to do and share with people. Last night, two of my students showed up, one by surprise. I felt a little nervous at first, uncertain about how to be myself the meditation teacher without somehow breaching some unspoken code as their math teacher. They were just undeniably amazing. They had amazing experiences, they inspired the people in the class, they inspired me. They have passion, enthusiasm and wisdom beyond their years and I just felt honored to the max to be able to share such tender truth with them.

When I was in 7th grade, I had an amazing math teacher named John VanAlstyne who we called VanA. He taught us algebra and baked us cookies to eat whenever we took math tests. This year, I've decided to experiment with following a version of his example. This year on test days, I'm going to lead my kids in meditation for two minutes, give them cookies and then pass out the test. I feel generous and abundant just thinking about it. The days that I actually bake will be the special ones. May there be many of them.

Lady Gaga & Exponents

2010-09-26T22:41:00.000-04:00

Ahh, it's been an interesting start of the year, eh? We started back in the beginning of September, but last week was the first week I met my two 9th grade classes and started teaching from my new textbook (more on this later). My student teacher turned out to be mis-assigned and left, I'm responsible for sorting out all the 9th graders who need to be in classes beyond or below our standard curriculum, I've been co-planning a professional development video club "pod" for some of our staff and co-creating a structure for our new school-wide office hours (which are not quite mandatory and not quite optional for our students). It's been flipping crazy pants.

I've been so excited to get back to writing and reading the blogs. I've been waiting for time, waiting for something smart or insightful or inspiring. In fact I've been thinking of quitting, remembering that 9th graders are 9th graders and enjoying the weekend pretending I am not consumed by this job.

Since that's where I'm coming from, I thought it might be nice to share something I saved from the summer, for when I teach exponents and distribution. This came from Bowen Kerins and Darryl Yong, of PCMI fame, although I did edit it.

If by some miracle you haven't watched this most watched video, go here. This song changed my life this summer.

I was inspired by this to write my own for what I thought was an ancient and outdated tune but which has been co-opted and recycled and might actually be familiar to your kids now.

If you're looking for some awesome math problems to do, go here. The problem sets that inspired this post are up there. Start with Day 1.

You all rock. For doing what you do, for taking any time to reflect and read and write. I hope this finds you all happy and rested and inspired. May this Autumn bring you joy and surprise in the mystery of it all.

Negative Space in the Classroom

2010-09-14T23:07:00.000-04:00

I thought this post by a new blogger was really inspiring...

Respect, Compassion, Courage

2010-09-11T09:22:00.000-04:00

"Respect leads to caring - a quality of impeccability in what we do...As we foster the quality of respect in our lives, we can also begin to see the world in a different light. The tone of caring that arises from giving respect can transform how we interact with society. We begin to explore the possibilities of service, of taking an active role in seeing what needs doing and lending our energy to those endeavors. Compassion motivates us to act and wisdom ensures the means are effective."
- Joseph Goldstein

"Compassion has nothing to do with achievement at all. It is spacious and very generous. When a person develops real compassion, he is uncertain whether he is being generous to others or to himself because compassion is environmental generosity, without direction, without 'for me' and without 'for them.' It is filled with joy, spontaneously existing joy, constant joy in the sense of trust, in the sense that joy contains tremendous wealth and richness.
We could say that compassion is the ultimate attitude of wealth: an antipoverty attitude, a war on want. It contains all sorts of heroic, juicy, positive, visionary, expansive qualities. And it implies larger-scale thinking, a freer and more expansive way of relating to yourself and the world...It is the attitude that one has been born fundamentally rich rather than that one must become rich. Without this kind of confidence, meditation cannot be transferred into action at all.
Compassion automatically invites you to relate with people, because you no longer regard people as a drain on your energy. They recharge your energy, because in the process of relating with them you acknowledge your wealth, your richness. So, if you have difficult tasks to perform, such as dealing with people or life situations, you do not feel you are running out of resources. Each time you are faced with a difficult task it presents itself as a delightful opportunity to demonstrate your richness, your wealth."
-Chogyam Trungpa, Cutting Through Spiritual Materialism p.115

"We each need to make our lion's roar - to persevere with unshakable courage when faced with all manner of doubts and sorrows and fears - to declare our right to awaken."
- Jack Kornfield

Dynamic Paper

2010-09-01T01:06:00.001-04:00

Hey Math Teachers!
Happy end of summer. I'm not quite ready to think about it, but 5 minutes ago Phil Dituri showed me something and now I can't wait to spread the word.
Check out this resource. Take 5 minutes to try out something under each tab just so you know what it does. If you don't have 5 minutes take 60 seconds. Make this a priority. It'll be fun, I promise.
August loves you but September loves you more. Be sure of it.

Group Dynamics for Teachers

2010-07-15T17:59:00.001-04:00

I gave a short presentation this morning at PCMI (during which, if you can believe it, I cried) on what I've learned about Group Dynamics in the last year. A few of teachers at my school have been working with an amazing and generous group therapist and expert on group development. It's been the best PD I've gotten at school, and I'm really excited to share and keep this conversation thread going. Hopefully the prezi stands on it's own enough to at least get people thinking about it. (just click the right arrow to advance the slide)

Groups Dynamics for Teachers on Prezi

Blackboards

2010-07-13T14:12:00.000-04:00

We just spent a second day looking at Blackboard use after reading Using Lesson Study to Develop Effective Blackboard Practices, Ch 10 by Makoto Yoshida, and I'm including below my three favorite ideas, followed by all the relevant notes I took today.

My three favorite ideas:
1) It's useful to simply be thoughtful about how I use my boards. What is the board for? (up until now, mostly random recording. after now, progression of a class, communication re: classroom structure see #2 below, etc.) What do I want the board to look like at the end of class?
2) If the board is intentional space, then the board structure supports the classroom structure. I.e., A board full of clearly organized notes can support student note-taking and receiving teacher dissemination of information. A board with lots of open space can support a classroom conversation where students are invited to communicate and contribute their ideas. Perhaps you could even say that the amount of open space on the board directly represents the amount of student voice that should be present in this lesson.
3) "You should not erase what you write if you write on the blackbarods and you should not write on the board if you are going to erase it." p. 95 This is just fascinating to think about.

What's important about blackboard use (we ended up inadvertently recreating this table from from Makota p. 97, Table 10.1)
- keep a record of the lesson
- help students remember what they need to do and think about
- help students see the connection b/w diff parts of the lesson & progression of the lesson
- compare, contrast, discuss ideas students present
- help organize student thinking & discover new ideas
- foster organized student note-taking skills by modeling good organization

What teachers can do to improve blackboard use
- Lesson plan using board plan (sequence)
- Think about what you place on the board and all of the connections
- Anticipate student contributions & responses and plan for how teacher weaves in and out of that

Things to consider recording on blackboard
- Student questions, words, pictures & math
- question/task
- resources/prior knowledge we’re using
- graphic organizer...outline, web, etc.
- misunderstandings
- correct examples
- clear teaching point, so when we get there we all know it. The punchline.
- vocab

Questions to ask when looking at sample blackboards (yours and others)
- What’s already made visible?
- How would you use the visual prompt/pieces for discourse?
- What are the consequences for these choices?
- What might help?
- What might hinder?

Other Ideas
- Transparent chronology allows ability to track back and forth through progression of class, idea, etc.
- Chart paper board work demands pre-planning but allows mobility, opportunity for reorganization
- Stickies, colors, underlining, boxing on board help code, organize to direct thinking
- Clear objectives, titles, headings seem useful even on the open ended boards.
- Dates?
- Photographing my board at the end of every class coud be really interesting!

Does it help/hinder to have the writing written in the moment or prepared ahead of time?
- Whatever the lesson demands…this choice communicates the values of that class.
- Can use live writing to give appropriate time for student notetaking (but could also just watch)
- Classroom efficiency

Jim Hiebert in person is awesome!!!

2010-07-13T01:51:00.002-04:00

PCMI was graced with the amazing Jim Hiebert, Jere Confrey and Denise Mewborn for a Q&A about pretty much whatever we could think of. These amazing researchers (who arrived to answer my questions from last week as if by my own personal request) couldn't actually answer most of our questions with any concrete certainty. Why is there no video documented research in high school math classrooms? What resources are there to teach for conceptual understanding? What's up with standards based grading, and what is good teaching anyway?

To my surprise, it reassured me: I don't know the answers, but it turns out the experts in the field don't either. Not because they haven't tried, but because it's that complicated and messy. I feel renewed in my enthusiasm for doing this job knowing that when I feel like there aren't clean-cut answers, it's because there really aren't any. Now I feel free to simply enjoy asking the questions and trying to find answers, without feeling like there's something wrong because things aren't already nice and tied up. When I feel like the job is too hard, it's because it really is. When I get confused about a problem I face in my curriculum development or my pedagogy, I can just sink my teeth into the discovery filled process of searching out a solution. Like doing mathematics. I totally didn't get this before.

Not to mention something else that I've been basking in since I got here: all my inspiration, intelligence, effort and creativity are shared; all my revolutionary tactics, all my "original" ideas. In the best possible way, there is no great idea I have had, no depth of loneliness or despair that I've wallowed in, and no question I have asked that others haven't pondered right along with me, maybe even before I was born. I can relax a little, knowing that there is a whole teeming world full of people who want to make things better, who are passionately and beautifully bringing their highest intelligence to bear on the most difficult problems. There is nothing I have noticed that has gone unnoticed, no problem I have had that other people haven't recognized and worked on too. Word.

I'm seriously going to bed right now, after having stayed up 2 extra hours talking to my roommate (aka Awesome) about how inspired she is about what she's learned here about pedagogy and discourse over the last two weeks. She told me that she hadn't known that she could be so much better in her teaching, and now that she does she's so excited to teach! My life is so cool.

Unconditional Enlightenment

2010-07-08T00:51:00.006-04:00

This is recently released video of my dear friend and meditation teacher Harshada Wagner. As a background note: through my meditation practice and my studies with Harshada, my enjoyment of life has intensified and the generosity and sincerity I bring to my teaching has deepened. I am more alive, more present, more joyful in all my activities, relationships and endeavors. I invite you to get a taste of Harshada's wisdom by watching this video. Enjoy!

And if that doesn't tie up all your loose ends, check this out.

Yes, more crying. The K.D. Lang but joyful kind.

2010-07-06T22:29:00.002-04:00

As you read, I invite you enjoy this blogpost multimedia style while you listen to K.D. Lang.

Walking home from dinner tonight, I was teased mercilessly by my comrades for crying and blogging about crying. While I am not the caricature they perform, I will admit that I have been crying with a lot more frequency of late. Back in May, of course, the crying wasn't so fun. But here at PCMI, it's been joyful bursts of awe and deep heart opening, mostly to do with mathematics. Not even talking about teaching, just straight up math. This has never happened to me before, and it's cool, even if people do make fun of me for it. I like it. I like being surprised in my own skin.

So I will tell you about two more recent tales of my mathematical emoting here at PCMI, where kids & families are welcome and all levels of mathematical experience will thrive and blossom. You yourselves are not guaranteed to cry, whether you want to or not: crying seems to be the way that this experience is manifesting my new levels of engagement and joy in my mathematical practice, but it would manifest differently for others, I'm sure.

In the afternoons, we all work in smaller groups to do some math and create a product that could be used by other teachers. I'm in the Discrete working group, and we've been looking at these jug problems, which are apparently iconically represented in mathematics curricula by hooking kids with the Die Hard with a vengeance clip. The basic problem: you've got a fountain, a 3 gallon jug, and a 5 gallon jug. How do you get 4 gallons? Last week, we worked, solved and extended this and other related problems, and enjoyed employing M.C.K Tweedie's graphical solution method on a triangular grid. I had been frustrated if excited by this method, because our leader just sort of showed it to us, and I couldn't figure out how on earth anyone would have just come up with it. But yesterday, our fearless leader gracefully and patiently talked us through how you can think of the possible states of the jug's as coordinates in three space, and when you do that, all those states lie in the same plane, which, if you connect the coordinates, makes precisely the triangular grid we had been working with. Let me tell you, I was the most surprised person there, but as soon as I saw it, I had to take my glasses off and wipe my eyes as I CRIED. Ridiculous, amazing, laughable, tender, wow. That's all I know to say.

Also, this morning the group had a nice conversation in our Reflection on Teaching Practice session about how to watch teacher videos, and I got to process a lot more about what Rob and Ben (in his comments here) said.

Here are the things I've been integrating from all this:
1- When we hang out with kids who haven't yet learned their times tables, do we ridicule and points fingers? Do we politely snub and dismiss them? Do we secretly feel superior because we have already mastered this amazing skill and they haven't? Mostly, I'm thinking the answer is no. When someone next to me is working faster than me, or straight up knows more math than I do, it is (mostly, at least) because they have spent more time doing it, they've seen it before, and/or they have already learned it. Ben Blum-Smith taught me this idea.
2- Learning to teach is like learning to do math. In fact (hats off to Ben here as well) you could say that the process is entirely parallel, within our own lives and between us and our students. In both, we need to be generous and kind to ourselves and our peers as we reflect and learn how to teach (do math) better, more fluently, more efficiently, more creatively. Just like we wouldn't ridicule the kid who hasn't learned something yet, we don't need to batter a teacher who hasn't learned to do a particular teacher move yet. It doesn't mean they are a bad teacher, or even that they are doing "bad teaching" necessarily. I won't even venture to say what it means, only that it seems worthwhile to hold back our judgment instincts and just practice noticing. Ben's comments address this specifically and beautifully. I'm linking to them again and again because they're so good.
3- In A New Earth, Eckhart Tolle writes, "In essence, you are neither inferior nor superior to anyone. True self-esteem and true humility arise out of that realization. In the eyes of the ego, self-esteem and humility are contradictory. In truth, they are one and the same." We can enter into viewing other teachers with the humility that we all have had many minutes in our teaching (maybe most of them) when we would, upon reflection, given more time or more experience, have made other choices. We can enter into viewing other teachers with the confidence that we are smart, capable, generous and qualified to be doing this job, which is one of constant learning and growth. A process.

I'm off to the ice cream social. I've been doing math what feels like 24-7, writing about it with urgency when I'm not doing it, learning it, eating it, sleeping it, teaching it, walking it, loving it. Mathematics has become my spiritual practice. Thanks for everyone who is holding space and supporting me through it. I am changing on the insides. May your nights be bountiful and delicious, whatever the weather.

Follow up: Distinguishing Teaching from Teachers

2010-07-06T09:13:00.000-04:00

Thanks to those who offered resources about this. I also got a riveting and satisfying response via email from my dear friend and mentor Rob Weiman about this and wanted to share it with you. Rob used to be my math coach, and is now getting his PhD at University of Delaware. Eat it up, people, this guys is amazing. Our facilitators here have done a great job with 2 and 3, but I'd love to see us do a bit more of 1 and 4.

"
I think that when you are looking at cases of teaching in a group, video or otherwise, one thing to do is to set up very norms ahead of time about how we talk about the teaching. In situations where I have been with groups looking at video, the facilitators took great pains before showing the video to spell out very explicitly a few basic ideas:

1. These teachers have given us a great gift to learn from their practice. It is a privilege that we have through their generosity, we need to be thankful and respectful to them, and appreciate their making their practice public so that we all can learn, them included.

2. In general, when looking at video, the facilitators have not asked for general critiques, or evaluations of the teaching, but have asked specific questions about teaching moves. For instance, they would ask, what moves did the teacher make that pushed for justification? Or how did the problem advance, or inhibit sense-making for this group of students, or What moves did the teacher make that effected the ways students talked about the math?

3. Whenever people made comments in answer to these specific questions, they had to provide evidence to back up their claims, So if they would say something like, "when they said "good job", to Johnny, that really shut down the conversation." The facilitator would respond with, "Where is your evidence? How do you know that that shut down the conversation" The facilitator would also ask for the specific transcript line (if there was a transcript, and generally there was) so that everyone was working from he same instance of evidence.

4. Alternatives were presented as wonderments, not as "better" methods. If they were not, the facilitator would rephrase them. So if someone said, "If kids were in groups, this would have gone a lot better", the facilitator, might respond, we don't know how it would have gone. We can ask ourselves what might have happened if this had been a group activity, rather than an individual activity, but we have no evidence to support the claim that it would have gone better. One thing we could do, in this situation is to try it out and see how it goes.

In general, the ravaging of teachers comes from an ideological standpoint. i.e. I know what good teaching is, and this is not it. What we really want to encourage is an evidence-based orientation. We simply cannot say what is good or bad, we can say that this move at this particular point seems to have had this effect based on this evidence. And we can wonder what a different move may have produced. However, we simply cannot know what a different move may have produced because that different move did not happen. Our job is not to critique other teachers, or champion one particular mode of teaching, but to learn about teaching based on this example of practice, an example that we are privileged to witness through the generosity of this particular teacher.

If this is all happening in a class, the facilitator, or some other person in the class can make a big difference by saying these things over and over. At the very least, we can all imagine some instances of our teaching that could be ravaged, and the scariness of making our practice public, so a little empathy can really go a long way to changing the discussion.

As for teaching versus teacher distinction. Attached is a large review of the research literature about the effect of teaching on student learning. One of the authors is my advisor, who is pretty big on this distinction between teachers and teaching. Pages 377-378 of this review (it is part of a much larger book, it is not a 400 page review!) address this distinction specifically, and give evidence of its prevalence. Of larger interest, perhaps is the conclusion of the review, that we know remarkably little about how teaching effects learning. This review addresses some of the reasons why people who try to use evidence to support their claims find it so difficult to claim that specific teaching techniques are effective, and does attempt to say that despite the difficulties, there appear to be two big ideas, that if procedural fluency is the goal, then clear modeling, immediate practice with immediate feedback is effective, and if conceptual understanding is the goal, then struggle with meaningful mathematics is effective. This struggle also seems to help with procedural fluency.

The point here is that folks who are so ready to critique their fellow teachers should know that the best researchers in the business have had trouble making the claims they are so ready to use as blunt instruments to level the well-intentioned efforts of their colleagues, but that is just my take.

Other sources for this idea:
I. James Hiebert
- Hiebert's introduction to Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development (fondly known as "The Purple Book") by Mary K. Stein, Margaret Schwan Smith, Marjorie Henningsen and Edward A. Silver, is nice short (3 pages) description of this idea as well.
- Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world's teachers for improving education in the classroom. New York: Free Press.
This is a bit of a classic, it's main thesis is that the difference between the US and other countries is the teaching practices that are culturally embedded in the US mathematics classroom, and the lack of any kind of institutionalized structures to improve this instruction. Thus there is a ever-widening gap between the US and some other countries that have practices that are not only more effective, but also evolved and improve, sometimes by specific design.

II. Deborah Ball
Deborah Ball writes about teaching being an unnatural activity. Although this is not really about separating teachers from teaching, it does separate a teachers personality and skills in the adult world from the kinds of personal skills teachers must learn and cultivate as teachers. These personal skills may be seen as "teaching skills" rather than "teacher attributes" since these attributes simply are not the kind of attributes that people have naturally in the real adult world. (One example she gives is that cultural survival depends on people assuming shared meaning in most of our discourse, but teachers need to often drop that assumption. So, for instance, in math class, it is a very good move to ask a student what they mean by bigger, but that a guy in a bar talking about sports would soon find himself drinking alone if he asked what somebody meant by bigger, every time it came up in a discussion of the Jets and Giants offensive linemen.)
Another thing Ball writes about extensively is that we measure teacher knowledge through all these "proxies" rather than the knowledge that they need as teachers. So for instance, we look at how many college courses they took in math, or what their SAT score was, or whether they measured in math, rather than actually identifying and testing the specific knowledge they would need for teaching mathematics. Indeed, her whole research program, for which she has received huge recognition, is directed toward trying to identify and develop measures for the kind of mathematical knowledge specific to teaching mathematics. She calls this mathematical knowledge for teaching (MKT)
- Hill, H. C., Sleep, L., Lewis, J. M., & Ball, D. L. (2007). Assessing teachers' mathematical knowledge: What knowledge matters and what evidence counts? In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111-156). Charlotte, NC: Information Age Publishing.
This is another huge review, somewhat dry. However, there is a smal section at the beginning where she talks about how historically we have used proxy measures to determine how knowledgeable teachers are.
- Ball, D. & Forzani, F. (2009) The work of teaching and the challenge for teacher education, Journal of Teacher Education, 60(5), 497-511.
Her point about teaching as an unnatural act is on page two of this article. You might also want to check out her website, she has lots of stuff to read there. Not connected to this topic, but a nice read and the thing that launched her is:
- Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373-397.
"

If you guys have difficulty finding any of these articles, let me know. I've digital files for most of them.

Crying in May

2010-07-05T22:44:00.000-04:00

On May 4, 2010, I wrote the following:
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Today I cried again at school. I used to do this every Wednesday after my lunch. I thought I would have stopped by now. It felt horrible and at the same time it was somehow a relief to let it all out. I was throwing a pity party; a very sad pity party. It is exhausting feeling like a failure: so much responsibility isn't good for people. I knew intellectually even at the time that it wasn't real. But lord.

This afternoon I found out that one of our highest flying math students, a kid who tutored for us last year, takes 12th grade math in the 10th grade, devours new ideas, loves a challenge and excels in math, is failing all his other classes. Failing. In danger of repeating the year. WTF!?

I've never actually taught this kid, but we say hello when we see each other. He's got that rare maturity that chicken or eggs the mathematical discipline and resilience. I just happened to see him in the office right after school and grabbed him.

"Do you mind if I get in your business?"
"No."

We talked for 30 minutes about what motivates him, why he's failing, what he wants. He wants to be interested, he wants competition among his peers, he wants people to care about him, he wants to please those people, he wants to go to college. Maybe he wants to transfer schools, maybe he wants to do his homework, maybe not. He said no one had talked to him like this before. He didn't realize anybody noticed or cared. Crap.

It's not important figuring out how to get what you want. It's important simply to figure out what you want, and to keep your eyes there. Then it's easy taking one step and then another.
---

I'm going through my blog, taking my free evening to sort and edit and finish. I liked finding this record of time. It reminds me how hard I have been on myself (omg, please let me be just a little bit more patient with myself next year; let my presence and joy sustain me) and how time passes and things change. I kept crying till school ended, that didn't get better really, but I ended up teaching meditation after school on Wednesdays after I wrote this post, and this same kid came every week. It was great. I learned a ton. We had a good time meditating at school. I got to offer something to a kid who accepted it without a fight. I am so grateful. I am so grateful. I am so grateful. That's what it's all about even in the toughest moments: gratitude for simply being able to serve.

Math Reflections: What I've learned so far...

2010-07-05T22:22:00.001-04:00

Last Friday, our incredible math teachers Darryl Yong and Bowen Kerins asked us to reflect about our first five days of problem solving. Let me begin by saying that this part of PCMI is the most mathematically satisfying thing in my memory, and follow that up with three sections:

I. Mathematical ideas that I was supported in discovering on my own:
Sequences can be added together piecemeal like linear combinations to create new sequences. I (think I) can think of sequences like elements of a group under addition or multiplication (this is an idea I haven't explored, but it seemed like one worth exploring...sequences as generators, etc.) What happens when you multiply the terms of a sequence by the terms of another sequence. A new way of finding two numbers given their sum and product, which deserves it's own blogpost. How to express a sequence recursively given it's closed rule (like the sum of two different bases both raised to the x power). What kinds of starting values will make a recursive function (like J(n) = 7J(n-1) - 10J(n-2)) exponential and why. How to express a sequence given recursively in closed form (over and over again, which was amazing every time.) How to use a TI-Nspire. The rational representation of .001001002003005008013... which is really neat. Finally, the closed form of the Fibonacci sequence. This literally made me cry. I did it. I wrote it down. I started crying. I stopped writing and looked around the room. Nothing had changed that I could see. But I felt like someone had just opened up a window to God. It's one of the most beautiful things I've ever seen. I have never cried about math before.

Other ideas:
Rich and rewarding mathematical environments arise in groups when problems have a low threshold and a high ceiling. Humor helps. I like to work near people but be able to move at different paces. When I feel satisfied in my own mathematical exploration, I don't care whether I'm ahead of or behind those around me. I don't want to be told the wow: I want to find it on my own.
My peers are superlatively hard on themselves when they feel behind or make mistakes or don't understand. It is of supreme value to be generous and loving with myself, (even if I didn't deserve it!) because then I am able to put my full attention on the mathematics, rather than the distraction of berating myself. I have had a supremely enjoyable time doing math here this week, and feel really blessed. It's really hard (so hard!) to continue doing math when you're feeling stupid and behind. My admiration soars for those people who are willing to be here and keep trying even when they feel stupid. My admiration soars for my students who are willing to keep trying even when they feel stupid. It's got to be one of the most painful day to day kinds of experience we humans can have.

My dance teacher Nancy Stark Smith says this wonderful thing about a way to approach improvisation, and of late it has become my life mantra in every context I'm in: "Replace ambition with curiosity." I have been practicing this for about a decade, and it's freaking amazing to see some concrete results. Life is so much more fun this way.

I wish for everyone on the planet to experience the freedom and relief of being let off every hook they hang themselves on. I want to tell them, "Enjoy it. Be as good as you can, but enjoy it. For goodness sakes. You are all so bright, so deserving. Be easy on yourselves."

My big question:
What are the brilliant Darryl Yong and Bowen Kerins really up to? What genius is required to make problem sets like this? What do they think about when they're planning? What will I have to keep in mind as I try to follow their example? Because that's why I'm here, and I intend to bring their model to my classroom or fail trying.

Distinguishing teaching from teachers

2010-07-05T14:21:00.000-04:00

I'm in this great reflection on practice class at PCMI every morning, and we've been looking at a bunch of videos of math teachers teaching. It's great: the videos are thought provoking and we are having all kinds of cool ideas inspired by the teaching (as well as what we want to do to avoid repeating that kind of teaching). People seem to be thinking about teaching in new ways, having a paradigm shift in understanding the value of being a 3 on the rubric for levels of discourse that I posted last week and getting excited as they identify how to get there. Their desire to be a 3 is palpable. (I haven't shared Ben's suggestions for adding 4s and 5s to the rubric yet!) It's exciting.
In the midst of this, I am noticing how quick we are to critique these virtual "peers": we don't know them personally of course, but they are our conceptual colleagues. We are so ready to dismiss what they're doing, and I'm not sure if we're saying that what they're doing isn't teaching, or if we're saying that they are not teachers. What's the difference?
I like our high standards and I wish us caution in judging the teachers we're watching, both because we're watching literally minutes of their careers, which must be limited in it's capacity to fully represent them as teachers, but also because I think even if these short videos of their teaching were representative, that there is some value in distinguishing the teaching from the teachers.

I am guilty of blurring the line between these two in my own career. It's the reason that I ever feel bad about myself when I reflect on my teaching. It's a new distinction in my life, and I'm really interested in how other people think about it and what they've read about it:
How do we distinguish between teaching and teachers? While teachers have the power and responsibility for teaching, what can we do to focus our critiques on the teaching rather than the teacher? My first math ed mentor, Rob Weiman, was the first to point out this possibility to me. I think it's worthwhile to be mindful of how hard we are on the people doing the teaching so that we can focus on the techniques themselves and the models they provide.

I'm curious if anybody has read anything about this? I'd love to read more.

An article, a blog, a performance, a realization...

2010-07-05T00:00:00.000-04:00

My friends and colleagues at PCMI are going beserk for blogs. The amazing Sam J Shah did a perfect presentation about blogging and tweeting with other math teachers and I think it must have indirectly sent a bunch of those amazing people here. Saturday morning I found a comment on a recent post from the talented, kind and newly blogging Clint Chan. He recommended some articles and I have been happily reading math ed articles in bed since.
The first I read was Lessons learned from detracked mathematics departments by I.S. Horn.

In the first paragraph of this article, I read the following:
"Mathematics is an academic domain often perceived as beyond the reach of educational reforms...This is due to the conventional wisdom that mathematics is unique among the disciplines in its lack of adaptability to more open-ended styles of teaching and learning. How can we teach mathematics for understanding, for example, if the subject is made up of discrete facts that need to be memorized?"

This first paragraph stopped me reading: this introduction is just not my perception, nor the framework that I have been trained within. Specifically, the idea that mathematics is a "subject made up of discrete facts that need to be memorized" felt *almost* unfamiliar. So while the question of how to teach mathematics for understanding is a huge question in my life, my paradigm is utterly tied to the assumption that mathematics makes sense and so largely doesn't need to be memorized. I need to express my gratitude for the lifetimes of amazing and daily work of many (100s? 1000s?) researchers and educators who have helped shift the conversation about mathematics education within my life and across the globe. I want to thank them right here and now - with sincerity, emotion and endless repetition - for shifting the conversation before I arrived and for the momentum they created that has kept me moving forward. Among the many questions I get bogged down by, the question of how to teach a subject that is defined by memorization is just not among them.

Today, I finally got back to the article and found it worthwhile and thought-provoking. Horn analyzes some of Jo Boaler's research, trying to figure out what makes de-tracking work. The two most compelling points to me were: 1) teach curricula based on big mathematical ideas, connections and meaning, rather than a sequential progression through procedural skills, and 2) distinguish between teaching kids how to do math and how to do school. According to the article, these are two key ingredients to higher student performance, deeper understanding, and more enjoyment of mathematics. Duh. But awesome.

Here are some of my favorite nuggets:
1) A characterization of "group-worthy problems"
(a) illustrate important mathematical concepts,
(b) include multiple tasks that draw effectively on the collective resources of a student group,
(c) allow for multiple representations,
(d) have several possible solution paths.

2) A new definition of math: "A tool for sensemaking: Students need opportunities to understand mathematics through activities that allow them to make sense of things in the world."

3) A useful distinction: "Teachers avoided commonly used terms like canceling out to describe the result of adding opposite integers such as –3 + 3. Instead, they preferred the phrase making zeroes, as it more accurately described the mathematics underlying the process."

4) A HW accountability structure:
"At the front of each classroom was a homework chart laid out much like a teacher’s roll book, with students’ names in a column along the side and the number of each homework assignment across the top. Although actual grades were not posted, completion of homework was represented by a dot."

5) A nice detail:
"All...math teachers had a large sign with the word YET placed prominently in their classrooms. In this way, when a student claimed to not know something, the teachers could quickly point to the giant YET to emphasize the proper way to complete such a statement."

6) An important acknowledgement:
"Figuring out how to operationalize slogans like teaching for understanding is a challenge when teachers have not had opportunities to develop understanding themselves; are pressed toward the competing goal of curriculum coverage; work in isolation from their colleagues; and work in systems that value summative over formative assessments."

7) A refined idea:
Noticing whether or not students respond "sensibly" - I haven't integrated this one totally but it feels juicy. Teach for sense-making and hopefully kids will respond with some sense?

8) A great suggestion:
Looking at "fast" student's weaknesses. Are they just doing what they have to do get through the work, or are they making connections, trying to understand the purpose of the activity.

---
Finally, I wanted to mention Blackboard Jungle, this movie I watched the last half hour of a few nights ago. Apparently the first movie which employed the delights of rock n' roll in it's soundtrack, this 1955 education flick tells the story of Mr. Dadier, an English teacher at a tough boys school. Mr. Dadier rails passionately against his colleagues' complacency and strives to get the young men in his class excited about stories. His kids harass him, harass his wife, even threaten him with physical violence. Of course even in the worst moments (as with so many ed movies) only one kid spoke at a time (so they can really deliver those lines, I know), but there was something more tender and honest about the lonely struggle of this teacher who was trying to shift the paradigm he'd entered into.

Imagine what it would have been like to start teaching in 1955 rather than 2005! Wow. Despite the 55 years difference, I recognized the frustration, the fear, the despair, the passion. I, like Mr. Dadier, do feel like I'm trying to change the conversation, still pushing against the grain, still trying to do what feels impossible, and sometimes I even feel very alone. In the hardest moments, I scold my colleagues, I get discouraged and I feel sorry for myself.

Reading math ed research, being at PCMI, being welcomed by math teaching tweeters I've never met, watching Dan Meyer's Ted Talk, writing this blog, all reminds me 1) that I'm not alone, 2) that I'm not the first, 3) that this is somehow how it's supposed to be (at least for the last 55 years) and 4) that the reason I get to struggle with this stuff is thanks to all those who came before and laid the groundwork: it's a privilege to be able to fight and think and despair about all the stuff I do. I've just got to remember to enjoy it.

May your Mondays be refreshing and delightful. Happy Independence Day!

Student Publishing

2010-07-01T17:18:00.000-04:00

Great session today with my fellow teachers at PCMI on student publishing. We talked about tools for publishing student work and the various pros and cons for those tools. Within this context, it seemed that there were three subcategories to consider: 1) the object itself 2) the technology used to collect the object 3) the activity/structure to present/share the object. By the end of the conversation, I had collected some really exciting questions and some really exciting ideas.

Questions:
What is publicizing work?
Who is the audience?
Who is the publisher?
(students publishing for themselves, vs. for me, vs. me publishing their work for them, etc.)
What is the work that gets published? Problem, solution, answers?
Is it artistic?
Is the work best work or just any work?
Is publicizing work always a visual thing?
What is the right amount of info?

Ideas:
- “fix the problem” aka “math hospital”
- student created problems become class activities
- student created instructional videos?
- audio "posters"
- notetaking/secretarial role in discussions…post those notes (Give them as notes for kid’s binders? Post on class blog/website?)
- groups record selected parts of their conversations for grade, either for class or for teacher.
- give kids a microphone. could be that they actually talk live to some classroom somewhere else, or could just be a structure for the conversation. we have one here, to communicate with our satellite in new mexico, and even though we aren't amplified we feel like we're performing when we sharing our ideas
- Voicethread.com
- Twitter
- "On the spot" (kids solve new problem live)

Did I mention that last night I had one of those terrible dreams about school starting and there not being any board space and I hadn't prepared anything and ugh. I'm so tired.

PCMI is amazing though. Ya'll should all come.

oh, and ps. I am now officially tweeting as a math teacher? It's really exciting.

Rock Paper Scissors Lizard Spock

2010-06-28T17:00:00.004-04:00

To improve your game, go here.

To buy the t-shirt, go here.

Productive Discussion in the Classroom

2010-06-28T14:01:00.004-04:00

Everyday at PCMI, we're doing an hour-ish of reflecting on practice. Today, we talked about productive discussion in the classroom and watched a video of Cathy Humphries teaching a 7th grade class about dividing fractions. We spent the whole time watching, re-watching, reading the transcript, discussing and even arguing (with evidence) how "good" the class conversation was. After all that, they showed us a rubric to help us discuss this in our future conversations, and I thought it was really awesome.

The rubric is from an article in the November 2007 issue of Mathematics Teacher called "Let's Talk: Promoting Mathematical Discourse in the Classroom." Enjoy! I'm blogging for speed rather than depth, but I think this alone could be food for months of study.

Graduation

2010-06-23T18:48:00.002-04:00

My first freshman graduated today. We grew up together, me a first year teacher when they were freshman, and now as they graduate I feel I have experienced some kind of rite of passage as well. I wrote little blurbs for a few for when they come up to the stage, after saying their name, but before they actually take their diploma in hand. Here is one I wrote for one of my dearests:

"WR. In the hallway, in the classroom and on the basketball court, your smile and intelligence have lit up our community. Your quiet and fierce determination to do things your own way has helped you achieve success today: may it continue to do so. May your life bring your joy and deep satisfaction. You deserve it. Congratulations, WR."

As a staff, we wished them success, joy, satisfaction. We admired them, acknowledged their accomplishments. Some of us sang a song to them, which got everybody clapping and energized. It was a simple, sweet graduation ceremony. I felt (I thought strangely) a little numb.

Then I went to the staff after party down the street, where we were going to be celebrating and relaxing together, and also toasting the staff that are leaving this year to have babies, pursue PhDs, move to Miami and get married, become lawyers. We spent over an hour singing the praises of the people who are leaving us. We sang, cried, told stories, expressed gratitude, cited evidence of the amazingness of each person and all the ways they would leave us with holes in their absence but also with the teachings and memories to inspire us and continue to help us grow.

After about the 5th person (we had 8) I realized that we could do this for every single one of us. There wasn't one person I could think of that we would have to fake it for. We genuinely love, admire, appreciate and learn from each other. It was at this point that I started crying, really feeling the grief and upsurge of energy that perhaps needs to come on graduation days, on days when we say goodbye.

I realized that we had done the same sort of thing (albeit with a little less specificity or at least at less length) for our graduates that morning. This specific expression of love and appreciation, admiration and encouragement, is amazing. It was amazing to hear, to say, to feel indirectly, knowing that I would get it directly if I decided to leave too.

I thought it was deeply nurturing for our community, even as we say goodbye to so many, to reinforce and make explicit our deep wells of gratitude. Statistics aside, I believe today that what we're doing matters, that we're being human together in amazing ways, and that we can do amazing things, that I can do amazing things as a part of our community.

May we all know our own worth, which is extraordinary, downright priceless. May we weep with abandon when we say goodbye and welcome the relief and hope that comes of endings and new beginnings.

kids thinking

2010-06-16T07:11:00.005-04:00

We had meetings all day yesterday. I got discouraged. Thinking about the past year has left me a bit deflated. I have grown more confident in my own thinking about teaching mathematics, but remain discouraged about my ability to teach it well.

In the math department meeting yesterday, I heard my colleagues describing how our students simply don't think when we give them mathematics to do. Even when they have all the necessary skills, they don't engage their minds. We were in agreement that this is not because they can't do it. We all believe they absolutely can. I believe that it's one of the things their brains are designed to do naturally. My new idea is that somehow they are not deeply thinking in math class because they haven't found it useful to do so.

I know that I have a tendency to go too hard on myself and my colleagues in moments like this. I asked myself, "Do I want a revolutionary miracle in every class?" If I do, I am probably setting myself up for failure. Is getting kids to use their naturally pattern-seeking powerful minds such a huge demand? What is the part that gets kids really deeply involved, not just taking a class for the sake of passing this adolescent rite of passage?

How do I get kids to value the power of their own thinking?

I remember this day a few weeks ago when I was at a loss of how to teach finding the slope of a line in a way that demanded this from them. Of course I could just show them, and they could do it, no problem. Maybe even some of them would think about what they were doing while they were doing it and notice some patterns and sense in the whole thing. Maybe even they would all be so successful that if I gave them a quiz they would all ace it and I could pat myself on the back feeling very successful because my kids were successful.

I talked with Ben about it and he reminded me: "But it wouldn't be math."

Right. So I'm trying to teach math.

This last year I feel like in many ways I've been doing math for the first time. Truly discovering, playing, exploring and sense-making about actual phenomena. Maybe I just need a little more practice to teach this way well.

I'm looking forward to next year. I will be better at classroom management, at cultivating positive open relationships with kids, at organizing and structuring transparent systems and routines in my classroom, and at unit planning. So I'm excited that having those ducks more in a row might mean that I have more time to think about inviting my kids to really think, figuring out what they are thinking and celebrating and honoring and valuing that so that they do it more.

Yeah.

Ben Peled Saves the Day

2010-05-20T06:27:00.002-04:00

My math teacher friend Ben Peled sent me this yesterday and it made me gape, smile and feel a bit better.

"Often at this time of year, surveying all the kids (may not) have learned, I feel overwhelmed by the gap between the teacher I am and the teacher I'd like to be. That's why it's always good to be reminded that, however often I screw up in the classroom, there's someone out there effing it up much, much worse."

For all of you who have been having a challenging week, doubting the value of your practice, feeling discouraged, or frankly just need a little eye-popping.

May you be reassured and rejuvenated for these final days.

Jascha on Sims

2010-05-16T15:29:00.002-04:00

My brilliant friend Jascha Hoffman sent me an awesome interview he recently did with John Sims, who creates and curates mathematical art, and who has a year long series of exhibits at the Bowery Poetry Club.

Sims is articulate about the intersection, even union, of math and art. At last. I'm interested to learn and see more!

2 Additions to my Conceptual Understanding

2010-05-15T14:47:00.005-04:00

I just graphed these inequalities for the first time. I was filling out the end of year survey for Math for America and the last section asked us to identify the common misunderstandings that might arise and how we would address these misunderstandings.
I've been thinking a lot this week about how to teach for conceptual understanding, how to get kids to use skills as their inherent pattern seeking mechanism activates. How to create curricula that gets them generalizing useful and true patterns (rather than, for example, that every function is linear), and how to offer accessible depth in mathematical thinking.
So I was excited to play with these inequalities, partly because they were new to me, partly because they demand conceptual understanding and resist procedural memory. Rather than go on, I leave you to play with them yourself if you're not already familiar. I'm going to use them with my 10th graders next week!

Then I discovered that jd2718 came up with this activity back in March, and there's a nice discussion of them in the comments on that post. Check it out! Thank you jd2718 for being so creative, brilliant, infectious!

Also, I did some catch up last night on Ben's blog and was really inspired by his discussion of revising how we teach and present negative numbers. He writes beautifully and at length about this, and I seriously encourage you to take 10 minutes and read his post. He's been reading some awesome primary texts and the history they tell has convinced him that we should teach negatives a bit differently. First, we can we ask the question (perhaps often) do negative numbers even exist? Then let's introduce negative numbers first as solutions to things, like 5-7. Let arithmetic necessitate these new creatures. Once we're comfy, we can start using those things as objects with which to do arithmetic, as solutions to equations, etc. Last we can use them as coefficients (and exponents?!). I love this!

Kuta Software

2010-05-07T09:42:00.003-04:00

Yeah, Kuta Software!
I realized this week that I hadn't mentioned the online worksheet resource I use most for arithmetic and algebra. My colleague and old co-teacher Cristina found it. The site has tons of free worksheets, great for practice and exercise, the kind of thing that I like to use so that I can put my creativity to activity planning rather than writing problems or formatting worksheets for simple practice.
They have also recently added a Geometry section, and if you have a PC, I bet it would be AWESOME to buy their program that helps you design and create your own worksheets.
Check them out!
Happy Friday.

The Learning Network, Teaching with the NYTimes

2010-05-05T20:34:00.002-04:00

I was honored to be a part of a meeting with the lovely and tremendous women (both former teachers) who are writing the The Learning Network, the education blog for the New York Times. They "curate" the Times for educational purposes, encouraging readership, literacy and global awareness for anyone who can go online.
We (I, 5 other math teachers, two awesome folks from MfA and the writers of the blog) were there to talk about how they can develop the math education portion of what they do. Apparently the most searched phrase on the entire blog is test prep, and so the remarkable Patrick Honner wrote some. Check out his "quiz" on financial literacy. Give comments if you have feedback.
I also recommend just generally checking out the website. They put up all kinds of cool stuff for teachers and kids, daily lesson plans, comments from kids, cool questions and interactive surveys and stuff. It's a great resource.
I'll keep you posted as we (hopefully) contribute some more mathy stuff too.

The Math Worksheet Site

2010-05-02T08:26:00.002-04:00

My coach showed me this website which has good basic resources. For free, you can get blank graphs, number lines, and worksheets on integer arithmetic or telling clock time. You can also subscribe and get access to a whole bunch more stuff that I can't tell you about.
The site is designed for early ed, but I use everything I listed above in my 9th and 10th grade classes as well when kids are struggling with their basics. You can choose (to some extent) what kinds of features you want included in your worksheet. So it's a little DYO but still very simple and straightforward.

By request, Superstar Pictures

2010-05-01T11:06:00.004-04:00

Unfortunately, I have found none with the three together in one shot (alas, sitting at different sides of a round table) but here's a little something.
Aren't they beautiful?!

Ben, Sam, Kate

2010-04-29T22:56:00.006-04:00

I had the tremendous pleasure on Wednesday night of doing math and eating dinner with some spectacular math educators and true superstars of the math education blogosphere. I feel tremendously lucky to know these people personally, to have done math with them and had their inspiring guidance both live and online.

Ben's most recent post has changed my whole thinking about math this week. Not to mention his amazing NYMC Dinner & Math evenings, which were super sweet and brought us all together Wednesday night.
Kate's presence in my classroom today felt like an honor beyond measure, a dream come true, actually. She's such a rockstar.
Doing math with Sam was like going back to the best parts of all my math classes ever. He is oh so good in person, people. I hope you know. Funny, humble, smart, and with the nicest handwriting.

Oh, and both he and Kate own t-shirts that say "I only twitter with math teachers"
Amazing.

I cavalierly told my mom on the phone this morning that mathematics was an activity, not a body of knowledge. Whether this is entirely true, it seems to me that doing math with other people, especially ones who are kind and generous and sincere in their curiosity, is just about the best thing ever. The play of it, the improvisation and interaction of ideas, is wonderful. Sharing a passion for high school education takes the whole thing off the charts. Thank you Ben, Sam, Kate. You keep my dream alive.

Writing Systems of Quadratics to Solve by Graphing

2010-04-29T07:07:00.005-04:00

This week I was making up systems of quadratic equations for my kids to solve by graphing. In my experience this is the sort of task that seems like it should be easy but which I have spent a number of years doing poorly, carelessly, and at length!
Here's my new trick:
1. Choose two binomial factors whose products will have integer roots, i.e. (x + 3)(2x - 4)
2. Multiply those factors.
3. Set the product equal to zero.
4. Use inverse operations, your own creativity and the principle of equality to move some part of each term to the other side.
5. Those are your equations, and the solutions will be {-3, 2}, the roots of your original quadratic.
Voila!

At this point you get to play with how nice the roots and vertices of your quadratics are, but it doesn't matter much to me for solving systems by graphing. Of course then you can plug your equations into Wolfram Alpha to check yourself.

Hope this helps!

New York Math Circle Summer Workshop 2010!

2010-04-28T07:19:00.003-04:00

If you look at my first posts on this blog, you'll see that many of them are my notes on the PD I received at the New York Math Circle Summer Workshop I went to last summer. This summer's workshop proposes to be even better, with a week of study of the Pythagorean Theorem on Bard College's beautiful campus.
The NYMC instructors and Bard Professors that are leading workshops are tremendous mathematicians and experienced teachers and I highly recommend attending.
They have recently extended registration to April 30, which reminded me to let you know about it.
Go here for more information and registration.
There's also an information sheet.

Go and enjoy!

Go to the Kaplans' Math Circle Institute!

2010-04-16T13:28:00.003-04:00

After reading Out of the Labyrinth and meeting Bob and Ellen Kaplan last year, I had the immeasurable pleasure of attending their week long intensive at Notre Dame last summer. It was the best professional development I had attended up till that point: I learned a ton of math, I got to teach a bunch of math circles, and I built professional relationships that continue to sustain my enthusiasm for teaching today. I am so inspired by the work that the Kaplans do, and highly encourage anyone interested in cultivating more enthusiasm and fascination in themselves and their students to read their books and go to their intensive this summer.

"Since 1994 The Math Circle at Harvard and Northeastern Universities has made math a source of intense delight and collegial enjoyment for students from 4 to 70. Our approach is to have the students do the discovering and proving for themselves, in friendly conversation.

Branches have now opened elsewhere in America, and abroad, thanks to the Summer Institute we hold for a week on the campus of Notre Dame. This summer it meets from July 5th to 11th. The cost is $800 for room, board and all expenses (except travel). If you are interested in running a Math Circle yourself or using its approach in your classroom, please contact Bob and Ellen Kaplan: kaplan@math.harvard.edu."

Math for America deadlines approaching...

2010-04-14T14:27:00.003-04:00

Having benefited immensely from Math for America's financial, educational and professional community support for 5 years, I am delighted to have the opportunity to share the info about two of their amazing fellowships for NYC public school teachers.

Spread the word and check out their website.

"Math for America has an exciting new Early Career Fellowship beginning the 2010-2011 school year for first, second and third year math teachers of secondary mathematics in New York City public schools. The Fellowship provides four years of professional support and growth opportunities for teachers early in their careers. The deadline is May 7, 2010. Benefits, eligibility requirements and application deadlines are available at Math for America.

Math for America's Master Teacher Fellowship rewards exceptional New York City public secondary school math teachers (with over four years of experience) with a four-year Fellowship. The deadline is May 21, 2010. Further details on the program, including stipends and professional development opportunities, can be found at Math for America."

Inappropriate Models

2010-04-12T18:28:00.004-04:00

We're trying to come up with the generalized process for solving equations. Now that we've got combining like terms and distribution and variables on both sides and sometimes no numbers, just a bunch of different letters and all the combinations of that stuff. What is it that we're really doing when we solve equations?

The supervising mentor that observes and works with our student teachers is a cool old guy, used to teach math, was a principle, still works in schools, and has no shame and no hesitation. He's smart and cheesy in the most compelling way and he fascinates and weirds out the kids when he comes. It's great. He's awesome. Last time he was here, I overheard him talking to a kid about how when we solve equations, we are trying to get x naked. That's why it makes sense to do inverse operations in the opposite order you use to evaluate, because when you're undressing you take off your shoes before your socks.

So I was trying to go with that, and here's what I came up with:

1- Who do you want to get naked? What do you want to solve for?
2- Focus their attention on you. Simplify the environment. Simplify the sides of the equation.
3- Get 'em in a room. Get all the variables (you're into) on one side of the equation.
4- Take off their clothes (shoes before socks, remember!) Use inverse operations to get variable alone.
5- Do they look good naked? Check your answer: does it make the equation true?

I almost told my kids this today, and ended up garbling it in an attempt to make it somewhat appropriate and talking about changing baby's diapers. I actually said "OK, wait, I'm thinking naked kids. No, I mean little kids," out loud. It was hilarious and memorable but pretty gauche.

I thought you guys might have some good ideas, either from experience or improving on this one. I want to laugh this much at school everyday.

Heroism

2010-03-24T06:38:00.003-04:00

I had a transformative weekend.
I was singing when I got to school on Monday morning. Out loud. Singing.
Then something happened first period, I felt disrespected, ignored, disappointed, took it all personally, got pissed, got more disappointed in myself, etc. I told my principal that I wanted to quit. It didn't make me feel better. That afternoon at our school professional development I was late and he actually thought I might have left.
I was mad the whole day. I still wanted to punch and kick things that night.

Before I left school I told my principal that I felt like I was Luke Skywalker trying to get the ship out of the mucky pond. It just felt too big. But then I knew that this little green guru could do it with no problem. Why couldn't I? I'm so impatient to be doing what I can envision. I'm so impatient.

Yesterday morning on my walk to the subway, that conversation returned to me like a dream, and I saw this key point that I had missed the whole day before:

I am Luke freaking Skywalker! I am learning superpowers! It doesn't matter which movie I'm in, I know how the story ends and it's awesome. Out of the mouths of babes.

This is not to say that I'm saving the galaxy. That would be cool too. This is just a shift in perspective that reawakens my enthusiasm. Whew. Close one.

Your job working with kids is hard, riveting, ego-smashing work. I hereby proclaim that we are all the young Lukes of our universes. So what if you meet Darth Vader in the cave? You are the opposite of disappointment to the universe, which depends on you and your work. Plus you're in training with the coolest puppet in the galaxy. You're the hero. You are learning how to fly. Just relax and enjoy.

Humility, student teachers & chess

2010-02-11T21:27:00.004-05:00

The math department has five amazing student teachers this semester. They eat lunch together and we talk math. They are like a team, and they provide a sort of community of support for each other and for us. Our tutoring program has a new feel because we have the presence of an army. Partly by luck, partly I think because of the safety that so many of them can provide, they are more confident and comfortable in our school than any student teachers I've ever seen. They are going to learn a lot because they are already so willing to throw themselves into this. It's so beautiful to see their generosity, their new bright eyes, to hear their arguments and passionate brainstorms.

It's my second time having a student teacher. Last semester when I dropped off my blogging, it was in part because I was so humbled by the constant witness of my student teacher that I hardly ever had anything positive to say about what was happening in my classroom. In many ways, having a student teacher brought me back to the ego bashing of my first year. Who was I to teach anyone how to do this? Lord oh lord.

This time around, I know what to expect a bit more. I know that it can be hard to have someone watch me do this job every minute. It's hard just to have someone else that I've got to be around all day long. It's hard to have anyone else that I'm responsible for teaching. But I'm ready, I'm brave, I'm humble, I'm honest. I'm trying to trust that just by doing the job that I do and being the person I am I can support this new addition to our field. I know a bit better what it is that I'm good at. It's worth it to try to share that.

This feels better from my perspective of course, but it also makes it easier to encourage him to critique me, to ask me why I'm doing what I'm doing, to question my motives and offer his own ideas to improve my classroom. That's half of what I can do for him, and it's a wonderful thing to let go of my own pride enough to let him do that without letting it get under my skin or make me question myself on a fundamental level.

Of course the other half of what I can do for him is to be the best and clearest model of passionate, mathematical, thought-provoking, community-supporting teaching that I can. That practice, that aim, can only be great for me and my students.

--
This afternoon my student teacher and I went to chess club together. He reminded me how to set up my pieces and then I got to play LOJ, an 11th grade kid I taught when he was in 9th grade. This kid is truly one of the most distracted students I've ever taught. He's spent three years wandering around in the halls and talking through his classes, mostly about whatever the class wasn't. He's more mature now, and I don't get frustrated with him anymore, but I also don't know how useful school has been for him. I've always known he was capable and smart but today was the first time I had ever seen him focus on one thing for more than 2 minutes. And the kid was freaking awesome. A great player, schooled me in a serious way, beat the crap out of me, and was utterly focused on his game when he needed to be. He was quiet, thoughtful, generous in his advice to me. Beautiful.

Also beautiful was my explicit willingness to lose this game to a student. I haven't played chess more than four times in my life because I hated feeling so bad at something. Today was the first time in over a decade that I've touched a chess piece, and I feel like I let go of something adolescent that was holding me back. Hoorah.

As always, may this find you all happy and inspired, able to love yourselves and your students.

Slide Rules with John Ewing

2010-02-10T21:57:00.001-05:00

The Math for America President John Ewing lead a workshop on slide rules last week, and I was really excited to have the opportunity to get to know him better and learn about slide rules. At 29, I had literally never seen a slide rule. I had no idea how they worked. I will be forever grateful to John for changing this.

Caution: I was confused off and on throughout this workshop because, never having seen a slide rule before, I had no idea which indices to look at when, or how to keep track of decimals when I chained operations. If you are like me you will need to play around with these a bit to figure them out.

Classroom benefits of integrated slide rule use:
- Estimation
- Scientific Notation
- Complex arithmetic
- Number sense
- Meta understanding of accuracy (how close to actual), precision (how reproducible), significance (related to precision…how many digits?) and difference b/w those.
- What makes an answer foolish. (Don’t need 100 digits to decide how much paint to use.)
- How functions work: increasing, decreasing, even concave up and down all were natural notions

John's questions:
How did the advent of technology and the handheld calculator affect the way that people approached and thought about calculation? How does it affect us? John is interested to think about why some things are harder or just different to teach today without slide rules.

Interesting History:
First handheld calculator, the HP-35, cost $395 in 1972 (~$2000 today)
People used to carry around book of log and trig tables…had interpolation tips as well to increase accuracy by 1 digit or so.
The Regents exam provided log tables

Gauss complained about the time consuming annoyance of calculation, and he’s famously good at it.

John Napier’s 1614 invention of logs was revolutionary. Before every calculation was done by hand. Napier got a really good feel for what the log function looked like (do high school students know what it looks like?)
Log xy = log x + log y (converts mult into add)
Log x/y = log x – log y (converts div into sub)
This is what was revolutionary, since addition & subtraction are way easier than multiplying and dividing.

Reverend William Oughtred (www.oughtred.org) used this trick to manipulate logs:
- Label rulers with numbers 1-10 but at log distances
- Developed within a decade of Napier’s work, but took two centuries to catch on
- Need increased in 19th century with engineering and war (cannon aiming) and so got popular in 1850, with the addition of the middle slider.

Tricks with slide rules:
- easy to chain calculations
- squaring things just means double log
- geometrically, just double the scale
- reverse to find square roots! (sq on A scale, root on D scale)
- cube the number on the K scale (and cube root backwards!)

Problems:
1. Want to paint a large sphere with radius 12.5. Paint label says 1 gallon covers 450 sq ft. How many gallons do you need?
2. Cylindrical tank has radius .82’. How tall should it be to hold 65 gallons? (NB: 1 cubic ft = 7.48 gal)
3. Have a tank 5.2’ high. Want it to hold 63 gallons. What is the radius? (uses sq root)

My inspirations after this:
- Teach decimal addition with just regular sliding rulers, play with measurement and perimeter. Use as a way of building meta-cognition about the sense of answers.
- Do a calculator correction activity, e.g.: Fix the problem: 42/85 = 2.0238
- Adapt what John did and run a math circle style workshop in which we re-discover the concept of sliding rulers to do arithmetic, the amazing transformation that happens when the rulers have logs on them instead of our regular rulers (I’d be interested in constructing a log table for that matter), how to chain operations, etc. Apparently real engineering slide rules have log log scales, so play with those too.

This pdf is what John gave us (kindly already cut out, just not folded) to make our very own slide rules.

students are human?

2010-02-08T20:48:00.003-05:00

One of my most promising 9th grade students made a C- in my class last semester, Ds in her other classes, including the 10th grade science class she was advanced into.
Today we had an intervention. Four teachers. One student. A little over an hour.
We shared her strengths as we saw them. We asked her what her obstacles to success were. She was candid and honest with us. I think she felt glad to have such attention paid to her. She described her history of being bullied (for whatever reason anyone is bullied) and ridiculed for being faster/more focused on her work. She explained what it was like for her to work in a group, both at school and at home, where people had already given up. She talked at length about all the reasons she can't work at home, her brother who takes her books and throws them across the room, her mother who is always asking her to help out, her grandparents who need her to run errands, the chaos of her family life. She explained why she couldn't work on the subway (she'd get so into the work she'd miss her train stop) and spoke almost dreamily about how "sweet" it would be to have just an hour a day to do her work without being distracted or distressed.

There was nothing in her reasons that seemed particularly surprising, but somehow I recognized the humanity in her. She was stuck in that thing we all get stuck in, where we accept that our lives are just difficult and we have to survive them, at best understand why they are crappy, but not change them. How many of us don't exercise more, eat better, fail at New Years resolutions, get stressed out on Sunday nights when we haven't done as much grading or planning as we mean to... Of course she does that. Of course she gets lost and overwhelmed. I don't know anyone who doesn't.

It was exciting to use concrete structural interventions to try to help empower her to make her life how she wants it: to realize that "sweet" feeling she has when she imagines having time to study.

Cool, huh?

Here I was thinking I was the one whose humanity gets overlooked. Amen for awakening.

hugging

2010-02-06T19:27:00.005-05:00

I've been hugging all my students. Can I remind you that I teach high school students? The kind that are busy with their hair and their technology and being cool and impressing their peers. They have that haughty insistence on independence, sometimes even a resistance to connection.
But I tell you what: I hadn't seen them for real class in two weeks (Regents, Intersession) and when I opened my arms to hug them, one after another they came at me. No hesitation. Full on hugging. Like we love each other. Like we're in something together. It was amazing. I didn't know it was possible.

I've decided to love them more. I know, I know, this is what I'm saying in every blog post. Where is the math in my mathbebrave? I promise, it's there. I will talk about it more. But at the moment this just seems so important. Because I think I'm pretty damn loving, but each time I push myself just a little bit more into this job, that's the thing I learn. Not work more, or even more time: just love them more. Or rather, feel the love I have for them more freely, more purely.

This is softening my edges. I don't take the stuff they throw at me personally the way I used to, even two weeks ago. Things go badly in class and I practice loving them, seeing their humanity. I don't know how to extend their knowledge of Combining Like Terms tomorrow, but I know that I'm going to love them. Sunday night I was excited to go to work. I like my job, but I've never been excited on Sunday night before.

Here are the goals that I gave them for the week:
1. Have Fun
2. Do Tons of Math
3. Work in Groups
4. Do some problem solving
5. KIds talk more in class than teachers
6. Practice combining like terms
7. Quiz Friday

Then they checked the box that best fit how they fit about my goals: awesome, eh, or no way!

Hyperbolic Space, Global Warming & Crochet

2010-02-06T18:49:00.003-05:00

If you're thinking to do something with hyperbolic space or global warming, you might want to check this out:
http://www.ted.com/talks/margaret_wertheim_crochets_the_coral_reef.html

Khan Academy

2010-02-06T17:18:00.004-05:00

My stepmother sent me this today:
http://www.khanacademy.org/
I've only looked at half of one video, but I've already got ants in my pants to share it with you. The one I watched on linear equations was anti-climactic: procedural, superficial, kind of the opposite of what I want to do with my students. But I bet it's helping somebody, maybe lots of somebodies, and maybe it's of interest to you.

---
My friend Amy just clued me in to this:
"this dude was on NPR recently. i think it's pretty awesome, especially considering the whole thing was borne out of his distance-tutoring for his cousin."
for the NPR story:
http://www.npr.org/templates/story/story.php?storyId=121978193

Love is...

2010-02-05T18:14:00.002-05:00

After 3 days of making art with my students for our annual Intersession, I was delighted to return to classes today. New semester, new student teacher, new me. I felt rejuvenated by our art-making, and inspired by the PD with Jennifer Abrams I had on Monday, which I'll write about over the weekend.

Then someone emailed me this quote:
"Love is patient, love is kind. It does not envy, it does not boast, it is not proud. It is not rude, it is not self-seeking, it is not easily angered, it keeps no record of wrongs. Love does not delight in evil but rejoices with the truth. It always protects, always trusts, always hopes, always perseveres." 1st Corinthians 13

I don't know the Bible very well, nor have I looked to it before for guidance. But I am so happy to have received this today. It sums up exactly the kind of love that I want to have with my students, every day, every minute. When they are distracted, frustrated, discouraged, disrespectful, perfect. Protect them, trust them, hope for them, persevere on their behalf. Even when I'm disillusioned and discouraged, even when I'm tired and frustrated.

May you all experience this kind of love in yourself while you work. May you receive it from those around you too.

Happy Friday!

Assessments, Problem Solving, Questions for you!

2010-01-27T15:54:00.000-05:00

In other news, this morning I had a math department meeting. We talked about the newest aspect of our DYO assessments: the Grade Level Competency Exams (GLCE). These exams are supposed to reflect the minimum a student should be able to do and still pass a class. They're about basic skills. 80% is a passing grade. (I find it really hard to make these exams with confidence. Should reading a clock be on the 9th GLCE? I want them to read a clock, sure. I recently discovered that they don't all know how, yikes. Does that mean it should become part of my 9th grade curriculum? That they shouldn't be in 10th grade without it?)

In the midst of all of this, we're trying to be more transparent about how our classes intersect, to talk about what we value, what we expect of each other and how what we teach works together. As a group, we have difficulty doing this when it's called vertical planning, but it's organic in this context. Interesting.

Then we answered the questions: Why is it useful to study math? What is the most important thing you teach?
All of us, unanimously, said problem solving and critical thinking in answer to both questions. We all were also clear that our students don't know that. They don't know that everyone in the department cares more about problem solving and thinking than anything else. Why not?
Maybe it's got something to do with the fact that we're so busy making these basic skills exams?

So I'm looking for some advice here.

1- What do your students think you think is the most important thing that you teach? What do you do to make that clear? In other words, what do your students know that you value, and how do they know it?

2- How do you all plan in a way that makes your big values transparent?

3- How do you assess problem solving and critical thinking?

4- Is it ever useful (and if so how and how much) to assess straight up skills without problem solving or critical thinking?

Affirmations

2010-01-27T09:49:00.003-05:00

Hope your Wednesdays went by easily and that you felt appreciated and loved.

I was discouraged the other day about one of my 9th grades, and my colleague asked one of the students from that class to give me a hug and tell me how much she appreciated me. This student (code name MD) is the least likely of all of my students to want to do such a thing, and I cringed as I heard the question. MD was predictably resistant, making faces and "um"-ing. My colleague pushed her: "Jesse's having a hard day. She's been talking to me for the whole last 45 minutes about your class. She's worried about you guys and she really needs to know that you appreciate her." MD interrupted at this point, stamped her foot, and raised her voice to say, "She knows how much we appreciate her!" as if appalled that anyone could doubt it. It was the best affirmation I may have ever received.

I just realized that maybe you guys don't all do this self-doubting thing as much as I do. But if you do, let this be a gift to you all: the relationship that you have with your students does not go unnoticed, no matter who you are, no matter how inadequately you (think you) do your work. You work hard, you love hard, you get up every day and show up whether it's raining or you feel grumpy or whatever and you offer whatever it is you've got. They notice it. They appreciate it. They have no idea that you don't know that. Trust it.

Rejuvenation via PD

2010-01-21T21:21:00.002-05:00

Happy Winter! I know for many it can be a dark period of the school year in more ways than one, and I hope that for this moment if not in general that you are happy, enthusiastic, hopeful, grateful.

I went to a mediocre PD workshop last night and it made me miss you all, and appreciate again how you have created your own PD, DIY style. You are so good at it! I've learned so much from you, been inspired and rejuvenated over and over.

Despite the anticlimax of last night's PD, I had three really cool experiences today I wanted to share:

The first: a new precision of compassionate humility.
Part of the PD last night was this 5 minute game where we were asked to locate consecutive numbers, arranged a la Where's Waldo, at speed. I felt, I think for the first time, utterly disoriented and confused doing an activity with numbers. My friends were being hilariously but half-seriously competitive. There was no reason to what we were doing, no sense to it, no one could help me do it better or faster, and by the end of the activity I felt the impulse to skip numbers. No one would have noticed. Why not?
Because I was not on the hook in any way, I could laugh: at the competition, at myself, at the anxiety. But it would not have been funny if I had been 15. And this gave me a window into what it might be like to be 15 and doing math when it doesn't make sense.
With this fresh in my mind, I was more sensitive to this possibility when my 10th grade classes today went into revolt mode when I asked them to evaluate expressions that I thought would be total review, easy-peasy. I changed my whole lesson plan on the fly to accommodate their actual readiness for the material, and it was awkward because I wasn't prepared but I didn't get frustrated with them because I was able to see it wasn't their fault and they were so appreciative. They just needed me to back up about 3 paces further than I had realized.

The second: remember how good it feels to stand up!
In my sweet 9th grade class today, there was a minor revolt. One kid raised his hand and asked for things to be more fun, to have races and competitions and make posters. OK. I hadn't realized how long it had been since we did that. So I passed out the worksheet I had planned and asked them to pick their favorite problem, do it, compare their answers with their group, then rewrite the problem as clearly and beautifully as they could on mini-posters and tape it up wherever they wanted in the room. They had a blast, made posters that say "my favorite math problem" and now they can see all around them what they already know about combining like terms, not only today but all next quarter. Duh. It was so freaking easy. For the 15 minutes they were making the posters they weren't doing new math. But I think they might remember what they did do a little better, and most certainly we had more fun than if I was forcing them to just do more worksheets. I can totally do this.

The third: pattern seeking machines
Bob & Ellen Kaplan say in their book Out of the Labyrinth that human beings are pattern-seeking machines. They talk about how students don't always see the patterns we intend for them to. For example, if the quadratics that students plot always have vertices with integer coordinates, then students are (understandably) likely to assume that all quadratics have vertices with integer coordinates, and then get baffled and thrown when they get one that doesn't. This phenomenon reflects well on students' cognition. And unfortunately for them, this happens all the time through well-intentioned scaffolding and careless textbook writing alike.
I tutored one of my students for two hours after school today. She could show me step by step what to do to solve two step equations. Her procedures were near perfect. I would not have noticed a problem had I just been looking at her work. But she couldn't explain anything, and her reasoning turned out to be entirely based on repeated exposure to those visual patterns, as opposed to actually understanding equality, inverse operations, solutions and true statements. Hopefully she's a bit clearer now.

The upshot:
Do whatever it takes to stay inspired. Be reflective because it's fun. Try new things. Love them. Love yourself. Be patient. Be honest.

You all rock my world. Happy 2010. May this year be the best yet for everyone.

Toothmallows!

2009-11-17T19:25:00.002-05:00

Today I tried something I'd never done before. I did it with my 9th graders, the class that is made up of all these awesome minds, hard workers, smarties, funnies, goodies, but somehow doesn't feel like a working group. We made towers of cubes out of marshmallows and toothpicks, looking for patterns (edge & vertex counting), making functions.

I gave them almost no direction at all, just showed them a couple of small starts that I'd made and they went nuts. I should have brought my camera. It was awesome! They worked so well together, building stuff, talking about what the building blocks were (these dangly things, not fully formed cubes) and didn't eat anything or poke each other at all.

Once they were supposed to start actually counting stuff, finding patterns and rules, the whole thing deteriorated. I left class feeling sort of defeated but determined, sure that I should feel some sense of accomplishment. After all: trying something new, messy, unpredictable. That's success right there, in a way.

So next, I'm trying to figure out how to get it to feel like that more often, how to connect the work they do in their minds to the work they do with their hands. How to get them to really bite into the concept of the variable, which was my motivation to do the lesson in the first place.

Also, what to do tomorrow. Ah.

Good Teaching

2009-11-11T07:58:00.004-05:00

Yesterday in class, my coteacher Phil Dituri (amazing, amazing, amazing) was asking our kids to be silent for the start of the class. We've been trying this out for the first three minutes of class the last few weeks just to get them settled in and working, so that when group work starts they can really bring themselves fully into that conversation.
Yesterday proved to be on the difficult side. The kids had had a rough morning (last day of the quarter, NYPD metal detector surprise that morning, typical season affective disorder type blues) and weren't getting quiet. After a couple of minutes of reiterating instructions, and individually trying to help kids focus, Phil went around with big blue paper tape, drew smiling mouths, some with tongues out, and gave it to kids to tape their mouths closed, which they all wanted to do.

This looked ridiculous to anyone who came in at that moment. Comical, maybe even inappropriate. But the thing is, Phil could have yelled, made the kids feel bad, expressed his frustration, disappointment, he could have sent kids outside, called home, whatever. Instead kids were "smilingly" (both on the tape and underneath) trying to be quieter than their neighbor. They ate it up.

The tape didn't exactly turn our kids into models of focus and undivided attention. But kids got some individual attention, which they seemed to need, they got a structural intervention to help them follow instructions and get quiet, and the room stayed positive and focused on the math.

I realized after I watched it happen that this kind of thing is something they didn't teach me in grad school. I've never even (consciously) heard anybody talk about it before. But it seems like a vital part of that magic that good teachers have: to be able to keep the forward momentum, the mathematical focus, the positive energy, in the face of whatever the kids throw at us. It can be so easy to take a momentary failure personally, to get frustrated, feel like the kids aren't listening, whatever. Or just to run out of ideas. What Phil did in that moment was magic because he didn't drop the expectation: the kids were still supposed to be quiet. But he didn't get nego either, on them or himself. He gave them a new reason to follow directions, perhaps clarifying the directions for some kids in the process, and then back to the math.

Phil transformed what could have been a power struggle into a collaboration. He turned what could have been rule-following into a game. He was able to honor the kids, acknowledge their state and their need, give them total respect, and still help them transition and move forward towards our mathy goals. It was beautiful.

The whole thing took maybe 60 seconds. It wasn't such a big part of classtime. But I think it's those moments, the decisions we teachers make in those moments, that set the tone and culture of the class. We are in this together. We teachers aren't going to pull the authority card just because we can. We want to have fun with you while we work. For me, cultivating this feeling in the classroom is the best leverage I've got, for getting a class working, and it's way more fun to be in a room where that's the vibe.

I love teaching with Phil. May you all be so lucky in your teaching collaborations.

Inspiration: Kate Nowak

2009-11-11T07:33:00.004-05:00

Kate Nowak's posts are optimstic, pragmatic, generous and immediately useful. I used her row game in class yesterday and it was the best new group work structure I've tried since I started teaching.
My whole class is (ideally) based on group work. My students should nearly always be talking to each other, comparing answers, helping when someone needs help, asking questions when they're confused, helping each other stay on task, etc. Sometimes this is natural, and mostly I see that this is how kids want to do math. Some training is helpful though.
While doing the row game yesterday with order of operations and signed arithmetic, I heard the best conversations I've heard in my new 9th grade classes. I witnessed serious debate, authentic and sustained engagement, passionate investment in each other's work, and evidence of a compelling motivation to deeply look at one another's work, since they hadn't themselves already done one another's problems.
FREAKING AWESOME!
Simple to create and direct, this activity is going to change my class. THANK YOU KATE!

Kate's current post is also beautiful and I'm going to use it in class imminently. I'm excited about the accessibility of the visuals and the ease with which I think kids can interpret and make meaning of them.

While I'm thanking Kate, I should also say that she is the reason I know about blogging in the first place, and the primary support for me starting this here blog, which I'm so glad I did.

3 cheers.

Nearly Quarter 2.

2009-11-08T18:15:00.003-05:00

According to one of my 9th graders, our first unit was due the title:

"Educational Revolutionary Mathematics"

That felt like the arising of my own personal cheerleading team.
That was also about three weeks ago.
Since then, I've been negotiating parent teacher conferences, failed experiments in the classroom, incorporating a student teacher (my first), dealing with longer darker evenings, the storming phase of group dynamics, and my own personal dramas around all of the above. Truth is it's felt like such a hard couple of weeks that I didn't want to look at the reality of it in my own head, much less write about it to anybody else. However, I did learn some things.

Highlights that I'll share:
If I feel disrespected, discouraged, frustrated, like a failure...
it's worth considering how my students are feeling. Because they are probably feeling a bunch of those things, and I want to be conscious and sensitive to how I respond to such volatile and hot emotions. Also because I feel immediately compassionate for them in a way that I struggle to feel for myself, and that's a useful button to push in me.
Because when I'm feeling pissed off at myself or them or the situation or whatever, I get distracted by all those feelings and forget to actually look at what's happening, pay attention to the math and the kids who are learning (or struggling to learn it). But once I hit my compassion button, and start looking out for my kids' feelings, then I'm paying attention again, and it's just a problem solving activity, rather than an ego trip or an identity crisis.

I can't be sure, but I think it's safe to guess that I'm not the only teacher who takes stuff personally when they shouldn't, and for all of you out there, I just want to offer this:
Whatever your worst feelings, your students' are worse. Focus on that and it's easier (for me) to let go of my own head trip and get back to thinking about something I can actually do something about (tomorrow, or right now) rather than impossibly fretting about the past.

I want to be your cheerleader now, assure you that somehow it's all about the process, all about showing up and doing your best and working hard and enjoying it as much as possible and in fact doing whatever you need to do so that the hard work feels good. Remember to sleep enough. Remember to eat three square meals every day. Remember to love yourself. Remember to tell them how awesome they are, whenever they are. Look for those moments so you can jump up and down and cheerlead them. We all need it, eh?

Hope your last three weeks have been fantastic. Hope that your Mondays are awesome.

Larry Zimmerman strikes again

2009-11-08T18:13:00.001-05:00

I wanted to share another few excerpts from Larry. He gave a talk last Tuesday at the NY Math Circle's PD for Middle School Teachers, and I was impressed again by how he asks questions to extend thinking about a problem, offer more entry points, and deepens my interest. He does it so quickly, I can't help but get interested and then my mind is thinking about 10 different but connected things at once, and whatever I get into relates to the bigger picture, and it's like magic differentiated instruction and guided inquiry. It's really awesome. We did a bunch of problems, and here are three that I wrote down his questions for:

Show that the sum of two odd numbers is an even number.

What is an odd number?

What is an even number?

Is 4 even?

Is 0 even?

Are there any even primes?

What’s the next even prime?

Are all prime #s greater than 2 odd?

Are all #s greater than 2 prime?

Is -4 even?

Is -6 even? Why?

So now, what is an odd number?

Is every even a multiple of 4?

Show that no perfect sq ends in 3.

Do you believe it?

Do any end in 6?

Did you know that every sq ends in 6 iff the number’s 10s digit must be odd?

Would anyone like to see a proof of that? Good! Go do it.

(I was curious to observe here that every sq is

- a multiple of 5

- one less than a multiple of 5

- one more than a multiple of 5)

Show that x² – y² = 2 has no solutions in integers

Translate this into words.

Say this in words: x² + y²

(x + y)²

x² + 5x + 6 = (x + 2)(x + 3) Find the value of x that makes this false.

Factor it using diff of squares: (x + y)(x – y) = 2

Note: remember given info

Integers factors must be 2 & 1

“I wanna savor that for a minute. To some people that sounds trivial, but to me it sounds profound. Never dismiss the obvious as being trivial. Even if it’s obvious it may be very very important.”

x + y = 2 and x – y = 1 (or other way around)

“Do something, do anything, and if it doesn’t work, SO WHAT?”

quotes of the week

2009-10-15T21:33:00.004-04:00

i don't know if any of these translate without audio and facial expressions, but here's hoping:
"jesse come look come look! i have my own rule for doing this now, let me explain it to you!"
"this class is so sexy...erhm, i mean, i just really love math class."
"when you sing, it makes angels cry. CRY!"
"there are 10 days in a week"
"there are 12 hours in a day"
and, finally, no one asked why their math teacher asked them to draw a picture of a flying monkey.

smarter than 2

2009-10-06T16:52:00.005-04:00

this whole conversation started when ben blum-smith read me the preface of mariana cook's new book of photographs of mathematicians:

"Mathematicians are exception. They are not like everyone else. They may look like the rest of us, but they are not the same. For starters, most of them are a great deal smarter."
i totally nodded right along with him reading! what is that? confession: i walk around thinking that my math friends ben and justin and all these people are just smarter, more capable, better than i am.
and the thing is, it doesn't matter if that's true. the thought that it is, the possibility that it is, is so debilitating that i have felt (in the last three weeks!) that i should stop teaching, despite the fact that i know i'm doing this about as well as people do it and the people that have seen me do it would hate to see me stop. or whatever, even if i was bad at it, i won't get better if i think i'm just not smart like mathematicians are smart. or whatever. the problem is the very idea that true hierarchy exists. the notion of hierarchy is limiting because then some people are just smarter, and then the thought "i am one of them" or "i am not one of them" are equally problematic.

why do we (or a big portion of the general we) see mathematicians as different? and how is it affecting my students? kids (plus me and unapologetic too, it seems) are walking around feeling defined by either having smarts or a lack of them. i question the truth of these ideas in the first place, but definitely the usefulness of these beliefs when kids get attached to these self-definitions. i wonder how to keep the identity question balls up in the air during high school, rather than supporting kids in pinning themselves down.

i'm proposing that ALL kids can feel empowered and smart, not because all kids are the same, but because kids aren't all one thing. i want to give kids opportunities to perceive themselves in different roles so that they don't ever feel limited, in anything. i want my kids to be able to bite into the experience of success (which will look different for different kids) and struggle (a very useful thing for everybody to know how to work through). i want them to see each other as resources, sometimes surprising and unexpected. i want anyone in the room to be able to be the one that has the insight, no matter what has happened in their pasts.

good luck, right?
well i'm trying.

seriously, big thanks to you big debaters out there. i love it.

smarter than...

2009-10-04T18:15:00.004-04:00

two big ideas came together for me today in a beautiful way:
idea 1: from the psychoanalytic standpoint, group theory suggests that students take on different roles: monopolizers, hiders, harmonizers, bullies, etc. this is natural, even useful. problems in the group arise not when kids take on roles, but when kids are always playing the same roles! if the same person is always doing all the talking, maybe at first a bunch of folks are content to sit back and let them do the heavy lifting. but eventually the group is going to get pissed because they aren't getting their say. so it's important to keep roles in the classroom (or whatever group you're leading) fluid, and how to do that is what i've been studying and thinking about lately. just trying to watch it happen is pretty cool.

idea 2: ben blum-smith made a comparison for me today between racism and the idea that some people are better at math than others. he was pointing to both as culturally insidious ideas of inequity that have maintained their strength despite obvious uselessness and inaccuracy because the structural privilege they provide benefits too many people. i couldn't quite grab onto this parallel until he explained that the benefits of the idea of mathematical talent (let's call it) aren't just to the elite mathematical class, but to anyone who has ever felt like they were smarter than someone else. ever.

huh.

that hit me harder than anything all week. and i've had a big week. is it actually true that some people are smarter than others? certainly the power of "i'm smarter than..." rang loud and clear for me. there's a lot of meat in here for me to consider, but the thing i wanted to share was this:

what makes sense to me is that kids need to experience themselves in different ways in order to learn, in order to learn how to learn, in order to get good at living. and so it's not ok for the same kid to always be the one that's good at it. who gets it. who explains it. who aces it. it's not ok for the same kid to always feel like they're just not good at it. the good at it and the not so good at it are ok, maybe, as long as those labels don't get stuck in the same place.
this idea feels like the most radical one i've ever had. that we none of us "deserve" to feel like we're smarter all the time. or that we're less smart all the time. i don't believe it, but even if someone had research to show that it was, i'd still advocate for the structural intervention of pretending that it's not.

practical applications for tomorrow:
- see kids as changing beings. really observe, listen, watch, each day with conscious but open eyes (as opposed to permanently labeling them in my own mind as high fliers and low.)
- give them lots of different kinds of things to do, and to highlight the success and struggle of everyone in those contexts, so that it's clear that there will probably always be folks in those roles, that the roles in and of themselves don't necessarily mean anything, because those roles are changing all the time.

quickly...

2009-09-30T18:59:00.001-04:00

"replace ambition with curiosity" - paraphrasing nancy stark smith
let go of assumption.
listen.
my ideas of the day.

September 2009

2009-09-17T19:04:00.002-04:00

I want to say a word about hope here.

I want to say a word about how the most important thing is just to show up, to have faith, to be present, to fuck up and know it and still love yourself completely, and show up the next day, still with faith and a willingness to be present, to really see what is happening in this moment, now.

I want to say a word about how a person can spend 4 years teaching and feeling incompetent and frustrated and like it doesn't matter, and then out of the blue, for relatively unimportant reasons, ten kids that you taught when you were just starting, tell you without hesitation or qualification how you changed their lives, and then go back to talking about pizza and what they're going to do after school.

I want to say a word about the importance of uncomplicated friendship that rejuvenates and inspires, clarifies, answers, supports.

I want to say a word about gratitude. To be able to notice any of this is the whole point, I think.

We are revolutionaries, all of us, each of us doing the best we can, loving and hoping and serving every day, even the ones that feel like a total sham. Eat it. Enjoy it. Live it. Trust it. Take it. It's yours.

Why does it matter?

2009-08-04T12:50:00.006-04:00

I just read an interesting post , addressing a student's question "Why does this matter?" and it's got me thinking.

As a teacher, being prepared to answer the question of how whatever we're teaching is relevant is important. In fact, I hope that it drives our planning, that we are riveted, fascinated, engaged in the usefulness and application of what we teach. If we are clear about the context, meaning, beauty and application of a given lesson, being transparent about the topic preempts the question.

In my experience, whenever this question does get asked it's not because they actually want to know why a lesson or topic is important. It's because they're not learning. If they have time to ask this question, either they are not experiencing enough challenge or they are not experiencing enough success and one or the other is arresting their learning.

"What's the point" is code for "I'm bored and I don't want to do this because it doesn't matter" or "I'm lost, and I've been lost, and I don't want to do this because it sucks to feel lost." In either case, an explanation of the value of the topic doesn't actually address the real concern: if actually doing the math is not interesting or engaging or challenging enough to capture their interest, no amount of verbal explanation is likely to help; if they are too confused to do the math in the first place, no amount of verbal explanation is going to get them to "get it."

I think it's our job to figure out what to do to get the kids learning again. Even with an awesome explanation for the worth of algebra, if they're asking why it matters then something more basic is missing for them. When they are learning, both feeling successful and being challenged, the question doesn't come up.

Larry Zimmerman Plus

2009-07-30T18:18:00.003-04:00

Larry Zimmerman is an extraordinary man, teacher, mind; one of those math teachers that infectiously inspires creativity and enthusiasm. As a teacher, he is alert, industrious, sensitive, clear, direct. He presents an entirely different model of teaching than anything I’ve seen before, and it made me wonder, if a little desperately, how my life would change if he had been my coach. He has recently retired from a long career at Brooklyn Tech and is pretty involved in the New York Math Circle (http://www.nymathcircle.org/)

My notes and quotes on him came from a workshop he taught on problem creation yesterday. My take on this part of his teaching is that it taps into students' (& teachers) metacognition (thinking about thinking, thinking about the fact that they are doing math while they're doing math), pattern seeking abilities and natural curiosity. I'm trying to approach planning by simply stating the questions I want to be prepared to ask during a lesson, as well as return to through the course of a unit. Here's an example of the list of questions that came from one initial problem (which I didn't initially even think was all that interesting.) I found the questions we came up with fascinating and pretty surprising.

“Theme”: How many distinct positive integer factors has the number 36?

“Variations” in no particular order:
- How many distinct positive integer factors has the number 37?
- List the factors of 36.
- What is the smallest positive integer which has the same number of factors as 36 (including 1 and itself)?
- How many distinct positive integer factors has the number 40?
- How many distinct positive integer factors has the number 49?
- How many distinct positive integer factors has the number 944?
- If x < y, and x divides y, can the number of factors of x be greater than or equal to the number of factors of y?
- Is there a positive integer for which the integer is less than the number of its factors?
- Can you find the number of factors of a number without enumerating them?
- Which positive integers have an even number of factors?
- Which positive integers have an odd number of factors?
- How many distinct positive integer factors of 36 are even/odd/perfect squares/multiples of 6/etc.?
Other inspired questions, not particularly related to the theme:
- Is zero an even integer?
- Is zero a factor of 5?
- Is 5 a factor of zero?
- Is 2 prime?
- What is the next even prime?
- Define a definition.
- What is the purpose of a definition?

Larry Zimmerman Quotes & Notes

2009-07-29T17:34:00.003-04:00

“It is perhaps more important to be able to compose problems than to solve them.”
“Problem creation is the essence of mathematics.”
“The first effort is rarely if ever the final product.” (we must emphasize editing)
“Creating problems is theme and variations.”
“Determine what you are looking for and WRITE IT DOWN.”
“I don’t know what I want them to realize.”
“Beware of reinforced suspicion,” which is not a substitute for problem solving.
“If I’m 100% I will not forget it, I write it down.”
“There is nothing wrong with clarity by redundancy.”

Types of Problems
• Surprising
• Novel
• Fruitful
• Charming, imaginative, alluring
• Historically significant
• Haunting, musical
• Beautiful, elegant, sublime

Elements of Problems
1. a goal (construct, prove, maximize, minimize, classify, compare, compute)
2. given information
3. special rules (sometimes)

Some Big Ideas
- Plant a seed and then walk away until someone says something. Students and teachers struggle with silence, but it’s important and it saves time.
- Keep a notebook (both teachers and students) of interesting questions.
- We are well versed at turning a lot of words into symbols. Do the reverse.
- There are 3 varieties of equations: identities (always true), conditionals (sometimes true), and contradictions (never true)

Sites of the Day

2009-07-29T17:33:00.000-04:00

Arbitrary digits of pi in base 16
http://www.sciencenews.org/sn_arc98/2_28_98/mathland.htm

Online Encyclopedia of Integer Sequences: http://www.research.att.com/~njas/sequences/Seis.html

Mary O'Keeffe Best Games Ever

2009-07-29T17:25:00.004-04:00

Number Bracelets Game (adapted from Marilyn Burns at
http://www.geom.uiuc.edu/~addingto/number_bracelets/number_bracelets.html)

Imagine that you have lots of beads, numbered from 0 through 9, as many as you want of each kind.

Here are the rules for making a number bracelet:

Pick a first and a second bead. They can have the same number.
To get the third bead, add the numbers on the first and second beads. If the sum is more than 9, just use the last (ones) digit of the sum.
To get the next bead, add the numbers on the last two beads you used, and use only the ones digit. So to get the fourth bead, add the numbers on the second and third beads, and use the ones digit.
Keep going until you get back to the first and second beads, in that order.
How long (or short) a bracelet can you make?

Other Questions:
How long is the longest?
How short is the shortest?
How many number bracelets are there?
- How many different starting points?
- How many different bracelets?
Does it necessarily repeat?
Extensions:
Change the mod/cutting number
Change the rule (from Fibonacci to some other sequence)
Why no bracelet of 6?
Why no bracelet of 5, 10, other factors of 60?
What if only use even numbers between 0 and 9?

Prime Number Monopoly
Materials: 2 dice, 5 cards (one for each of the first 5 prime #s) with the following costs and deed contracts:

Prime Number Deed:
Cost:
2 – $250
3 – $167
5 – $100
7 – $71
11 – $45
The owner of this deed
splits the winnings
when another player gets a multiple of:
(prime number here)

Game: Roll two dice and can choose the amount of either number formed by those two digits. Can buy prime numbers. Then whenever another player chooses a number that is a multiple of your number, they split the $ with you.

Goal: Get as much money as possible (there will be times when it will be to your advantage to choose the smaller number). Play for a set amount of time, clearly stated in advance.

Questions:
How did I work out the pricing?
What’s the best prime real estate buy?

Mary O'Keeffe Quotes

2009-07-29T17:24:00.002-04:00

“Math can and should be a social experience.”
“Get people comfortable with making mistakes and taking intellectual risks in public.”
“Give people powerfully compelling activities they will want to share with others, who will want to share them with others, etc.”
“Best way to learn is to share things with others.”
“I like to blurt out things when I’m not at all sure that I’m right.”
“We respect respect and we correct mistakes.”

Mary O'Keeffe Book Recommendations

2009-07-29T17:22:00.003-04:00

About Teaching Mathematics by Marilyn Burns
Aha! Gotcha by Martin Gardner
Aha! Insight by Martin Gardner
The Annotated Alice by Martin Gardner
Feynman and Asimov
Melisande by E. Nesbit
Mathematical People (interviews)
More Mathematical People (interviews)
Raymond Smullyan logic stories
Calculus By and For Young People by Don Cohen http://www.mathman.biz/

David Hankin Questions, many of them good & big ones

2009-07-28T22:36:00.002-04:00

• What is geometry?
• When we study geometric figures, what are we concerned with?
• When we study geometric figures, what are we not concerned with?
• What kinds of motion (transformation) keep a figure “the same”?
• What is the maximum number of these motions (isometries) are necessary to get from one figure to another if they have the same orientation? If they have different orientation?
• What minds of motion change a figure?
• What is an angle?
• Given a diagram, what do you think might be true? (rather than given a diagram and some true things, prove this other thing)
• What is symmetry?
• Why do we study almost exclusively symmetric figures in geometry?
• Is a line segment symmetric?
• Is an angle symmetric?
• Are triangles symmetric?
• What can we say about the points on a perpendicular bisector?
• What happens when we find the perpendicular bisectors of the sides of a triangle?
• When is the circum center ON (/in/out) a triangle?
• What happens when we find the angle bisectors of a triangle?
• How many lines of symmetry does a quadrilateral have?
• Is it possible for a figure to have more than four lines of symmetry?
• How many lines of symmetry does a circle have?
• How do I draw a symmetry line through a circle?
• What happens when we draw the diameter through a point on a tangent line?
• What happens when we draw perpendicular chords through a diameter?
• What happens when we draw progressively smaller and smaller perpendicular chords?
• Problem: Given a triangle and side lengths, find the lengths of the segments formed by the incircles.
• Problem: Given a quadrilateral and side lengths, find the lengths of the segments formed by the incircles.
• Must a quadrilateral have a circumcircle?
• Reflection Problem: Given two points A & B, and a line (below them), what is the shortest path from point A to the line and then to B?
• Reflection Problem: In pool/minigolf, where do you aim so that A bounces once (twice, 3 times, 4 times back to A!) before hitting B? When are these possible/not?
• What has translational symmetry?
• Translation Problem: Two towns on opposite sides of a 1 mile wide river. Where should we put the bridge (to minimize the distance)? What if there are two nonparallel rivers?
• What letters are preserved under 180 rotation (/half turn/point reflection)?
• What geometric figures are preserved under point reflection? (which have point symmetry?)
• What geometric property follows from the fact that the letter Z has point symmetry?
• Point Reflection Problem: two circles intersect. Draw through the point of intersection a line which creates congruent chords in each circle.

David Hankin Quotes

2009-07-28T22:28:00.000-04:00

1. Majority rules in math right?
2. Did we prove it or did we just all agree?
3. Math is not about computation, but about proving things.
4. When something works well in math, you should milk it.
5. Geometry is something dynamic, not a bunch of stationary objects. Transformations help us see this. If you’re good at geometry, chances are you moved stuff around in your head.