Thursday, July 15, 2010

Group Dynamics for Teachers

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)

Tuesday, July 13, 2010


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!!!

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.

Thursday, July 8, 2010

Unconditional Enlightenment

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.

Tuesday, July 6, 2010

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

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

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.

Monday, July 5, 2010

Crying in May

On May 4, 2010, I wrote the following:
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?"

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...

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

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...

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!

Thursday, July 1, 2010

Student Publishing

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.

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?

- “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
- 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.