Thursday, December 16, 2010

Doodling in Math Class

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.

Thursday, November 18, 2010

Bieber, yeah

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.

Monday, November 1, 2010

Voting Math

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.

Wednesday, October 6, 2010

Just a little patience

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.

Saturday, October 2, 2010

The 5th year

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.

Wednesday, September 29, 2010

It's Wednesday

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.
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.

Sunday, September 26, 2010

Lady Gaga & Exponents

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.

Saturday, September 11, 2010

Respect, Compassion, Courage

"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

Wednesday, September 1, 2010

Dynamic Paper

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.

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.

Monday, June 28, 2010

Rock Paper Scissors Lizard Spock

To improve your game, go here.

To buy the t-shirt, go here.

Productive Discussion in the Classroom

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.

Wednesday, June 23, 2010


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.

Wednesday, June 16, 2010

kids thinking

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.


Thursday, May 20, 2010

Ben Peled Saves the Day

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.

Sunday, May 16, 2010

Jascha on Sims

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!

Saturday, May 15, 2010

2 Additions to my Conceptual Understanding

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!

Friday, May 7, 2010

Kuta Software

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.

Wednesday, May 5, 2010

The Learning Network, Teaching with the NYTimes

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.

Sunday, May 2, 2010

The Math Worksheet Site

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.

Saturday, May 1, 2010

By request, Superstar Pictures

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?!

Thursday, April 29, 2010

Ben, Sam, Kate

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"

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

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.

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!

Wednesday, April 28, 2010

New York Math Circle Summer Workshop 2010!

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!

Friday, April 16, 2010

Go to the Kaplans' Math Circle Institute!

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:"

Wednesday, April 14, 2010

Math for America deadlines approaching...

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."

Monday, April 12, 2010

Inappropriate Models

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.

Wednesday, March 24, 2010


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.

Thursday, February 11, 2010

Humility, student teachers & chess

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.

Wednesday, February 10, 2010

Slide Rules with John Ewing

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 ( 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!)

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.

Monday, February 8, 2010

students are human?

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.

Saturday, February 6, 2010


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

If you're thinking to do something with hyperbolic space or global warming, you might want to check this out:

Khan Academy

My stepmother sent me this today:
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:

Friday, February 5, 2010

Love is...

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!

Wednesday, January 27, 2010

Assessments, Problem Solving, Questions for you!

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?