Purpose and Meaning in a Final Exam

Ever since @CmonMattThink tweeted out this poster that I'm just nuts about, I've been thinking about all the different uses I have for this saying:

Today, as we're starting our final exam week, I'm thinking about it with regard to assessing the meaningfulness of a final exam. 

We give common departmental finals, which is why, when my normally independent 9th graders felt compelled to pepper me with questions about the majority of questions on the test, it made me notice and wonder about the test itself:

Is this test a scavenger hunt for right answers? Or does it measure students' ability to provide evidence of their understanding?

There was a previous conversation on Twitter a while back in which some of us were debating the allowability and rationale for allowing students to have a reference sheet on a test. Darryl Yong (@dyong, who you should follow if you're not already) said something that I 100% agree with: if I am measuring higher-order thinking and problem-solving, then THAT'S what I should be measuring and a reference sheet makes sense.

If I am measuring students' effectiveness at memorizing things (such as vocabulary terms), then a reference sheet doesn't make sense because memory and recall are what are being tested.

So right now I'm sitting in a ditch, frustrated by the fact that my Algebra 1 students have been run off the road because the current test is merely a scavenger hunt for right answers to memorized algorithms... when what I REALLY would like to be measuring and understanding is, Do my students know what to DO with their knowledge in both routine and non-routine situations?

And I hate this particular ditch.

As Beckett's Molloy says, "I was out of sorts. They are deep, my sorts -- a deep ditch and I am not often out of them. That is why I mention it."

Building a Feel for 'Major Moves' in Proof

This year I'm experimenting with developing students' intuition for and sense-making about what we call 'major moves' in proof.

Rather than ask students to buy into the illusion that two-column proofs emerge spontaneously and fully formed from their brow, we are inquiring into how we mortals can better brainstorm and use our reference tools to create the sub-assemblies that we can use to build our rough draft proofs. Then we'll be better able to polish our final proofs and present our work.

This has meant that we are developing students' intuition that that these sub-assemblies are knowable and predictable. We call these our "major proof moves." Some of our major categories of major proof moves include:

• the relationships between parts & wholes
• a sense of bisectors and "half-ness"
• parallels and the results of parallels
• perpendiculars and their results
• right angles and their results.

It's working out surprisingly well.

Today we started experimenting with using these higher-order concepts to work on harder, multi-stage proofs. The kids were quite excited to be able to figure things out.

Every year I am amazed at how many times students have to repeat an experience before they get that "click." This is giving us a much wider field to wander in as we master the art of proof.

The First-Ever Block 5 Math Department Pot Luck Lunch

Today we had our first-ever block 5 Math Department pot luck lunch.

We have a very large department (24 teachers) and not all of us have the same block for lunch. The block 6 lunchers had a pot luck last week, and so fueled by the competitive spirit, the block 5 lunchers were mobilized by one of the least social people in the department to host our own pot luck. The sign-ups were on the corner one of the office white boards, so the menu was shaped over the last week. And since we have by far the largest crew, hopes were high that we could pull this off. And we did so -- with style.

I made the Lemongrass and Ginger Roast from the Field Roast cookbook because I've been wanting to try it and we have a surprisingly large number of vegans and people with significant food sensitivities. I rushed home from school yesterday so I could make it and set it on the stove to simmer. Sarah made a salad which I had been planning to use as a base for my new favorite school lunch (chef salad surprise). Ernie made an incredible, silky hummus and pita bread. Lisa made a shrimp and avocado ceviche. Raymond made a wonderfully spicy red lentil dal, which I still don't understand how he heated up but it was delicious. Tyler brought donuts. Scott baked a boule of crusty sourdough with cheese and sausage.

Not to scale

But the runaway winners of the day -- and in my opinion, the offerings that raised this pot luck to the level of Artistry -- were Robert's mother's fried chicken wings and Alex's made-to-order waffles.

Alex brought in his waffle iron and a killer cinnamon syrup that he had discovered on the internet. His TA worked the waffle-making station at the standing desk, making waffles to order and generally supervising (don't worry, we fed her).

That's right.

We. Had. Chicken. And. Waffles For. Lunch.

In. The. Math. Office.

And fresh-baked sourdough bread. And vegan charcuterie. AND DONUTS.

Who can turn down a good donut?

At one point, Art strolled in from the computer lab and felt bad that he hadn't brought anything. Everybody jumped in and chided him, "DON'T FEEL GUILTY -- WE ARE DROWNING IN FOOD." So he loaded up a plate and joined us.


I don't know why we never did this before.

As we were winding down, Lisa said, "I kind of have a math problem I wanted to ask about, but I don't want to spoil it."

And everybody jumped in again and said, "DON'T FEEL GUILTY." She sketched the problem and most of us started tinkering on scratch paper, but as usual, Robert saw straight through to the core of the simplification. We all put down our forks and pencils in shock at his surprising yet unsurprising clarity.

Then, of course, we needed another round of waffles.

It was the best community-building activity I've ever done. And it was basically free. I hope we do it again soon because Raymond's grandmother made these killer dim sum things for our end-of-school pot luck last year that nobody knows the name of but everybody devoured them. They look like little fried footballs with some kind of mystery meat or veggies inside their cavernous pockets. And they are TO DIE FOR.

Summer of Healing and Self-Care

I am sitting on the sofa in our dining room, deep in avoidance. I need to pack for the red-eye back east to see my family. There is thick silver cloud cover all the way down the hill to the bay and across the water. I can see a freighter which seems to be anchored on the water, but the East Bay hills, which I can usually see, are completely missing. Lost in the fog.

This has been my summer of self-care and healing. There have been too many deaths in my life this year. I needed time alone to grieve and to heal so I can turn the page and carry on. Fred always said, when it gets like this, it's time to turn inward. Focus on yourself. Ask, what is the best way I can use this situation for growth?

A big part of my work is focused on equity, but it has become a much more in-person thing and less online than ever before. I became the co-sponsor of our Black Student Union (BSU) and started raising money for them and advocating, mentoring, and learning alongside them. I started receiving awards this year for my equity work in my school and my community. On my end-of-year survey for the district's BSU program, the form asked, What is something you are most proud of this year? I wrote, "I raised over $8,000 for the programs and initiatives my Black students wanted to bring to fruition." The district BSU coordinator set up a phone appointment to follow up on our questionnaires. He asked me, "How did you DO that?" I told him, "I used my privilege and my connections, and I wouldn't take no for an answer."

I held my community accountable and I started to break through. And it made everybody feel much happier.

I won some actual awards for my work, but I don't like to focus on that. The actual ability to do this work is its own reward. I got some wonderful nods for an NPR commentary I did about what I am learning. And I realized I have found my path.

It was hard losing TMC because it got caught up in ripcurrents that were only partially about what they claimed to be about. That left an emptiness in my calendar in late July, but the harder thing was the loss of friendships I have cherished and come to rely on. I had to pack away gifts from former friends, wrapping them in tissue paper and tucking them into the back of a cabinet. It made me too sad to look at them. I received more hate-tweets and actual hate mail than I'd ever thought possible. I discovered Twitter's block function, but I learned it is only a mesh, not a wall. Hate still gets through.

I am learning how to develop a thicker skin.

I poured my energy into building our garden. Nothing heals like growing things. I spent days and weeks on end building and learning. It's become a sanctuary, a sacred space for growth. Now I'm studying hydroponics so I can grow lettuces and basil and tomatoes in the garden. Whole new parts of my mind and body are coming to life.

Now I'm headed east to visit family and to spend time down the shore, as we say in South Jersey. The garden is my West Coast bracket for healing. The warm waters of the Atlantic are my East Coast bookend. Connecting with family gives that healing burst of nutrients, and with my feet in the warm waters, I find nutrition for my next chapter of growth.

I think I will create a living wall in my classroom this year. A living wall and a Peace Corner. Some things are more important than math.

How I Started the Lowell Faculty Seminar on the Climate Emergency (and how you can too)

Towards the end of school, there's the narrowest of windows between the end of AP testing and the beginning of final exams when teachers at my school can sort of catch our breath. It's a week or two of relative down time for review and synthesizing and studying which reminds me of my beloved Reading Period at Princeton.

This year, I decided to put a stake in the ground and create an event so that our faculty community can make better use of this moment to collaborate in an interdisciplinary way on how we teach and learn about the climate crisis.

I have been reading and thinking about Project Drawdown's web site and NYT best selling book Drawdown, so I wrote a grant for a pilot project to get fifteen of our 200 faculty and staff together to help each other integrate climate change into our teaching.

The response was tremendous! So I wrote this blog post as a guide for myself and for anybody else who is like-minded about how I organized this faculty-led PD program -- and how you can do this with your faculty too.

THE LAUNCH
In May, I ordered 15 copies of Drawdown and sent out this e-mail blast to the faculty inviting them to the Lowell Faculty Seminar on the Climate Emergency:

Hello colleagues -- 

   A new NPR/Ipsos poll shows that 86% of teachers say that students should learn about climate change... but only 42% teach it.  

   Because of this, it seems like the #1 thing we as a faculty can start doing about climate change is to talk about it -- and help each other understand it better.  

    So if you would like to read, learn, & discuss climate change with other Lowell teachers in a friendly, interdisciplinary way, then please join us (K__, C__, and E__, to start with) in the new Lowell Faculty Seminar on the Climate Emergency (aka the climate change book club).  

   SO, AFTER AP EXAMS -- To start things rolling, we will read and discuss Project Drawdown's NYT best selling book, Drawdown: The 100 most comprehensive plan ever proposed to reverse global warming: 

   This is a global problem so all are welcome! 

   Our short-term hope is to develop a common understanding, vocabulary, and framework for talking about climate change among the faculty. Our longer-term goal is to evolve some form of ongoing PD to help us integrate climate change across the curriculum. 

   GOOD NEWS! Thanks to A__'s and S__'s expert budget planning, we have 15 copies of Drawdown for teacher-participants who would like to join in this effort.
Please reply to this e-mail and let me know if you would like to join us (and if you would like to reserve a book).  

Best,
Elizabeth (Dr. S -- room 274) 

I received about 20 yeses, of which 15 turned out to be viable. During my prep blocks, I went around distributing books and introducing myself to staff across the faculty and across the campus who were interested. It was great to meet colleagues in English, Social Studies, Ethnic Studies, Peer Resources, World Languages, Academic Counseling, and Physical Education who were thinking about the climate crisis too, in addition to us nerds in math and science.

We used a Doodle poll to find an available after-school hour and I sent out a message confirming time and place and logistics. Everyone committed to reading the three introductory sections of Drawdown in preparation for our kickoff meeting.

OUR FIRST MEETING
The first meeting focused on introductions, intentions, and the basic science of the climate crisis. Our goal was/is to map out topics, strategies, and projects on which we could collaborate over the next year in our work together. Many other teachers who were not available for our kickoff meeting also expressed interest in these conversations over the next year.

In an interdisciplinary context, it felt essential to make sure that everybody -- including colleagues in non-quantitative fields -- felt equally empowered by the science rather than intimidated. I served as the seminar leader and ran the agenda, even though it was intimidating to do so in front of our two National Board Certified AP Environmental Sciences teachers and a number of other accomplished science teachers.

One benefit of my being the one to lead the meeting -- as a math teacher, meditator, writer, Black Student Union advisor, and member of the school's equity leadership team -- was that I could bring slightly different lenses to the process of discussing the climate.

AGENDA

I. OPENING 
I asked for the group's indulgence to begin with a slightly different opening ritual than we usually use at our PWAI (Predominantly White and Asian Institution)  -- a LAND ACKNOWLEDGMENT.  We stood and I read this draft of our statement:

Before we begin...

We pause to acknowledge that Lowell sits on the traditional lands of the Ohlone people, who are the indigenous stewards of this beautiful place we teach and learn on. We acknowledge their ancient and federally unrecognized claim on this land, their enslavement in our city, and our own uncomfortable role in this history. We do this to pay respect to all Ohlone people, past, present, and future.

This led to an amazing and heartfelt discussion. We talked about the fact that we have Native Americans in all parts of our school -- on the faculty, in our staff, in our alumni community, and in our student body -- and that this is something we need to acknowledge in order for everybody to feel seen and included.

The faculty of color at our meeting expressed surprise that they had not previously experienced this kind of ritual at an official school event, and everyone expressed interest in having us integrate these acknowledgments into our future events as well.

II. INTRODUCTION TO THE SCIENCE 
For many of us, it has been years since we've formally studied the science of climate change, so it seemed important to set our initial level of understanding. I used this video by noted climate scientist and explainer, Katharine Hayhoe, of Texas Tech University to kick off our discussion of the science:

We followed this up with an open discussion as well as questions from the non-science teachers. The "1.5 to 2 degrees" limit proved to be a very fruitful hook which everybody was eager to use in their pedagogical thinking.

III. THE LANGUAGE OF CLIMATE CRISIS 
A big part of introducing the climate crisis into our pedagogy involves using current and appropriate language for framing our discussions. This brief opinion piece by the Environment Editor of The Guardian newspaper gave our English teacher member an on-ramp to lead the next part of our discussion:

https://www.theguardian.com/environment/2019/may/17/why-the-guardian-is-changing-the-language-it-uses-about-the-environment

What role do our language choices -- both conscious and unconscious -- play in how we teach and learn about the climate crisis? Is it sufficient to call it "climate change" or is it more accurate to use terms like "climate emergency"?

T__ was particularly interested in using this op-ed with her Critical Writing classes, although everybody wanted to do more thinking about the effects of the language we use, so we stuck a pin in this as a topic we would like to collaborate on over the next year of our meetings.

IV. DEALING WITH CLIMATE SCIENCE DENIERS 
Even though we teach and learn together in liberal San Francisco, which is an admittedly friendly context for mobilizing about science, we all admitted frustration over the frequency with which we encouter climate science denialism.

I introduced the free, self-paced MOOC (massive open online course) I have started taking from edX, offered by the University of Queensland in Australia, called Making Sense of Climate Science Denial:

https://www.edx.org/course/making-sense-of-climate-science-denial-2

In particular, their chart on the five characteristics of science denial found immediate fans and a home in our soon-to-be-revised Critical Writing course:

Several colleagues promptly expressed their intention to take this course over the summer.

V. USING THE NEWS AS A SPRINGBOARD FOR INQUIRY 
The New York Times provided a timely article from the previous day about how the Trump administration is doubling down on its attacks on climate science:

https://www.nytimes.com/2019/05/27/us/politics/trump-climate-science.html

 This prompted a discussion about how we can integrate current news stories into our curriculum.

VI. CLOSING ASPIRATIONS & NEXT STEPS 
We set our intention to meet monthly during the next school year and also to set up an e-mail alias for ease of communications.

Over the next two weeks, I heard from so many participants and soon-to-be participants about how valuable they felt this was. It really drove home the message of Katharine Hayhoe's viral TED talk: that the most important thing we citizens -- and particularly teachers -- can do about the climate emergency is to start talking about it.

Trust is built in very small moments

D stopped by my room after 7th block. He came by to just hang out before our BSU meeting. Usually he drops by for a quick hello during passing period to work on my handshake. Spoiler alert: I'm still terrible. Today, he just felt like hanging out. He kept me company while I purged papers from giant piles on my desk.

D is a big, handsome, talkative, brilliant and witty young guy, a very strong student, solidly built, with dark skin, a ready laugh, and the brightest black eyes I have ever seen. He asked me how I liked his new twists, tilting his head so I could get a good look. I liked them a lot. He explained the process of setting them up and caring for them. His girlfriend really likes them.

We just shot the breeze about everything and anything in the late-afternoon light. He asked my advice about two gift options he was considering for his and his girlfriend’s three-month anniversary. He showed me some pictures on his phone and I gave my opinion (I liked both, but had some thoughts). We talked about climate change, the school-to-prison pipeline, manners and the lack of manners, Flat Stanley (“Flat who?” I could see the wheels turning in his head. Finally he quirked an eyebrow at me and said, “White people have some crazy-ass ideas about education”), course selection for the fall, summer plans.

It felt like such a blessing.

I never taught him as a student, so it felt like an extra helping of miracle for us to be just sitting there in my classroom after school, hanging out.

One of the books I’ve been reading in my personal school equity work this year is Dr. Joy DeGruy’s Post Traumatic Slave Syndrome: America’s Legacy of Enduring Injury and Healing. I have found it most helpful in understanding the legacy and issues that come with this transgenerational trauma. Much of the current equity focus among teachers and teacher-educators in math education has been from a sociological lens, which I honestly have not found that helpful in addressing the systemic issues in my teaching life and in our school. I come to my teaching work from a more psychodynamic point of view. It’s not that I don’t appreciate the sociological work; I do. I just find it the psychological perspective more clarifying.

The most important learning I’ve had this year came from a passage in Dr. DeGruy’s book. The section on the preeminence of relationship in Black culture really made an impression on me and has been foundational. She writes, “In African American communities, relationship frequently trumps everything else. Consideration of relationship permeates all of our interactions. For example, when Black students feel they have been disrespected by a teacher, they often feel completely justified in rebelling and shutting out the offending teacher, even if it means failing the class and sabotaging academic aspirations “ (p. 19).

This makes enormous sense to me, especially in light of what I know from the work of Drs. John and Julie Gottman in their work on repairing and rebuilding marriages. Trust is the foundation of everything in a close relationship, and as John Gottman says, “Trust is built in very small moments.” So this has become my mantra in all my equity work this year at school. Without trust and relationship with my Black students, there is nothing.

It doesn’t matter how many book chats I do on Twitter, how many times I get called out for my own internalized racisms and make changes, how many times I support my BIPOC adult colleagues. What matters for my equity work with my students and with my school is how much trust and relationship there is in our shared well.

D and I ran the backstage happenings at our Black History Month assembly in February. Actually, he was the stage manager and I was his assistant. What truly mattered, it turns out, was the fact of weaving that relational web together.

I noticed the time and said, “Hey, would you help me close the windows? It’s time to go to BSU.”

He took his BSU jacket out of his backpack and carefully pulled it over his head. Then we flipped off the lights and headed down the hall together to our meeting.

Dismantling Privilege Up Close is Personal, Part 1

How do my teaching and learning practices as a white, middle-aged, math teacher with a great deal of privilege either support or disrupt the unconscious processes of systemic racism that underlie every aspect of my classroom?

At my very large, very diverse, high-achieving, high-poverty urban magnet school, this has been a question each of us 180+ teachers has been investigating daily, both individually and together, over the past three years.

We just finished Year 3 of our five-year program of anti-racist transformation and I tell you, it has been (and still is) grueling. It is exhausting to inquire into the tiniest corners and crevices of your teaching practice to root out systemic oppression. Looking into the mirror of your practice like this every day is just draining. But the only way out of this systemic mess is through, and so I slog on.

Year 3 of our guided process has focused on pedagogy and personal practice and transformation. Every teacher is part of a 10-15 person interdisciplinary inquiry group. The group I am in includes teachers from math, English Social Studies, world languages (Chinese, Japanese, Romance Languages), Special Ed, orchestra, English, and PE. This has been a deep year as we have begun to really get at the in-class and in-person ways in which white supremacy, privilege, and the systemic parts of systemic racism manifest in our classrooms, in our school/community, and in our personal teaching practices.

Being able to talk in a trusted and trustworthy way about our distinct experiences with the same kids across different parts of their academic and social experience has opened a lot of minds. It's like really seeing the differing facets of the same crystal in the light. It has also honed our individual and collective abilities to see the systemic parts of our racist culture playing out in our personal lives and practices. Nobody just yells out, "See? White privilege!" any more. We notice it more quietly, more deeply. The learning grinds away at us.

One of the most recent news stories around all this that is still playing out on the national stage has been the college admissions cheating/bribery scandal. This is only the latest and most publicized instance of how privilege and white supremacy in our culture of systemic racism are operating, but it has affected me and others in my group profoundly. It has focused our attention on the issue of opportunity hoarding -- the many ways in which privilege reproduces itself by taking advantage of leaks in the educational system which can be turned to the advantage of the privileged classes.

I need to add in a clarifying note here. The dominant culture at my school has its own unique flavor and manifests in ways that are uniquely Californian and uniquely San Franciscan. At 41% Asian and 23% white, our dominant culture is a strange fusion of Asian and white cultures. We already are a "majority minority" institution. But we also reflect the unholy history of racism in California and San Francisco -- a dominant culture that codified its anti-Black and anti-Asian racisms in law during the years leading up to the Civil War and statehood. The anti-Black, anti-Asian, anti-Native American, and anti-Latinx racisms we have inherited were institutionalized in a racialized way in the California Constitution. And they were crafted with intention to pit racialized minority groups against one another in an ongoing battle for "second place" after whiteness. My school was founded in 1856, in the middle of all of this, and our culture today retains many of the structural and institutional traces of the pre-Civil War and Reconstruction-era racisms that continue to impact our school, city, and state to this day. In a nutshell, Reconstruction-era California was really weird -- and completely distinct in its inheritance of Jim Crow-style anti-Black racisms from the Southern states and of aversive-style anti-Black racisms from the Northern and Midwestern states. California's forms and history of systemic racism are truly weird and unique. If you are interested in learning more about this, I recommend the late D. Michael Bottoms' history, An Aristocracy of Color: Race and Reconstruction in California and the West, 1850-1890.

I say this because when I talk about the dominant culture at my school, I am not just talking about the kind of white supremacy I grew up with in New Jersey. What we experience here daily at my school is a direct legacy from the free-soil movement before the Civil War, as well as the segregation of Chinese and Chinese schools into the largest Chinatown outside of Asia, plus Reconstruction, the Chinese Exclusion Act, and every other toxic legacy that white migrants brought with them from other parts of the country. Black students and families were integrated into our school long before Asian students were. In fact, Black students were first integrated into our school in 1875, then re-segregated and re-de-segregated multiple times before the Civil Rights era. When I look at our ancient yearbooks, I see Black faces, Asian faces, and Native American faces sprinkled in for years at a time, and then suddenly disappeared. I can see the pitting of racialized minority groups playing out across the pages of our historical yearbooks, working backwards into the late 19th century and probably into Reconstruction and the Civil War era.

What I'm trying to say is that my school's dominant culture of Asian and white cultures pitted against Black, Native, and Latinx cultures to compete for status and resources has a long and well-documented legacy. At least our unique dominant culture comes by its deep and toxic weirdness honestly.

So when I talk about my school's dominant culture, I am speaking of a terrible alliance of legalized striving in California that was put into place to put racialized minorities at odds with each other as they fought to win second-place in an unholy alliance with the white power structure.

My school's legacy of this strife in is that our dominant culture reproduces this alliance between Asian and white cultural groups. Our district takes off both Christian holidays and Lunar New Year, although lip service is paid to Native cultures through the renaming of Columbus Day as Indigenous People's Day. But the power structure strongly reflects this alliance of white and Asian cultural groups. Majority rules with a velvet gloved fist.

Asian cultures in our school retain a lot of their unique distinctions. There is a status hierarchy of cultures that I have learned about. But the power structure is definitely a streaming arrangement, with the majority Asian community flowing in unquestioning currents with the white community through an unspoken power-sharing dynamic.

All of this plays out every day in my classroom, in our hallways, in our PTSA, in our alumni association, and in our school district.

I could not have explained any of this four years ago because I had only the shallowest understanding of it. But now I see it operating everywhere I look.

What I am still learning from Twitter Math Camp

Here's what I have learned and am still learning from the birth, life cycle, and death of Twitter Math Camp.

1. Begin with an abundance mindset. See your own blessings. 

2. Create wonderful things.

3. Don’t be afraid to create new wonderful things.

4. Share them. Be generous. Give without expecting anything in return.

5. Search for what you need/want.

6. If you don’t find what you need/want, reach out.

7. If you still can’t find what you need/want, create it.

8. If you get stuck, ask for support.

9. Try your best to avoid giving in to negativity. 

10. Don’t let other people’s scarcity mindsets stop you from creating wonderful things. 

11. Don’t be afraid to get criticism, but don’t let it shut you down either. People’s opinions are just their opinions. Even if they are opinions from the most righteous and celebrated human beings in our world, they can still be wrong. In no case were they carried down on stone tablets from Mount Sinai.

12. If people insist on killing off what you have created or loved, grieve your losses; then dust yourself off and create something new.

13. Remember to check in from time to time with step 1.

14. Remember that all organisms have a natural life cycle. Things are born, they flourish, and they die. It’s not the end of the world. Something new will arise in its place.

15. Keep creating new and wonderful things that nourish your soul until we finish healing the world.

Make It Stop -- Toward a Conservatory Model of Math Ed

I've been thinking a lot about why I hate the distinction that gets made in math education between a so-called "conceptual emphasis" and a so-called "skills emphasis."

Maybe it's because I grew up with a conservatory model of musical performance education, but after 10+ years of keeping quiet, I'm ready to propose a different distinction.

In a conservatory model of musical performance learning, there are multiple different dimensions from which a learner approaches their learning. The basic dimensions of focus are:

Repertoire: There is repertoire for your instrument or ensemble (the composed or improviseable pieces of music which the student is learning to play and/or perform. There are compositions for every level of player: as a very young child, I played pieces from Bach's Anna Magdalena Notebook that were appropriate for beginners because the technique they required was accessible to my level of technical and interpretive skill. More accomplished players might work on pieces from the Two- and Three-Part Inventions and even up to the Goldberg Variations and beyond.

Technique: A student also learns how to do focused regular work on technique, without which one lacks sufficient skill to support the playing of the repertoire. A steady diet of technical work (scales, arpeggios, Hanon exercises, etc) would probably lead to world wars because taken alone, these are boring. But they are the skills out of which we build our playing and understanding. No one wants to hear me play the Goldberg Variations. My technical skills are just not there. However, they are quite sufficient to not cause a casual listener pain while I work on Two-Part Inventions.

Music History & Theory: We study the context of the music we play because it gives us important insight into the composer's thinking. Understanding contemporary preferences, styles, beliefs, and historical context enable us to make some better sense of the pieces we play.

All three of these strands help us to make meaning & sense as we play, perform, practice, teach, and learn together. These are not the extent of everything we do in a conservatory model of teaching and learning, but they feel like a reasonable basis for comparison.

My Essential Question is: Why do we not take this kind of woven approach in math ed?

Modeling and problem-solving are the heart of our "repertoire." I want my students to be able to think about how to solve problems which may be "real-world" or not, but they are authentically problems that require thinking, activation, and transfer of prior learning in a new or novel way.

In music education, technique is the difference between musical performance and music appreciation. If I'm taking time to further develop technical skills that will enable me to access (i.e., to perform) more complex compositions, then am I not weaving together both conceptual and procedural forms of learning?

And if I'm simply listening and understanding musical compositions as a listener, then I am definitely accessing the concepts to be sure. But I am not at that moment engaged in the struggle of problem-solving/mastering/practicing/performing that composition. It's not that this posture in this moment doesn't inform my generative/productive performance/practice/learning as a performer. But I'm not acting in the role of performer in that moment. I'm acting in a receptive capacity.

By the same token, I have many times in my life had a major light-bulb moment while doing warm-ups like Hanon and realizing that THIS piece of technique can be used to improve my performance of THAT compositional passage. This is an authentic moment in which practicing skills can lead to true conceptual and performative insight.

So when I read Dan's latest blog post about how everything is modeling, that feels as true to me as saying that everything is based on skill-building. The kinds of mathematical thinking I want my students to be able to access includes powerful, flexible productive/generative mathematical modeling as well as sensitive and receptive mathematical listening/reading. I don't want my students to JUST unthinkingly repeat skill practice, but I also know that without a deep and flexible number sense and other forms of fluency, they will be cut off from the kinds of problem-solving I value most for them.

This is the question of "access" that I truly wrestle with.

I'm wondering if anybody else thinks of the nexus of conceptual understanding and procedural fluency in a related way.

Dreaming is Hard, But It's the Only Way to Make Something — BSU Design Thinking, Part 1

This is the first of a series of posts I am writing for myself so that I can remember how I've been using design thinking in my work with our Black Student Union (BSU) program.
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One of the things Steve Jobs always valued about me was my ability to mobilize a huge group of people to do impossibly large things. In college, I ran an opera company. In the 1990s, I started NeXTWORLD Expo (which signaled the inflection point that would eventually lead NeXT to save Apple). In the 2000s, I was a co-founder of a women's wilderness retreat. And in the 2010s, I helped to start Twitter Math Camp.

Now I am one of the faculty leaders of the equity steering committee at my very old, historically entrenched, very large, racially and ethnically complicated academic magnet school. Three years ago, we embarked on a five-year process of school culture transformation to make it a more welcoming place, both for our students of color and for our faculty and support staff of color. Our students' families and communities are engaged, our PTSA is an active participant, and our 40,000-member alumni association is also involved. We are up to our eyeballs in the process of waking up to the toxic soup of systemic racism that we all swim in. I have often used the Zen metaphor of fish not noticing the water that they are swimming in, but I've come to realize that systemic racism is a polluted condition of that water. Our goal is to learn how to notice the pollution and to stop polluting the water further. We need, as the Tao Te Ching says, to let the mud settle and the water run clear.

The goal is not only to disrupt the trance of inequities that we all walk around in; the goal is also to  create a healthy and sustainable school culture in which learners from marginalized communities are entitled to thrive.

The progress is slow. The work is grueling. But there is also a lot of heart in our community, and sometimes that opens up an occasional moment of grace. So when our Black students told us what they needed, I took them seriously.

Raising money and helping them organize to achieve their deep dreams are things I know how to help with. So back in September, I got busy doing what I knew how to help with. The BSU officers and I met in our classroom to work together on envisioning and fundraising and mapping out our plans.

The envisioning is always harder than the fundraising, but when you are discouraged, it looks like the opposite is true. That is part of how the status quo and the power structure maintain their hold. The status quo wants you to believe that money is the problem, but I'll let you in on the secret: money is almost never the problem. The reason nine out of ten start-up efforts fail is not for lack of money; they fail for lack of imagination. This is one of the illusions that Design Thinking can help thinkers to break out of. The imagining is the hardest part because it requires us to go against the psychological and emotional defense mechanisms that keep us locked in our "safe" but disempowered crouch. They keep us thinking and dreaming small, when what we really need to do is to take the risk of thinking and dreaming big.

So the BSU officers and I engaged in a first draft envisioning and imagining conversation. This was some of the most exciting but complicated — and emotionally exhausting — work of my teaching career so far.

"What do you want to accomplish?"
"We want jackets."
I scribble on my list. "Done. What else?"
They were stumped.
"What else?"
"But we can't afford jackets. We need to have more bake sales."
"Don't worry about the money. That's my job. What do you want to  do??"
They thought for a minute.
Um, maybe a camping trip."
I write this down. "Good. What else?"

For thirty minutes, I pushed them and insisted they just toss out crazy ideas. No self-defeating talk about implementation, just make a list. Brainstorming rules. No judgment. Don't think.

And they starting coming up with ideas. College visits. Service projects in their neighborhoods. A Senior Showcase. An assembly.

"Good, good, good. What else?"

Every time they used the word "But..." I cut them off. "We're not talking about that right now. We're talking about deep dreams and making a list. What else?"

It was bewildering to them. They've been conditioned not to dream and I was prodding them to reconnect with this basic human capacity.  Honestly, I don't know if we could have gotten anywhere if there had not already been a years-long foundation of mutual trust woven between and among the five or six of us. They know I'm a little nuts, but they also know that they can trust me. They know that when I screw up, they can call me on it and I will own it and apologize for it. We've been blundering along together for years. I taught all of these girls as freshmen and sophomores, and I've been mentoring them, tutoring them, coaching them, writing letters of recommendation for them, and pushing them to reach high for all the time they've been at our school. And now they are on their way to becoming persons of power — the scientists and artists and politicians and engineers they are determined to become.

But the academic stuff turns out to be the easier piece. Dreaming is a different kind of path, and it's one that usually only the kids with privilege get mentored into. I was breaking this boundary, and it felt dangerous. That is almost always the way I know when I'm doing the right thing. As George Lucas once said, "When people tell you it's impossible, you're on the right track."

Once we had a huge list, we did some analysis on it. What are the must-haves? What are the nice-to-haves? What are the pieces we can live without if we have to? We prioritized. We negotiated. We gave things up; then we put them back onto the list. We organized them into categories. We identified the critical path. We flagged the dependencies. "This thing has to come before that thing. This thing can't happen unless that thing happens first."

It took a long time to get them into flow. There is a suspension of disbelief that has to happen during the envisioning phase. Otherwise nothing happens. They started getting involved in the process. Their body language loosened up. They leaned into the discussions. They started to lose their inhibitions about jumping up, grabbing a whiteboard marker, and drawing a matrix or a diagram on the board. Developing the comfort and safety and confidence to break the rules of compliant and oppressive forms of discussion is a giant step towards true empowerment. We began to make progress.

Finally, we got to a place where we could begin packaging up what we had thought of and started productizing it. Our first "product" would be a Black History Month program for the school. With each step, we asked ourselves, How will students benefit from this? How will the school benefit? What are the tangible and intangible outcomes? How will we be laying the groundwork so that future cohorts of the BSU will be able to replicate this program and grow it over time?

We started building a grant proposal draft and a spreadsheet on our BSU Google classroom. Parents requested permission to join the documents and spreadsheets and chats and I was thrilled. Request for access messages started popping up and I approved them as fast as they came in. Click. Click. Click. The parents mostly lurked and marveled at their children's boldness and imagination. They expressed their excitement and gratitude at Back To School Night, but I turned it right back on them. It was their children doing the amazing stuff. I was only the facilitator.

The kids occasionally panicked. "I don't have time to work on the grant proposal! I have a chem lab due and a mock trial event in LA and a basketball game on Wednesday. I don't know how this is going to get done."

And I did what I had been trained to do as a manager of large teams. I pitched in. I did whatever parts of the work I could help with. "Don't worry about it. I'll write a draft and people can edit it in the Google doc. We'll make it happen. This is the power of teams."

And we did it. In the end, I wrote the grants for their ideas. I interfaced with the power structure. And I got them the money.

Then the real work could begin.

If you believe you have achieved escape velocity from white fragility yourself, you are doing it wrong

Only by being ruthlessly honest with what we have internalized can we confront this problem.

We need to reimagine our whole-community conference from the ground up

We are trying to squash a new paradigm (a teacher-driven, free of charge, math ed professional development conference that locates equity at its center and foundation) into a too-small and outdated structure (TMC). We need to understand more about how the existing structure works so we can create a better plan.

We are still not anywhere near the point of identifying or understanding the unconscious, unspoken, equity-blind, harm-causing assumptions in TMC's structures. We're just not. This needs to be a whole-community effort. We're trying to set new goals but we're still walking around in the same old consensus trance.

There is no way that a small group of people could tackle this problem successfully. It's not realistic. It will take everybody in our community to surface and and interrogate the hidden assumptions in our structures. The problem with blind spots is that they are blind spots. If we don't work together on surfacing and transforming our assumptions, we will continue to just tinker around at the margins and that's not going to be satisfactory to anybody. It is also not going to unleash the liberatory potential in a a whole-community effort.

Here is an example of what I mean.

VERY SMALL-SCALE EXAMPLE
One of the toxic assumptions I stumbled on in my own classroom lately and am ruthlessly rooting out is the notion of individual attainment — the idea that each person has their OWN learning/mastery that is unconnected to the learning of anyone else in the room. This gives rise to a toxic kind of individual competition that is antithetical to sane and healthy learning for all.

This is the assumption that, once "I" understand the concept and "finish" an "assignment," then I am "done" and am on the hook for nothing more, leaving me free to play video games or do some other work.

But what if we were to start rejecting the underlying assumption(s) of individual attainment?

What if the goal were for the WHOLE CLASS to achieve fluency in a particular concept or skill as best we can, rather than for each individual to do it as best they can?

How would each of us teach, learn, listen, collaborate, help others? How would that change what I/you/we want? What would I/you/we need? What I/you/we would give?

What I realized in this thought experiment is that when grades are given for individual attainment, that is based on a socially Darwinian set of assumptions in which everyone arrives in my classroom with the same levels of everything. Every kid for themselves.

Once I saw this assumption, I couldn't un-see it, and I also couldn't help but challenge it. I am responsibility for every student in my room; therefore my goal needs be to get everybody over the finish line, right?

That forced me to look at the base-level operating assumption about individual attainment as the only measure. What if I were to change our goal to being one of, nobody wins unless everybody wins?

As I saw in the episode I wrote up on my blog, the kids took to this like ducks to water and our class averages were in the 90s rather than in the low 80s to high 70s.

IMPLICATIONS FOR A WHOLE-COMMUNITY CONFERENCE
How would we — each and collectively — think differently about our whole-community conference if equity were to be the absolute bedrock foundation upon which we were committed to build?

Wouldn't we have to start by analyzing together what the unconscious assumptions in our existing structures are?

Walking in the World With a Broken Heart – A Love Letter

Before I came back to teaching, I spent 25 years starting software companies in Silicon Valley. Starting organizations is a whole unique area of expertise. There is a huge amount of institutional knowledge that sprouts up about the life cycle of organizations. Newborn organizations become toddler organizations and if they survive, they become children and teenagers and adults.

Organizations are born, grow, split off, spawn new organizations, die, get acquired, fail to get acquired. Sometimes organizations become zombie organizations. A very few organizations get big. Some organizations stay the same size. Some organizations shrink. There are as many ways to thrive as there are to die.

In the corporate world, there are definitely different kinds of people who find they are suited to different kinds of organizations: there are start-up people and there are big-company people.

It takes a whole different skill set to start and run a start-up than it does to work in or run a large organization. The amount of infrastructure in a large organization is really impressive to me -- even in a poorly organized or run large organization. Anybody who has ever worked in a school district knows what I am talking about. There is a form for everything, a department for everything, a mission statement for everything.

Start-ups are completely different creatures. In a start-up, there's no infrastructure unless you create it. When you finally move a start-up organization out of the corner of your bedroom or your basement or your garage, you start encountering questions like, Where do ISBN numbers come from? Where do conference tables come from? How do boxed products get assembled and shipped? How do I calculate and submit sales tax to the state? Where does liability insurance come from?

I have always been a start-up person, descended from a long line of start-up people. Entrepreneurship is in my DNA. As I've done genealogical research on my family, it has been fascinating to discover that in between being attacked by pogroms and ethnic cleansing, my people as far back as the 1860s in Elizavetgrad in the southwestern Russian Empire were starting businesses and growing them and spinning off new businesses from old businesses.

So when we decided to start TMC almost ten years ago, it struck me as extremely exciting but also no big deal. I had done this kind of thing six or seven times before, plus I'd been watching my family members do it for generations. As Paul Hawken, the great environmental entrepreneur and author writes in his landmark book, Growing a Business, you don't start a new organization to overtake the competition. You start a new organization because there is no existing organization that exists to meet an identified need that is going unmet.

We started TMC because we felt like isolated individual math teachers who wanted to connect with other math teachers for free to find ways to improve our teaching practice.

And it's been amazing. My closest math teacher friends are people I would never have met otherwise. They come from all over the planet. They do not look like me or teach like me or have backgrounds like mine. And yet, I know their hearts in small ways and they know mine, and I love every single one of them dearly.

Starting a new organization is filled with risk. You are throwing in your fortunes with people you really don't know. What I think of as The Great Facebook Friending of December 2011 felt like a huge risk because it was. Everybody I know said, ARE YOU CRAZY? YOU ARE GOING TO MEET PEOPLE YOU MET ON THE INTERNET? HAVE YOU LEARNED NOTHING ABOUT DANGER IN THIS WORLD?

But I didn't care. I knew what the risks were, and I knew that I might need to go into Facebook Witness Protection if the 25 people I had just friended turned out to be psycho-killers.

They weren't. And I've never regretted it.

But something I learned by growing companies is happening now. within the TMC community and organization. If an organization succeeds, its mission starts getting stretched. People of good will have different ideas about what it means to live out the original mission.

This is normal. This is organizational growth. Sometimes it is possible to expand an organization's mission to meet differing needs. Sometimes it isn't. And that's when new organizations get born.

I have been wondering if this is why I feel less freaked-out about the evolution that is happening than other people do. I feel sad to see people I love leave this project I am committed to, but I'm also very used to seeing people I love leave my project to go off and start some other project that is going to be its own powerful, amazing force in the world.

The one thing I know about start-ups — and big organizations too — is that you can't just give them a personality transplant. A new organization is often needed to meet a need that is not being met by an existing organization.

It's important that we not view this as a problem. This is a great opportunity for our community.

There is so much institutional and organizational knowledge now in our MTBoS community about how to start, run, and grow a conference and a community that I see this as an enormous moment.

I am very sorry that people I admire and respect feel like they cannot live out their missions about how to create a teacher-led math conference/community focused exclusively on equity, but to me, it is very exciting that these incredibly creative, energetic, and dedicated teachers are spreading their wings and committing to figuring out how to bring their ideas about what they should look like to fruition in the world.

TMC is TMC. Something new will be something new. This is what exponential growth looks like. Today we are exploring the doubling function in our connected educator worldwide teacher community.

In the Tibetan Buddhist tradition, it is said that when a lot of things start going haywire all at once, it is because something large and beautiful and powerful is trying to be born.

This is what I see happening here. It doesn't work to try and do a brain transplant into TMC because nobody knows how to do that — and it's just not how organizational development and growth work. It just isn't.

But something new is in the process of being birthed, and I see it as a "both / and" moment.

The world is an enormous place and our country needs every kind of lens we educators can bring to it. And we have no time to lose.

I am excited to think that there could be TWO or more teacher-founded math conferences in the world started by people I love and admire and feel awe-inspired to teach and learn with.

Please let us know how we can help. My heart is with TMC from the start, but as you sail off on your next adventure, please know that I will be that tiny figure on the shoreline, waving and cheering for your mission and wishing you every possible success.

Growing Pains

The TMC leadership team just did a very useful blog post clarifying how they processed proposals for TMC19.

I found it extremely helpful in sorting through my own understanding of the confused and confusing messages in the response notices this year. I appreciate the leadership team's commitment to being accountable for its own stuff.

We are all on this journey together. We need to hold hands and stick together. #OtterNation

 I do think there was some inartful language in the acceptance letters — and I believe it caused more distress than was necessary. Communication matters, and sometimes we just get it wrong.

But this is why we need to be — and remain — in dialogue in this work. Alliances across difference are hard. They are impossible without receptivity on both sides of the dialogue.

Everyone in this community knows what to do when a lesson bombs. You analyze what happened, reflect, and try again. This is how we develop resilience.  This is how we improve our knowing.

We will all do better next time because all human beings are better than our greatest or our least. And this particular group of human beings is committed to doing things better, and to doing this work together.

Our community is rare because it is willing to accept feedback, to adapt, and to grow. Let's agree to do that now. It is natural — and necessary — to experience discomfort as we learn and grow. This is what our students experience. It is inevitable that we will experience it too.

This is one of those moments when we have to make a decision about how to love this community and our world.  My hope is that we can take what we know about powerful teaching and learning and use it to make TMC the kind of reflective, resilient, transformational community we envisioned right from the start.