Rooting Out Opportunity Hoarding and Perverse Incentives in the Math Classroom

tl;dr   The incentive structure of the math classroom is broken.

I live and teach in a community where opportunity hoarding is rampant. Students hoard points as if they were drops of water in the desert.

This leads to some perverse behaviors in the classroom. Students who have mastered a task or level want to take their attention to other parts of their lives. Their attitude is, I finished MY work; therefore MY obligations to math class are done. Students who have almost mastered a task or level of a topic become demanding of my attention in infantile ways. As soon as they run out of ideas, they tug on my sleeve, demanding that I re-teach them (or re-re-teach them) individually or in small groups. They value productive struggle only up to the point where they get stuck. The most challenged students feel so ashamed that they don't even know how to get started or even minimally unstuck that they try to hide in plain sight.

In a word, the incentive structure here is truly broken -- and perversely so.

I believe this is because the incentives here are all based on an assumption of individual attainment.

To allow a culture of individual attainment (what score /grade/mark did I get?) is to be complicit with the toxic culture of opportunity hoarding that pervades our whole society. I believe that the drive to hoard opportunity is one of the most powerful factors underlying the culture of systemic racism and oppression in schools.

Dylan Wiliam talks about how feedback needs to be more work for the recipient, yet every working classroom teacher I know knows that you can't force a kid to read or digest the comments. This is especially true when you have massive classes. With 37 kids per class, it's just not feasible. Kids look at the score and move on.

In my view, this is because the incentive structure of the math classroom is wrong. Not only is it wrong, it is sick and toxic. And we need to rethink these incentive structures if we truly want math class culture to heal.

If my grade means I personally have mastered or not mastered a topic, then once I get the score I want, my job is 100% done.

My problem with this is that, from the societal perspective, that is not my job as a classroom teacher.

My job as a classroom teacher is to get everybody over the finish line at the highest possible degree of mastery. For this reason,  my classroom's economy of achievement needs to become more collective, and less individual. I need to cultivate an incentive structure of positive interdependence -- "I" don't win unless others win too. Then we all win together.

There are times in my room when we're 37 individuals and there are other times when we are one classroom community. This is how things work on teams and in organizations throughout one's life in the U.S. So if we're one classroom community, then we need every individual to be as empowered as possible to achieve at the highest possible level.

For this reason, I've been expanding my whole-class skills quizzes. For a compound, complex skill such as solving a multi-step special right triangle problem (with interdependencies along the way), the quiz that I give is one that individuals take but each person's grade is an average of the scores of all the individuals in the class.

For two days leading up to the quiz, we do intensive collaborative work, including reciprocal doing-and-teaching practices such as speed dating. We also have unstructured time in which students identify as tutors or learners and then work to help each other improve the overall level of mastery in the room.

Our goal is a whole-class goal of mastery -- not an individual one. The goal is to raise the overall level of mastery in the room. Our goal as a class is to get everybody's level of understanding up. If you want to sit off to the side and work on your chemistry homework, then you're going to have to answer to your peers -- not to me. And if you don't like the grade that the whole class achieves, then too bad. Positive interdependence rules the day.

There are always one or two students who are so addicted to the toxic culture of individual attainment that they object, demanding, "If I understand it and they don't, then why should I be punished?"

And I have to explain to them over and over again. I tell them, "That's an infantile perspective. The better-prepared everyone around you is, the richer and more powerful your own learning experience is going to be -- both now and into the future. My job is to provide you with the richest possible learning experience so that you can go as far as you want to go. My job is to set the floor, not the ceiling. And this is how I, as the expert on learning, am empowering us to raise the floor of understanding."

Our school is unusual in that students get to choose their classes, their sections, and their teachers. My classes are very popular and are always among the earliest to fill up.

I choose to use this platform and my privilege to educate them. I'm blunt with the students who complain. "Listen," I tell them. "You chose this section. If you'd prefer a teacher who only gives individual scores on everything and lets you work on your chem homework when you're done, then we should talk to Counseling and get you into a course section where your desires are going to be met, because that's not going to happen in my class. There are plenty of other kids who'd be happy to switch with you."

I realize this may sound harsh, but they usually come around. And the fact is that my job is not to give them everything they think they want but to teach them and help them get aligned with the reality of things as they are.

The results bear this out. The lowest average on this first whole-class score of all my Geometry sections was an 87. The highest was 93.7.

The number of "free points" I provide in other parts of my class (through professionalism, home enjoyment packet completion, etc) makes this a wash. Nobody's grade goes down because of anybody else, but most people's grade do go up because their understanding improves. And as I tell them over and over and over again, what they need to do to raise their grades is to improve their understanding. The structure of the whole-class skills quiz empowers them to do so.

There's also less cheating and more cooperation because the incentive structures are aligned with our better, saner values.

There is still a place for individual attainment. Unit tests are individually graded as is the final exam. But individual attainment is demoted in my classroom and is put into better balance within our classroom community.

Individual attainment and opportunity hoarding are symptoms of our society's sickness. If we want to heal our learning environments and improve outcomes, we need to be open to revising the unconscious, unspoken incentive structures that keep reinforcing the systemic oppression we need to heal from.

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FOLLOW-UPS:

@KarenCampe asks:
Wow this is amazing. Kudos to you for implementing something that really changes the game.
Do you have parent pushback?
— Karen Campe (@KarenCampe) January 19, 2020

I'm fortunate to have a lot of support from both site and district administration. In my view, this is a moral choice. My job is to create an equitable learning environment. If a parent were to insist on an inequitable learning environment for their child, I'm not sure what there is that we could do to satisfy them, given that this is public education.

Thanks for the question.

@timteachesmath asks:
Thank you for sharing! 
You've detailed your conversations with those 'done early'; 
what do those still learning think? Is there pressure to catch up, or a super supportive community?
— Tim (@timteachesmath) January 20, 2020

They appreciate that there is time and support being made for them to master what they find challenging. They want to learn the skills, but they get to do so in a way that does not punish them for needing more time or practice. They appreciate being part of the solution rather than part of the problem. And they are better able to participate and achieve their ends -- which is the goal. We are trying to normalize high achievement for everybody -- not sort out who "got it" first and who didn't.

Thanks for the question.

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.

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.

Seating charts for equity

I can't imagine traveling to new lands and not wanting to try their cuisine. But there really are people who bring their own food with them. One of the best things about traveling in my opinion is being educated  in the sense of the Latin root word — being led out of my own ignorance.

The same is true for me about attending a large, great school. It always has been. From the moment I arrive in a great new school, I feel excited and open to meeting and learning with all different kinds of people from different cultures and backgrounds. I want to expand my own limited world view.

But it seems inevitable that, without outside intervention, I often end up knowing and hanging out with the other Buddhists and Jews in any room. Cultural affinity is a force that possesses a tractor beam all its own. Fortunately, I am not the first to have noticed noticed this.

Our amazing counseling department and our Peer Resources program noticed this phenomenon too, and when they did their most recent student survey of our very large, urban, diverse student body, they put in some questions about this in their student well-being section. And the results were very moving to me.

Students overwhelmingly reported that when they first arrived at our school, they felt enormous pressure to connect with their cultural affinity groups. And for this reason, they reported, they deeply appreciate seating charts in classes that take this pressure away. This practice overwhelmingly helped them to feel that they fit in here and that those who are different from them in some ways are more like them in other ways than they are inclined to believe. It also created a zone of psychological and emotional safety to explore social connections with others not as "Others" but as fellow explorers in a safe space.

These findings touched my heart. Our kids' deeper wisdom never fail to blow me away.

So I sit here on the Sunday before the first day of Spring term making up seating charts, making sure that everybody arrives in my classes in the same boat as everybody else, and with the same opportunity to experience connection with others in as safe a space as I can create.

I will also pre-make Seating Charts #2, #3, and #4 so that it's convenient for me to change the seating without having to think. Sometimes "don't think" is the best rule.

I don't have any scintillating conclusions to draw here. I just wanted to document for myself what I am doing and why so that when I forget, I can more easily remember.

Attending to Precision: INBs and group work (Interactive Notebooks)

I love new beginnings, but I am only so-so with early middles. Now that kids have started their INB journey, we've arrived at that crucial moment between the beginning and the first INB check. This, as the saying goes, is where the rubber meets the road.

I find that kids never understand at this stage why I insist on being so darned nit-picky about their notebooks. Every day someone new asks me why this or that HAS to go on the right-hand side or EXACTLY on page 5.

One of the many reasons why this is important, I have learned, is that it is all about teaching strategies for attending to precision — Mathematical Practice Standard #6, which is defined this way in the standards documents:

Mathematically proficient students try to communicate precisely to others.• They try to use clear definitions in discussion with others and in their own reasoning.When making mathematical arguments about a solution, strategy, or conjecture (see MP.3), mathematically proficient elementary students learn to craft careful explanations that communicate their reasoning by referring specifically to each important mathematical element, describing the relationships among them, and connecting their words clearly to their representations. They state the meaning of the symbols they choose, including us- ing the equal sign consistently and appropriately.• They are careful about specifying units of measure,• and labeling axes to clarify the correspondence with quantities in a problem.• They calculate ac- curately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 

The problem, I find, is that this description of precision is precise only at the theoretical level. On the front lines, it's unrealistic because most kids never get to this level of precision.

And that is because their notes and their work are generally quite a mess.

A big part of teaching students to attend to precision is giving them a structure for being an impeccable warrior as a math student — that is to say, taking and keeping good notes, noticing and keeping track of your own progress as a learner, preserving your homework in a predictable place that is not, let us say, the very bottom of your backpack, crushed into a handful of loose raisins.

It means stepping up your game as a student of mathematics and presenting your work in a way that makes it possible for others to notice the care with which you are specifying units, crafting careful explanations, describing relationships, and so on. And it means presenting your work in this way ALWAYS — in all things, in all times, wherever you go.

INBs are an incredibly low-barrier-to-entry, accessible structure for teaching attention to precision. There are no students who cannot benefit from having a clear, common, and predictable structure for organizing their learning. INBs are also a great leveler. For those of us who are focused on creating equity in our classrooms, INBs offer all students a chance to prove both to themselves and others that they are indeed smart in mathematics. As I saw the other night at Back To School Night, my strongest note-keeping students are rarely the top students computationally speaking. But they are the ones who can always find what they are looking for — a major advantage on an open-notes test.

INBs are also a phenomenal formative assessment tool. Flipping through a students INB gives me an incredible snapshot of where and when they were truly attending to precision and where they were fuzzing out. Blank spaces and lack of color or highlighter on specific notes pages give me a targeted spot for further formative assessment. In my experience, it is exceedingly rare for a student who thoroughly understands a topic to write no notes or diagrams on that page. If anything, they are the ones who are most likely to appreciate the chance to consolidate their understanding.

So I am sticking with it and zooming in on some of the areas where kids' understanding fell apart last week. We'll be reviewing how to convert from percentages to decimals and how to document and analyze the iterative process of calculating compound interest because that is where my students' notes fell apart.

I'll be astonished — but will report back honestly — if these on-the-fly assessments prove to have been inaccurate.